diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhwes" "b/data_all_eng_slimpj/shuffled/split2/finalzzhwes" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhwes" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThe nature of dark matter and dark energy is one of the most important issues today in physics. There are strong observational evidences in astrophysics and cosmology for the existence of these two components of the cosmic energy budget, indicating that about $95\\%$ of the Universe is composed by dark matter (about $25\\%$) and by dark energy (about $70\\%$), but no direct detection has been reported until now. The usual candidates \\textbf{for} dark matter (neutralinos and axions, for example) and dark energy (cosmological constant, quintessence, etc.) lead to very robust scenarios, but at same time they must face theoretical and observational issues. For recent reviews on the subject, see for example \\cite{Padmanabhan:2002ji, Peebles:2002gy, Sahni:2004ai, Bertone:2004pz, Sahni:2006pa, Copeland:2006wr, Frieman:2008sn, Martin:2008qp, Caldwell:2009ix, Li:2011sd}.\n\nThe strongest issue is perhaps the one regarding dark energy as the vacuum expectation value of some quantum field, which would be a natural candidate, but whose correct theoretical value could be predicted only in the framework of a complete theory of quantum gravity, which still we do not possess. Nevertheless, it is possible, at least, to guess some of the features of this theory. In particular, the holographic principle \\cite{'tHooft:1993gx, Susskind:1994vu, Bousso:2002ju} may shed some light on the dark energy problem. According to this principle, in presence of gravity the number of the degrees of freedom of a local quantum system would be related to the area of its boundary, rather than to the volume of the system (as expected when gravity is absent). Following this \\textbf{idea}, in \\cite{Cohen:1998zx} the authors suggested an entanglement relation between the infrared and ultraviolet \\textbf{cutoffs} due to the limitation set by the formation of a black hole, which sets an upper bound for the vacuum energy. We can then interpret the ultraviolet cutoff as the vacuum density value, but still we need an ansatz for the infrared cutoff. As a candidate for such distance, in \\cite{Li:2004rb, Huang:2004ai} the authors propose and investigate the future event horizon, tested against type Ia supernovae data and cosmic microwave background anisotropies in \\cite{Zhang:2005hs, Zhang:2007sh}. We shall present more detail on this in Sec.~\\ref{Sec:HolDE}. \n\nAdding new components of dark energy to the whole energy budget in order to explain the current observation is a way, but not the only one. Since General Relativity has been thoroughly tested up to solar system scales, it may be possible that the Einstein-Hilbert action contain corrections on larger, cosmological, scales thereby candidating as possible explanation of the evolution of the universe. Such modifications should be, in principle, composed by higher order curvature invariant terms (such as $R^2$, $R_{\\mu\\nu}R^{\\mu\\nu}$, etc) but also by non-trivial coupling between matter or fields and geometry. See for example \\cite{Nojiri:2006ri, Nojiri:2010wj, Amendola:2006kh, Amendola:2006we, Capozziello:2007ec, Sotiriou:2008rp, DeFelice:2010aj} for some reviews on the subject (especially on $f(R)$ theory). It is also worth pointing out that these terms should naturally emerge as quantum corrections in the low energy effective action of quantum gravity or string theory \\cite{Buchbinder:1992rb}.\n\nIn this paper we connect these two approaches, considering a $f(R,T)$ theory of gravity, where $R$ is the Ricci scalar, whereas $T$ is the trace of the stress-energy momentum. This modified gravity theory has been recently introduced in \\cite{Harko:2011kv}, where the authors derived the field equations and considered several cases, relevant in cosmology and astrophysics. As for the former, $f(R,T)$ models have been constructed describing the transition from the matter dominated phase to the late times accelerated one \\cite{Houndjo:2011tu}. \n\nOur task here, is to find out which form the function $f(R,T)$ has to have in order to reproduce the same properties of the holographic dark energy proposed in \\cite{Li:2004rb}. To this purpose, we employ the same reconstruction scheme proposed and employed in \\cite{Capozziello:2005ku, Nojiri:2006gh, Nojiri:2006jy, Nojiri:2006be, Wu:2007tn}. For reference, in order to track the contribution of the $T$ part of the action in the reconstruction, we consider two special $f(R,T)$ models: in the first instance, we investigate the modification $R + 2f(T)$, i.e. the usual Einstein-Hilbert term plus a $f(T)$ correction. In the second instance we consider a $f(R)+\\lambda T$ theory, i.e. a $T$ correction to the renown $f(R)$ gravity. In both cases, we consider dark energy accompanied by a pressureless matter component (which would determine $T$).\n\nThe paper is organised as follows. In Sec.~\\ref{Sec:HolDE}, the equations of motion are established and the holographic dark energy introduced. In Sec.~\\ref{Sec:Simpl} and \\ref{Sec:ComplCase} the above mentioned cases are analysed. Finally, Sec.~\\ref{Sec:DiscConcl} is devoted to discussion and conclusions.\n\nWe use $8\\pi G = c = 1$ units and adopt the metric formalism, i.e. \\textbf{the variation of the action is considered with respect to the metric quantities.}\n\n\n\\section{$f(R,T)$ gravity and holographic Dark Energy}\\label{Sec:HolDE}\n\nIn \\cite{Harko:2011kv}, the following modification of \\textbf{Einstein's} theory is proposed:\n\\begin{equation}\\label{actionfRT}\n S = \\frac{1}{2}\\int f(R,T) \\sqrt{-g}\\;d^4x + \\int L_{\\rm m} \\sqrt{-g}\\;d^4x\\;,\n\\end{equation}\nwhere $f(R,T)$ is an arbitrary function of the Ricci scalar $R$ and of the trace $T$ of the energy-momentum tensor, defined as\n\\begin{equation}\n T_{\\mu\\nu} = -\\frac{2}{\\sqrt{-g}}\\frac{\\delta\\left(\\sqrt{-g}L_{\\rm m}\\right)}{\\delta g^{\\mu\\nu}}\\;,\n\\end{equation}\nwhere $L_{\\rm m}$ is the matter Lagrangian density. We assume the matter lagrangian to depend on the metric, so that\n\\begin{equation}\n T_{\\mu\\nu} = g_{\\mu\\nu}L_{\\rm m} - 2\\frac{\\partial L_{\\rm m}}{\\partial g^{\\mu\\nu}}\\;.\n\\end{equation}\nVarying action \\eqref{actionfRT} with respect to the metric $g^{\\mu\\nu}$, one obtains \\cite{Harko:2011kv}\n\\begin{equation}\\label{Eqmod}\n f_R(R,T) R_{\\mu\\nu} - \\frac{1}{2}f(R,T)g_{\\mu\\nu} + \\left(g_{\\mu\\nu}\\square - \\nabla_\\mu\\nabla_\\nu\\right)f_R(R,T) = T_{\\mu\\nu} - f_T(R,T)T_{\\mu\\nu} - f_T(R,T)\\Theta_{\\mu\\nu}\\;,\n\\end{equation}\nwhere the \\textbf{subscripts} $R$ or $T$ \\textbf{imply} derivation with respect that quantity and we have also defined\n\\begin{equation}\n \\Theta_{\\mu\\nu} \\equiv g^{\\alpha\\beta}\\frac{\\delta T_{\\alpha\\beta}}{\\delta g^{\\mu\\nu}}\\;.\n\\end{equation}\n\\textbf{Planning} a cosmological application, we assume matter to be described by a perfect fluid energy-momentum tensor\n\\begin{equation}\n T_{\\mu\\nu} = \\left(\\rho + p\\right)u_\\mu u_\\nu - p g_{\\mu\\nu}\\;,\n\\end{equation}\nand that $L_{\\rm m} = -p$, so that we have \n\\begin{equation}\n \\Theta_{\\mu\\nu} = -2T_{\\mu\\nu} - p g_{\\mu\\nu}\\;,\n\\end{equation}\nand Eq.~\\eqref{Eqmod} simplifies as\n\\begin{equation}\\label{Eqmod2}\n f_R(R,T) R_{\\mu\\nu} - \\frac{1}{2}f(R,T)g_{\\mu\\nu} + \\left(g_{\\mu\\nu}\\square - \\nabla_\\mu\\nabla_\\nu\\right)f_R(R,T) = T_{\\mu\\nu} + f_T(R,T)T_{\\mu\\nu} + p f_T(R,T) g_{\\mu\\nu}\\;.\n\\end{equation}\nIn order to compare \\textbf{it} with \\textbf{Einstein's}, we cast the above equation as follows:\n\\begin{eqnarray}\\label{Eqmod3}\n G_{\\mu\\nu} &=& \\frac{1 + f_T(R,T)}{f_R(R,T)}T_{\\mu\\nu} + \\frac{1}{f_R(R,T)}p f_T(R,T) g_{\\mu\\nu} - \\frac{1}{f_R(R,T)}\\left(g_{\\mu\\nu}\\square - \\nabla_\\mu\\nabla_\\nu\\right)f_R(R,T)\\nonumber\\\\ &+& \\frac{1}{2f_R(R,T)}f(R,T)g_{\\mu\\nu} - \\frac{1}{2}g_{\\mu\\nu} R\\;,\n\\end{eqnarray}\nwhere $G_{\\mu\\nu} \\equiv R_{\\mu\\nu} - Rg_{\\mu\\nu}\/2$ is the Einstein tensor. Now we can identify\n\\begin{equation}\\label{Effmatt}\n \\tilde{T}_{\\mu\\nu}^{(m)} = \\frac{1 + f_T(R,T)}{f_R(R,T)}T_{\\mu\\nu} + \\frac{1}{f_R(R,T)}p f_T(R,T) g_{\\mu\\nu}\\;,\n\\end{equation}\nas the \\textit{effective} matter energy-momentum tensor and\n\\begin{equation}\\label{Effgeom}\n \\tilde{T}_{\\mu\\nu}^{(geom)} = - \\frac{1}{f_R(R,T)}\\left(g_{\\mu\\nu}\\square - \\nabla_\\mu\\nabla_\\nu\\right)f_R(R,T) + \\frac{1}{2f_R(R,T)}f(R,T)g_{\\mu\\nu} - \\frac{1}{2}g_{\\mu\\nu} R\\;,\n\\end{equation}\nas the energy-momentum tensor of a ``geometric'' matter component. \n\nWe now assume a \\textbf{background} described by the Friedmann-Lema\\^{\\i}tre-Robertson-Walker metric\n\\begin{equation}\\label{RWmet}\n ds^2 = dt^2 - a(t)^2\\delta_{ij}dx^idx^j\\;, \n\\end{equation}\nwith spatially flat hypersurfaces, and find a form for the function $f(R,T)$ which is able to reconstruct \\textbf{holographic dark energy}. \n\n\\subsection{Holographic Dark Energy}\n\nAccording to the holographic principle \\cite{'tHooft:1993gx, Susskind:1994vu, Bousso:2002ju} an entanglement relation between the infrared (IR) and ultraviolet (UV) cut-offs of a quantum theory, due to the limitation set by the formation of a black hole, sets an upper bound for the vacuum energy \\cite{Cohen:1998zx}:\n\\begin{equation}\\label{vacen}\n \\rho_{\\rm v} = \\frac{3b^2}{L^2}\\;,\n\\end{equation}\nwhere $b$ is a free parameter and the IR (large scales) cutoff $L$ needs to be specified by an ansatz. We are interested in the one proposed in \\cite{Li:2004rb}:\n\\begin{equation}\\label{anshol}\n L = R_{\\rm h} = a\\int_t^\\infty \\frac{dt'}{a(t')} = a\\int_a^\\infty \\frac{d\\bar{a}}{H(\\bar{a})\\bar{a}^{2}}\\;,\n\\end{equation}\ni.e. the future event horizon, that is the distance covered by a photon from now until the remote future. Note that the very presence of a vacuum energy component makes the above integration finite. We consider a model composed by holographic dark energy plus ordinary pressureless matter, i.e.\n\\begin{equation}\\label{Friede}\n 3H^2 = \\rho_{\\rm v} + \\rho_{\\rm m} = \\rho_{\\rm v} + \\rho_{\\rm m0}(1 + z)^3\\;,\n\\end{equation}\nwhere $H \\equiv \\dot{a}\/a$ is the Hubble parameter and the dot denotes derivation with respect to the cosmic time. Introduce the critical energy density $\\rho_{\\rm cr} := 3H^2$\\textbf{, we define}\n\\begin{equation}\\label{oliver15}\n\\Omega_{\\rm v} := \\frac{\\rho_{\\rm v}}{\\rho_{\\rm cr}}=\\frac{b^2}{R^2_{\\rm h}H^2}\\;,\n\\end{equation}\nUsing Eqs.\\eqref{vacen} and \\eqref{anshol}, it is easy to show that \n\\begin{equation}\\label{oliver16}\n\\dot{R}_{\\rm h} = \\frac{b}{\\sqrt{\\Omega_{\\rm v}}} - 1\\;.\n\\end{equation}\nThe holographic dark energy density $\\rho_{\\rm v}$ evolves according to the conservation law\n\\begin{equation}\\label{oliver17}\n\\dot{\\rho}_{\\rm v} + 3H\\rho_{\\rm v}\\left(1 + w_{\\rm v}\\right) = 0\\;,\n\\end{equation}\nbecause in Eq.~\\eqref{Friede} we have implicitly assumed the matter component to conserve separately. Now, using Eqs.~\\eqref{vacen}, \\eqref{anshol} and \\eqref{oliver16}, one can find\n\\begin{equation}\\label{oliver18}\n\\dot{\\rho}_{\\rm v} = -\\frac{2}{R_{\\rm h}}\\left(\\frac{b}{\\sqrt{\\Omega_{\\rm v}}}-1\\right)\\rho_{\\rm v}\\,.\n\\end{equation}\nComparing \\eqref{oliver18} with \\eqref{oliver17} one can read off\n\\begin{equation}\\label{oliver19}\nw_{\\rm v} = -\\left(\\frac{1}{3} + \\frac{2\\sqrt{\\Omega_{\\rm v}}}{3b}\\right)\\,\\,.\n\\end{equation}\nMoreover, combining Eq.~\\eqref{anshol} with Eqs.~\\eqref{vacen} and \\eqref{Friede}, the evolution for $\\Omega_{\\rm v}$ is determined by the following equation:\n\\begin{equation}\\label{OmegavEvo}\n \\Omega_{\\rm v}' = -\\left(1 + \\frac{2\\sqrt{\\Omega_{\\rm v}}}{b}\\right)\\frac{1}{1 + z}\\Omega_{\\rm v}\\left(1 - \\Omega_{\\rm v}\\right)\\;,\n\\end{equation}\nwhere the prime denotes derivation with respect to the redshift $z$. Testing this model against type Ia supernovae and cosmic microwave background anisotropies, $b$ turns out to be constrained around unity, with the case $b < 1$ favoured \\cite{Zhang:2005hs, Zhang:2007sh}. Note that, from Eq.~\\eqref{oliver19}, $b < 1$ means that the universe will end up in a phantom phase. For more comprehensive analysis of holographic dark energy models, we refer the reader to \\cite{Pavon:2005yx, delCampo:2011jp}.\n\nIn the next section we investigate a simple case of reconstruction of $f(R,T)$.\n\n\\section{A simple case}\\label{Sec:Simpl}\n\nWe now consider a single perfect fluid model with density $\\rho$ and pressure $p$, together with the following ansatz (one of the first considered in \\cite{Harko:2011kv}):\n\\begin{equation}\n f(R,T) = R + 2f(T)\\;,\n\\end{equation}\ni.e. the action is given by the same Einstein-Hilbert one plus a function of $T$. This is a particularly interesting choice since, from Eqs.~\\eqref{Effmatt} and \\eqref{Effgeom}, we get\n\\begin{equation}\\label{Tmtil}\n \\tilde{T}_{\\mu\\nu} = \\left(1 + 2f_T\\right)T_{\\mu\\nu} + 2p f_T g_{\\mu\\nu} + f(T)g_{\\mu\\nu}\\;.\n\\end{equation}\nFor $p = 0$ one has $T = \\rho$ and, choosing $f(T) = \\lambda T$ one can construct a model with an effective cosmological constant \\cite{Poplawski:2006ey}. From Eq.~\\eqref{Tmtil} one can read off the effective energy density and pressure of the universe content:\n\\begin{eqnarray}\n\\label{rhotot} 3H^2 &=& \\rho_{\\rm eff} = \\left(1 + 2f_T\\right)\\rho + 2p f_T + f(T)\\;,\\\\\n\\label{ptot} -2\\dot{H} - 3H^2 &=& p_{\\rm eff} = p - f(T)\\;,\n\\end{eqnarray}\nand therefore a dark energy component may appear, even if we are considering a single perfect fluid model. From Eqs.~\\eqref{rhotot} and \\eqref{ptot}, it is clear that we can pick out a ``fictitious'' component, due to $f(T)$, described by\n\\begin{eqnarray}\n \\rho_{f} &=& 2f_T\\rho + 2p f_T + f(T)\\;,\\\\\n p_{f} &=& - f(T)\\;,\n\\end{eqnarray}\nand, provided $f$ positive, it may well describe a dark energy component, since its pressure is negative. In order to reconstruct the function $f$ starting from the holographic principle, we note that the equation of state parameter of the dark component induced by $f$ is\n\\begin{equation}\n w_{f} = -\\frac{f(T)}{2(\\rho + p)f_T + f(T)}\\;,\n\\end{equation}\nand we identify it with $w_{\\rm v}$, the one provided by the holographic dark energy in Eq.~\\eqref{oliver19}:\n\\begin{equation}\\label{EqfT}\n \\frac{f(T)}{2(\\rho + p)f_T + f(T)} = \\frac{1}{3} + \\frac{2\\sqrt{\\Omega_{\\rm v}}}{3b}\\;.\n\\end{equation}\nFor the standard model given in Eq.~\\eqref{Friede}, consider the fluid component to be pressureless matter, i.e. $p = 0$. We are left to solve the following system of equations:\n\\begin{eqnarray}\n\\label{fevo} \\rho\\frac{df(\\rho)}{d\\rho} &=& f(\\rho)\\frac{b - \\sqrt{\\Omega_{\\rm v}}}{b + 2\\sqrt{\\Omega_{\\rm v}}}\\;,\\\\\n\\label{Omevo} \\frac{d\\Omega_{\\rm v}}{d\\rho} &=& -\\frac{1}{3\\rho}\\left(1 + \\frac{2\\sqrt{\\Omega_{\\rm v}}}{b}\\right)\\Omega_{\\rm v}\\left(1 - \\Omega_{\\rm v}\\right)\\;,\n\\end{eqnarray}\nwhere we have used $T = \\rho$, because we are considering pressureless matter. In order to solve the above system, we have to fix some initial conditions. Clearly, $\\Omega_{\\rm v}(\\rho = \\rho_0) = 1 - \\Omega_{\\rm m0}$, and we choose $\\Omega_{\\rm m0} = 0.3$, accordingly with current cosmological observation. For the initial condition on $f$, from Eq.~\\eqref{rhotot} (with $p = 0$) we write\n\\begin{equation}\\label{friedeqf0}\n \\left[1 + 2f_{T}(\\rho_0)\\right]\\Omega_{\\rm m0} + \\frac{f(\\rho_0)}{3H_0^2} = 1\\;.\n\\end{equation}\nEvaluating Eq.~\\eqref{fevo} today and combining it with Eq.~\\eqref{friedeqf0} we find the following algebraic equation determining the initial condition on $f$:\n\\begin{equation}\\label{f0cond}\n f(\\rho_0)\\left(2\\frac{b - \\sqrt{\\Omega_{\\rm v0}}}{b + 2\\sqrt{\\Omega_{\\rm v0}}} + 1\\right) = 3H_0^2\\Omega_{\\rm v0}\\;.\n\\end{equation}\nAs we expected, when $\\Omega_{\\rm v0} = 0$, then $f(\\rho_0) = 0$ and Eq.~\\eqref{fevo} implies that $f$ is identically vanishing.\\\\\n\\begin{figure}[htbp]\n \\includegraphics[width=0.45\\columnwidth]{Fig1.eps}\\;\\includegraphics[width=0.45\\columnwidth]{Fig2.eps}\n\\caption{Left panel: evolution of $f(\\rho)$. Right panel: evolution of $\\Omega_{\\rm v}$. The cases considered are $b = 0.6, 0.8, 1.0, 1.2$ (solid black, dashed red, dot-dashed blue and dotted green, respectively). We have chosen as initial conditions in $\\Omega_{\\rm m0} = 0.3$ the values $\\Omega_{\\rm v0} = 1 - \\Omega_{\\rm m0} = 0.7$ and $f_0$ given by Eq.~\\eqref{f0cond}. Note that $f$ and $\\rho$ are normalised to $3H_0^2$. The vertical lines in the plots represent $\\rho = \\rho_0$, i.e. the present instant.}\n\\label{Fig1}\n\\end{figure}\\\\\nIn \\figurename{ \\ref{Fig1}} we plot the solution of the system of differential equations \\eqref{fevo} and \\eqref{Omevo}. Note that we normalise $f$ and $\\rho$ to $3H_0^2$. As expected from Eq.~\\eqref{fevo}, for large values of $\\rho$, i.e. far in the past, $f \\propto \\rho$ because the dark energy component is subdominant. The actual difference among the various choices of $b$ takes place at late times, for small values of $\\rho$. Again from inspection of Eq.~\\eqref{fevo}, we can see that for large values of $b$ the linear evolution $f \\propto \\rho$ is again solution. That is why in the left panel of \\figurename{ \\ref{Fig1}} the curve seems to ``straighten up'' for increasing $b$. \n\nIn \\figurename{ \\ref{Fig2}} we plot the same quantities, but as functions of the redshift, in order to make clearer their cosmological evolution.\n\\begin{figure}[htbp]\n \\includegraphics[width=0.45\\columnwidth]{Fig3.eps}\\;\\includegraphics[width=0.45\\columnwidth]{Fig4.eps}\n\\caption{Same as \\figurename{ \\ref{Fig1}}, but with the redshift $z$ as independent variable.}\n\\label{Fig2}\n\\end{figure}\n\nA final remark about the future evolution. It is clear from Eq.~\\eqref{fevo}, that when $\\Omega_{\\rm v} \\to 1$, in the remote future, the solution for $f$ gets the asymptotic form\n\\begin{equation}\n f(\\rho) \\propto \\rho^{\\frac{b - 1}{b + 2}}\\;.\n\\end{equation}\nTherefore, we would have a future singularity for $-2 < b < 1$, as it appears for the relevant cases of \\figurename{ \\ref{Fig1}} and \\figurename{ \\ref{Fig2}}. The special case $b = 1$ implies an asymptotically constant $f$.\n\nWe now turn our discussion on a more general case, where the curvature $R$ \\textbf{comes into} the action \\textbf{as} a function to be determined.\n\n\n\\section{A more complicated case}\\label{Sec:ComplCase}\n\nNow we turn our attention to the special case\n\\begin{equation}\\label{complcase}\n f(R,T) = f(R) + \\lambda T\\;,\n\\end{equation}\ni.e. a $T$-linear correction to the class of $f(R)$ theories. With the ansatz \\eqref{complcase} the matter content \\eqref{Effmatt} is ``corrected'' as follows:\n\\begin{equation}\\label{Effmattcomplcase}\n \\tilde{T}_{\\mu\\nu}^{(m)} = \\frac{1 + \\lambda}{f_R(R)}T_{\\mu\\nu} + \\frac{1}{f_R(R)}\\lambda p g_{\\mu\\nu}\\;,\n\\end{equation}\nwhereas the geometry induced stress-energy tensor is\n\\begin{equation}\\label{Effgeomcomplcase}\n \\tilde{T}_{\\mu\\nu}^{(geom)} = - \\frac{1}{f_R(R)}\\left(g_{\\mu\\nu}\\square - \\nabla_\\mu\\nabla_\\nu\\right)f_R(R) + \\frac{1}{2f_R(R)}[f(R) + \\lambda T]g_{\\mu\\nu} - \\frac{1}{2}g_{\\mu\\nu} R\\;.\n\\end{equation}\nOur aim is now to reconstruct the form of the $f(R)$ which is able to reproduce the holographic dark energy paradigm. We again consider a pressureless perfect fluid with density $\\rho$ and again assume metric \\eqref{RWmet}. From Eqs.~\\eqref{Effmattcomplcase} and \\eqref{Effgeomcomplcase} the effective density and pressure are the following:\n\\begin{eqnarray}\n\\label{rhovcomplcase} 3H^2 &=& \\rho_{\\rm eff} = \\frac{\\rho}{f_R} + \\frac{3\\lambda}{2f_R}\\rho - \\frac{R}{2} + \\frac{f}{2f_R} - 3H\\frac{\\dot{f}_R}{f_R}\\;,\\\\\n\\label{pvcomplcase} -2\\dot{H} - 3H^2 &=& p_{\\rm eff} = \\frac{1}{f_R}\\left(\\ddot{f}_R + 2H\\dot{f}_R\\right) - \\frac{f}{2f_R} - \\frac{\\lambda}{2f_R}\\rho + \\frac{R}{2}\\;.\n\\end{eqnarray}\nFrom Eq.~\\eqref{rhovcomplcase} it appears that the energy density of the perfect fluid is rescaled by a factor $1\/f_R$. Looking at Eq.~\\eqref{Friede}, we can extract a form for $\\rho_{\\rm v}$ in the following way:\n\\begin{equation}\\label{rhovcomplcase2} \n\\rho_{\\rm v} = \\frac{\\rho}{f_R} - \\rho + \\frac{3\\lambda}{2f_R}\\rho - \\frac{R}{2} + \\frac{f}{2f_R} - 3H\\frac{\\dot{f}_R}{f_R}\\;,\n\\end{equation}\nwhereas the form of $p_{\\rm v}$ is already given in Eq.~\\eqref{pvcomplcase}, since our fluid is pressureless. From Eqs.~\\eqref{pvcomplcase} and \\eqref{rhovcomplcase2} we can write the following differential equation for $f_R$:\n\\begin{equation}\\label{freqcompl}\n \\ddot{f}_R - H \\dot{f}_R - \\left[\\rho + \\rho_{\\rm v}\\left(1 + w_{\\rm v}\\right)\\right]f_R = -\\rho(1 + \\lambda)\\;.\n\\end{equation}\nNote, as a cross-check, that for $\\rho_{\\rm v} = \\lambda = 0$ the above equation simplifies to\n\\begin{equation}\\label{freqcompl2simpl}\n \\ddot{f}_R - H\\dot{f}_R - \\rho f_R = - \\rho\\;,\n\\end{equation}\nwhich possesses the particular solution $f_R = 1$, i.e. $f = R + \\Lambda$, the original Einstein-Hilbert action plus an integration ``cosmological'' constant. We expect this solution to be the only one, otherwise there would exist an alternative $f(R)$ theory which would behave exactly as general relativity. Let us speculate a bit more on this point. If $\\rho_{\\rm v} = \\lambda = 0$, i.e. for a pure Einstein-de Sitter universe, we have from Eq.~\\eqref{Friede}\n\\begin{equation}\n H = \\frac{2}{3t}\\;, \\qquad \\rho = \\rho_0\\frac{t^2_0}{t^2}\\;,\n\\end{equation}\nwhere $t_0$ is the present cosmic time (i.e. the age of the universe). Considering the homogeneous part of Eq.~\\eqref{freqcompl2simpl} and looking for a solution of the form $f_R \\propto t^n$, we find:\n\\begin{equation}\\label{freqcompl2simplhom}\n n(n - 1) - \\frac{2}{3}n - \\rho_0 t_0^2 = 0\\;,\n\\end{equation}\nwhich gives:\n\\begin{equation}\\label{freqcompl2simplhomsol}\n n_{1,2} = \\frac{5}{6} \\pm \\sqrt{\\frac{25}{36} + \\rho_0 t_0^2}\\;,\n\\end{equation}\nand the general solution of Eq.~\\eqref{freqcompl2simpl} can be written as:\n\\begin{equation}\\label{freqcompl2simplhomsol2}\n f_R = 1 + C_1\\;t^{n_1} + C_2\\;t^{n_2}\\;.\n\\end{equation}\nNow, the initial conditions we adopt here are $f_R(t_0) = 1$ and $\\dot{f}_R(t_0) = 0$. The reason is essentially not spoiling the agreement between general relativity and solar system tests, see \\cite{Capozziello:2005ku, Nojiri:2006gh, Nojiri:2006jy, Nojiri:2006be, Wu:2007tn}. However, we stress here that these two conditions also imply $C_1 = C_2 = 0$ and therefore restore the general relativity limit $f_R = 1$ when $\\rho_{\\rm v} = \\lambda = 0$. \n\nChanging the variable to the redshift and employing Eqs.~\\eqref{Friede} and \\eqref{OmegavEvo} one can recast Eq.~\\eqref{freqcompl} in the following compact form:\n\\begin{eqnarray}\\label{freqcompl2}\n (1 + z)^2f''_R + \\frac{1 + z}{2}\\left[7 - \\Omega_{\\rm v}\\left(1 + \\frac{2\\sqrt{\\Omega_{\\rm v}}}{b}\\right)\\right]f_R' - 3\\left[1 - \\Omega_{\\rm v}\\left(\\frac{1}{3} + \\frac{2\\sqrt{\\Omega_{\\rm v}}}{3b}\\right)\\right]f_R = \\nonumber\\\\ - 3(1 + \\lambda)(1 - \\Omega_{\\rm v})\\;,\n\\end{eqnarray}\nwhere again the prime denotes derivation with respect to the redshift $z$. The curvature $R$ can be easily found as\n\\begin{equation}\n R = -6(\\dot{H} + 2H^2) = -\\frac{3H_0^2\\Omega_{\\rm m0}(1 + z)^3}{1 - \\Omega_{\\rm v}}\\left[1 + \\Omega_{\\rm v}\\left(1 + \\frac{2\\sqrt{\\Omega_{\\rm v}}}{b}\\right)\\right]\\;.\n\\end{equation}\nThe initial conditions $f_R(t_0) = 1$ and $\\dot{f}_R(t_0) = 0$, translated to the redshift variable, are\n\\begin{eqnarray}\n \\left.\\frac{d^2f}{dz^2}\\right|_{z = 0} = \\left.\\frac{d^2R}{dz^2}\\right|_{z = 0}\\;, \\qquad \\left.\\frac{df}{dz}\\right|_{z = 0} = \\left.\\frac{dR}{dz}\\right|_{z = 0}\\;,\\\\\n\\end{eqnarray}\nFinally, the initial condition on $f$ can be extracted by Eq.~\\eqref{rhovcomplcase}, being that $\\rho_{\\rm v0} = 3H_0^2 - \\rho_{\\rm m0}$. Thus, we have\n\\begin{equation}\nf(z = 0) = R(z = 0) + 6H_0^2\\left(1 - \\Omega_{\\rm m0} - \\frac{3}{2}\\lambda\\Omega_{\\rm m0}\\right)\\;.\n\\end{equation}\nNote the correction due to the $\\lambda T$ term.\n\\begin{figure}[htbp]\n \\includegraphics[width=0.45\\columnwidth]{Fig5.eps}\\;\\includegraphics[width=0.45\\columnwidth]{Fig6.eps}\n\\caption{Evolution of $f$ as a function of the curvature $R$. Left panel: $\\lambda = 0$, i.e. we are considering a pure $f(R)$ theory, and $b = 0.6, 0.8, 1.0, 1.2$ (solid black, dashed red, dot-dashed blue and dotted green, respectively). Right panel: $b = 1.0$ and $\\lambda = -0.2, 0, 0.2, 0.4$ (solid black, dashed red, dot-dashed blue and dotted green, respectively. The curves appear superposed.). We have chosen as initial conditions in $\\Omega_{\\rm m0} = 0.3$ the values $\\Omega_{\\rm v0} = 1 - \\Omega_{\\rm m0} = 0.7$. Note that $f$ and $R$ are normalised to $3H_0^2$ and the redshift interval chosen is $0 < z < 10$.}\n\\label{Fig3}\n\\end{figure}\nIn \\figurename{ \\ref{Fig3}} we plot the solution for $f$. Note that we normalise $f$ and $R$ to $3H_0^2$. In the left panel we fix $\\lambda = 0$, i.e. we are actually considering a pure $f(R)$ theory, and vary $b$. In the right panel, on the other hand, we consider positive and negative values of $\\lambda$. As one may note, $\\lambda$ has a poor influence on the evolution of $f$. We could expect this from inspection of Eq.~\\eqref{freqcompl2}. Indeed, $\\lambda$ only enters the source term on the right hand side and therefore, when $\\Omega_{\\rm v}$ grows to unity, its impact on the evolution of $f$ is weak. On the other hand, larger values of $\\lambda$ may have a relevant effect at early times, determining the slope of $f$.\n\\begin{figure}[ht]\n \\includegraphics[width=0.45\\columnwidth]{Fig7.eps}\\;\\includegraphics[width=0.45\\columnwidth]{Fig8.eps}\n\\caption{Same as \\figurename{ \\ref{Fig3}}, but with $-0.9 < z < 1$. Note, in the left panel, that the evolution of $f$ starts from below for all the cases.}\n\\label{Fig4}\n\\end{figure}\n\\newpage\nIn \\figurename{ \\ref{Fig4}} and \\figurename{ \\ref{Fig5}} we display the future evolution of $f$, both as function of $R$ or of $z$. For the same reason stated above, the effect of $\\lambda$ is not relevant.\n\\begin{figure}[ht]\n \\includegraphics[width=0.45\\columnwidth]{Fig9.eps}\\;\\includegraphics[width=0.45\\columnwidth]{Fig10.eps}\n\\caption{Same as \\figurename{ \\ref{Fig4}}, but with $f$ as function of $z$.}\n\\label{Fig5}\n\\end{figure}\n\\newpage\nFinally, in \\figurename{ \\ref{Fig6}} we plot the solution for $\\Omega_{\\rm v}$. Again, its evolution appears to be independent of $\\lambda$.\n\\begin{figure}[ht]\n \\includegraphics[width=0.45\\columnwidth]{Fig11.eps}\\;\\includegraphics[width=0.45\\columnwidth]{Fig12.eps}\n\\caption{Evolution of $\\Omega_{\\rm v}$ as a function of the redshift. Left panel: $\\lambda = 0$, i.e. we are considering a pure $f(R)$ theory, and $b = 0.6, 0.8, 1.0, 1.2$ (black, dashed red, dash-dotted blue and dotted green, respectively). Right panel: $b = 1.0$ and $\\lambda = -0.2, 0, 0.2, 0.4$ (black, dashed red, dash-dotted blue and dotted green, respectively). Note that the curves are superposed. We have chosen as initial conditions in $\\Omega_{\\rm m0} = 0.3$ the values $\\Omega_{\\rm v0} = 1 - \\Omega_{\\rm m0} = 0.7$.}\n\\label{Fig6}\n\\end{figure}\n\n\\newpage\n\n\\section{Discussion and Conclusions}\\label{Sec:DiscConcl}\n\nIn this work we have investigated a description of holographic dark energy in terms of suitably reconstructed $f(R, T)$ gravity theories. The latter have been recently introduced as modifications of Einstein's theory possessing some interesting solutions \\textbf{which are} relevant in cosmology and astrophysics \\cite{Harko:2011kv}.\n\nWe have \\textbf{considered two special} types of models: $f(R,T) = R + 2f(T)$, i.e. a correction to the Einstein-Hilbert action depending on the matter content, and $f(R,T) = f(R) + \\lambda T$, i.e. a simple $T$-linear correction to the class of $f(R)$ theories. \n\nSince we have assumed the matter content to be a pressureless perfect fluid, then $T = \\rho$, i.e. the corrections assumed are directly dependent on the energy density of the universe content. We \\textbf{have} constructed differential equations for the function $f$ under investigation and numerically solved \\textbf{them}, physically specifying the required initial conditions. Our simple analysis shows that holographic dark energy models are contained in the larger class of $f(R,T)$ theories, at least considering a given background evolution of the universe. \n\nIt would be interesting to investigate how the evolution of matter perturbations would change, depending on the description of dark energy. We expect, in principle, different results when using holographic dark energy or its $f(R,T)$ reconstruction and therefore there is possibility for discriminating between the two descriptions. For example, it would be interesting to adapt the recently proposed scheme for perturbations in $f(R)$ theories \\cite{Bertacca:2011wu} to the broader $f(R,T)$ class. We leave this as a future work.\n\n\n\\section*{Acknowledgements}\n\nMJSH and OFP thank Professor S. D. Odintsov for useful comments and also CNPq (Brazil) for partial financial support. \n\n\n\\bibliographystyle{unsrt}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{Introduction}\n\nCOMPASS is a fixed target experiment on the CERN SPS that uses tertiary high energy and high intensity muon or hadron beams with the aim of studying the nucleon spin structure and hadron spectroscopy. It is taking data since 2002, around seven months per year, and has shutdown periods in between, in which operations of maintenance and preparation of the following data taking period take place. The experimental setup is described in detail in \\cite{COMPASS01}.\n\nThe detector devices and the experiment's environmental parameters are monitored and controlled using an experiment-wide DCS. This system must ensure a coherent, safe and efficient operation of the experiment, by providing clear and prompt information for the shift crew and detector experts in the COMPASS control room. Some complex subsystems of the experiment have dedicated stand-alone control systems. These systems communicate with the DCS, providing it the most relevant parameters. Since 2003, the COMPASS DCS has been an exclusive responsibility of the LIP-Lisbon group participating in the collaboration. This structure and organization is at contrast with the one of the big LHC experiments, which have a hierarchical structure, with distributed responsibilities~\\cite{ATLAS,CMS,ALICE,LHCb}.\n\nThe DCS provides a graphical user interface for the shift crew in the COMPASS control room and detector experts to have access to all the relevant parameters monitored, their state (normal or in alert - indicated visually, by use of a color code, and acoustically) and their history, and a straightforward way to change their state, their settings, and the thresholds that define their state of alert.\n\nThe architecture of the DCS is shown in Fig.~\\ref{DCSscheme}. In the following sections, its different layers are described in detail: the supervisory layer, the front-ends layer and the devices layer.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.8 \\textwidth]{dcs-schema-3.eps}\n\\caption{The COMPASS Detector Control System architecture, comprising a devices layer, a front-ends layer and a supervision layer. The technologies used in each layer are indicated in the rightmost column.} \n\\label{DCSscheme}\n\\end{figure*}\n\n\\noindent\\section{\\uppercase{The Supervisory Layer}}\n\\label{Supervisory} \n\nOn the supervisory layer, all the data collected, managed and published by the different kinds of servers or made available in databases is gathered, analysed and displayed to the end user. This includes visual and sound alarms in case of states of alarm. It also provides archiving of the data in an external database. Settings and alarm limits are also managed by the system.\n\nPVSS-II\\footnote{Pro\\-zess\\-teue\\-rung und Pro\\-zess\\-vi\\-sua\\-li\\-sie\\-rung. See http:\/\/www.etm.com}, is the commercial SCADA system that was chosen by CERN to use in the LHC experiments, after a thorough evaluation process. Some of the aspects taken into account were: openess, scalability, cross-platform capability (i.e. to run in Windows and Linux) and long term support.\n\nThe COMPASS experiment has adopted PVSS early in its development phase and has been a benchmark for other experiments at CERN. In fact, before the LHC starting in 2010, COMPASS was the biggest experiment operating at CERN. Over the years, several versions of PVSS were used in COMPASS, that had been previously tested and validated both by CERN and by the COMPASS DCS group.\nThe installation of these patches during the data-taking periods requires a careful evaluation.\n\nThe JCOP Framework~\\cite{JCOP} is a CERN project to develop common software tools for High Energy Physics related equipment and operations, to be used with PVSS. It provides templates of datapoint types, panels, functions and mass configuration tools for different classes of equipments or functionalities, providing, for instance, tools for the management of priviledges, or for trending plots.\n\nThe objects provided by PVSS and the JCOP Framework have sometimes to be adapted to meet COMPASS' needs. In addition, other solutions had to be developed independently for non-supported custom devices. This includes the control of custom devices,\naccessed using their serial (RS232) interfaces or their web servers; \nor the monitoring of items from dedicated control systems, such as (EPICS, LabView, etc.\\footnote{See http:\/\/www.aps.anl.gov\/epics, http:\/\/www.ni.com\/labview}), which are made available by various means, including mySQL and Oracle databases. \n\nThe PVSS production system in use is both distributed and scattered. Historically, it started as a scattered project, meaning that it was constituted by a main PVSS project running in a Linux machine, and 3 associated PVSS projects running on Windows machines, that had PVSS processes running as OPC\\footnote{OLE (Object Linking and Embedding) for Process Control. See http:\/\/www.opcfoundation.org} clients (7 clients in total). As the DCS developed, the main project was split into two distributed projects, for performance reasons. \n\nPVSS works with objects called datapoints, which are structures, {\\it i.e.} they have a tree structure that can include branches and where the leaves are the the monitored and controlled parameters (and can have different types, such as floats, integers, booleans, strings, etc., or the corresponding array types). \n\nPresently, the project comprises over 20000 datapoints. Close to 17000 parameter values have alert handling, whereas almost 19000 parameters have their values archived. \n\nThe polling rates are adapted to the rate of variation of the parameters, and range from one value per parameter read every 1.5 seconds (for fast varying parameters, sensitive to the beam, such as high voltage channels' actual values) to 2 minutes (for slowly varying parameters, such as high voltage channels' settings, or detector positions). For any given type of equipment, the items are grouped in PVSS subscription data sets according to these rates.\n\nThe access to the PVSS project is made available upon login. There is a general user name for the shift crew, user names for each of the detector experts, and a username for guests. For each login, there is an authorizations policy associated: certain operations are restricted (such as switching on or off the high voltage channels for guest users), or specifically allowed (such as saving recipes or reference files of high voltage settings; see \ndetails later in this text).\n\nThe graphical user interface (UI) is the main mean for users to interact with the DCS. It is composed of multiple subpanels, organised in a hierarchical way, as can be seen in Fig.~\\ref{ui}. One can see, on the top, the alert table and, on the left, the buttons to access dedicated detector panels and, below them, a table with the summary status of the experiment. In the larger area of the panel, a synoptic view of the spectrometer is displayed. This area is also used for navigation in the subsystems controlled and to display the actual data.\n\n\nThe datapoints history is made available online. In fact, PVSS trending plots (namely values over time) are one of the more useful and more used features of the DCS. In the user interface panels, customised buttons are created for the items that have numerical values (generally, floating point), so that their history can be easily accessed by users. The JCOP Framework provides its own trending plot widget, that was further customised for COMPASS, to make it simple to use. Users can choose the time range they want to visualise, change the scales or zoom in or out the abscissa or the ordinate by using the mouse scroll button, or choosing a rectangular region for zooming in by simply selecting two opposite vertices of the region. \nTemplates can be created to allow, with one click, to see related parameters and adjust the ranges for each.\nIt is also possible for the users to include additional parameters to a predefined plot as well as to print the trends being displayed, or to save them to a file, not only as an image, but also in ascii format (comma separated values or CVS).\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.7 \\textwidth]{dcsUI.eps}\n\\caption{The graphical user interface (UI) of the COMPASS DCS. One can see, on the top, the alert panel and, on the left, the buttons to access dedicated detector panels and, below them, a table with the summary status of the experiment.}\n\\label{ui}\n\\end{figure*}\n\nOne of the most important functionalities of the DCS is the display of visual and audible alarms, when predefined conditions are met, namely, when parameter values beyond predefined thresholds are reached for datapoints with numerical values or when devices send alarm flags. The visual display of alarms follows a color code that indicates its severity. For the most relevant parameters, it is important to assure that the operator didn't fail to notice the alarm condition, even if it has disapeared in the meantime; in this case, it is requested that the alarms are acknowledged by the operator in the graphical user interface.\nUpon activation of an alarm, detector experts are warned by email or SMS of states of alert in their detectors. \n\nSince the DCS has both a relatively fast knowledge of the state of the parameters it monitors and the ability to send commands to the devices, it is used to ensure software-wise protections to several equipments.\nFor instance, some detectors have components that are sensitive to magnetic field gradients; thus, when a trip of one of the spectrometer magnets (SM1 or SM2) occurs, or when these are switched off with the detectors high voltage channels still switched on, the DCS issues a switch-off command, so that the time interval during which the gradient is felt is minimised.\nIn addition, for some detectors, the high voltage channels should only be switched on or off in pairs; hence, if a trip is detected in only one member of the pair, the DCS sends a switch-off command to the other member of the pair.\n\nFurthermore, some detectors have front-end cards that are refrigerated by a water circuit. When this circuit stops for some reason, the temperature of the cards increases and can reach a value above a predefined threshold. If this happens, the DCS issues a switch-off command to the low voltage power supplies that power them, thus preventing these sensitive and sophisticated cards to be damaged. A hardware interlock is activated at a higher temperature, but the recovery from the interlocked state implies access to the experimental area, and therefore represents greater beam time losses and is therefore to be avoided. \n\nA configurations database associated to the project was implemented. It is based in a JCOP Framework package, which was adapted for the COMPASS needs. It saves and retrieves data in an independent Oracle database. This database has two main purposes. On the one hand, it allows to save and to retrieve the so called ``recipes\", {\\it i.e.} sets of thresholds for alarms of groups of items. The recipes can be created, or its values committed in the PVSS project, using the DCS UI, provided that the user has the privileges to change the respective detector items.\n\nThe second important purpose of the configuration database is to store so cal\\-led ``con\\-fi\\-gu\\-ra\\-tions\", {\\it i.e.} the mapping of hardware names vs.\\ logical names ({\\it i.e.} PVSS datapoint element names and respective aliases), for snapshots of stable states of the PVSS project. These are used to keep track of changes of the aforementioned mapping. These changes can happen either because, for instance, a high voltage channel gets broken and the same part of the detector ({\\it i.e.} same alias) is then powered by a different channel ({\\it i.e.} different datapoint), or because channels are reused when switching between the muon and the hadron Physics program data-taking.\n\nFor storage of settings of high voltage channels (set voltage, maximum current allowed, ramp up speed, ramp down speed and trip time), ascii files are used, for convenience. Experts can access the files, edit them, and send the values to be used by the equipment. The shift crew can use these reference file to recover the normal state of the equipment in case of problems.\n\nOnly a subset of all the data that PVSS receives and manages is actually saved. For this to happen, the PVSS datapoints need to have an archiving policy defined. This is chosen according to the known changes of each datapoint and the relevance of its history. For instance, it may be useful to store the readings of a temperature every ten minutes or if the change with respect to the previous reading exceeds one degree. This smoothing condition, called dead-band, is adjusted for each datapoint group or even per datapoint, if needed. The generic rates of archiving range from one value for every $\\sim$40 seconds (corresponding to the beam supercycle time interval) for beam-related quantities, to one value for every half an hour (for the positions of detectors). \n\nCurrently, there are around $2\\times 10^9$ values stored ({\\it i.e.} around 300 GB of data, including indexes), comprising the project history since 2006.\nThe DCS historical data that had been saved in PVSS internal format (during its first years of operation) was copied to a CERN central Oracle database, and the new data produced was all stored in this database. This way of storing data has all the advantages of Oracle and makes their access independent of PVSS. The data is continuously replicated to a second database, to ensure that the access to the data never compromises the performance of the production database. The data can also be provided in other formats, such as ascii or ROOT~\\cite{ROOT} trees.\n\nThe DCS data is very important for studies of detector performances.\nSome particularly relevant parameters for offline Physics data analysis are regularly copied to the experiment's mySQL conditions database, using a cron job.\nThe history of alerts of all the datapoint elements that have alert handling is also saved and made available by PVSS. This includes the timestamps of their arrival and departure, and of eventual acknowledgements done, among other information.\n\nThe knowledge of malfunctioning of parts of the experiment relies substantially on the DCS, namely on the display of alarms. Hence, it is important to assure its integrity and availability, ideally, at all times. Some of the mechanisms used are heart-beats, watch-dogs, back-ups, a security policy and the issuing of regular ping commands.\n\nThe managers of PVSS's main project, which are independent processes running in Linux, may, for a number of reasons, either block or stop running. On the other hand, the servers -- either OPC, \\cite{DIM}, or other -- may stop delivering meaningful data. For this reason, for each manager in PVSS that acts as a client, a heart-beat item was created, that gives the timestamp of the last meaningful data it received.\n\nMoreover, to verify that PLCs\\footnote{Programmable logic controllers.} are sending meaningful data at all times, a mechanism of watchdogs is implemented. The OPC server marks as invalid values sent by the PLC in case the values of the items published for this purpose (which have, during normal operation, varying integer values), stop begin updated.\n\nThe communication with individual VME\\footnote{VERSAmodule Eurocard bus.} crates or power supplies is also monitored, by continuously checking for selected equipment items that the timestamp of the latest value read is more recent than a predefined time interval.\n\nTo ensure the integrity of the project if a software corruption occurs in the PVSS project and associated software, the data is copied every twenty-four hours to a central repository. Furthermore, local copies are periodically made. \n\nA thorough security policy is implemented. All the computers that integrate the DCS belong to a dedicated experimental domain, that communicates with the CERN network using dedicated gateways. All the PCs in use have firewalls implemented.\nIn addition, all the user interfaces, with the exception of the one in the control room that should be permanently accessible, have an auto-logout after one hour.\n\nThe project should be available in the network at all times, for instance to diagnose of eventual DCS problems remotely. For this to happen, the gateways of the COMPASS domain have to be switched on and accessible via the CERN network. To check that this is the case at all times, regular ping commands are issued (every fifteen minutes) from an external server and the response is monitored; a notification is sent to the DCS experts in case those gateways are not reachable. \n\n\\noindent\\section{\\uppercase{The Front-ends Layer}}\n\\label{FrontEnds}\n\nThe experiment devices that are monitored and controlled by the DCS are spread over nearly two hundred meters, including the spectrometer and the beam tunnel. To communicate with all the devices, different field buses and communication protocols are \nused, namely CAN\\footnote{Controller area network. ISO standard 11898, see e.g.\\ www.iso.org} bus (8 daisy-chains), CAENet (6 daisy-chains), ModBus\\footnote{See http:\/\/www.modbus.org}, Profibus\\footnote{See http:\/\/www.profibus.com} (4 daisy-chains) and Ethernet.The general baud rate used for monitoring in the COMPASS CAN buses is 125 kbaud ($\\simeq 34$ kbits\/s), which is the recommended baud rate for the length of the daisy-chains used. These field buses transmit the information about the measurements of sensors to the front-end PCs (and commands to actuators in the opposite direction). In the front-end PCs, standard PCI\\footnote{Peripheral Component Interconnect.} cards are installed to collect the information carried by the field buses. The data is transmitted to the supervisory layer using a server-client model. An exception to this model is the three-layer model which is used when a database is included as an intermediate between the server and PVSS. This happens for the monitoring of the calorimeters, beam and trigger rates, and part of the polarised target system.\n\nIn ad\\-di\\-tion, spe\\-cialised de\\-vi\\-ces are used as intermediates between the devices and some of the front-end PCs, namely ELMBs (Embedded Local Monitor Boards) and PLCs.\n\nThe ELMB, described in \\cite{ELMB}, is a multi-purpose multiplexed ADC with 64-analog input channels with 16 bit-resolution which was developed by the ATLAS experiment. The communication of the ELMBs with the front-end PCs is done with the CAN field bus, using the CANopen protocol. The ELMB was designed and tested to be radiation- and magnetic field tolerant: its tolerance ranges up to about 5 Gy and $3\\times 10^10$ neutrons\/cm$^2$ for a period of 10 years and to a magnetic field up to 1.5 T.\n\nThe PLCs (Programmable Logic Controllers) are stand-alone, very robust, reliable and relatively fast devices that allow, among other operations, to regulate flows of gases and their percentage in mixtures, according to predefined settings and tolerance intervals, as well as to regulate cryogenic systems. The measurement of gas flows or gas percentages in mixtures is provided by the PLCs by ModBus to the DCS front-end PCs. \n\nManufacturer's OPC servers are used when available and stable. This is the case for the modern CAEN equipment and for Iseg equipment. In order to communicate with PLCs, an OPC server from Schneider\\footnote{See http:\/\/www.schneider-electric.com}, is used. Moreover, an OPC server was developed at CERN to control relatively old Wiener equipment, as the one used in COMPASS. To communicate with ELMBs, a CANopen OPC server, described in \\cite{CANopen}, is used.\n\nThe Distributed Information Management system (DIM) was developed at CERN and allows the implementation of a server-client model of publishing of lists of items and their actual values. The SLiC\\footnote{See http:\/\/j2eeps.cern.ch\/wikis\/display\/EN\/SLiC} DIM server developed at CERN, allows the control of the six CAENet lines used for the older type of CAEN crates. Each server has different groups of items with individually tunable speed reading cycles, thus permitting the separation of fast reading cycles (comprising voltages, currents and channel status) with reading frequencies as low as 1 Hz, thus allowing a fast detection of high voltage trips and failures; and of slow cycles, used for the read-back of settings.\n\t\t\t\t\nThe DIM protocol is also used to monitor other systems, namely temperatures and disk occupancy of servers, and processes of data transfer from the DAQ machines to CASTOR. \n\nDIP\\footnote{See http:\/\/en-dep.web.cern.ch\/en-dep\/Groups\/ICE\/Services\/DIP} is a protocol developed at CERN, based on DIM, but allowing exclusively read-only parameters, which are, in practical terms, those related to the CERN infrastructure (such as beam line magnet currents and the last beam file loaded, the primary target head inserted, the parameters to allow the monitoring of the CEDAR detectors, and data relative to the liquid nitrogen supply).\n\nOne PLC from the polarized target system is monitored using the S7 driver provided by PVSS, thus avoiding the use of an OPC server.\n\nPVSS provides functions to access relational databases such as mySQL and Oracle. This allows the access of information from the experiment conditions database (a set of mySQL databases), such as the calorimeter calibration event amplitudes, the beam and trigger rates, and parameters related with the polarised target.\n\nThe high voltage system of some of the detectors (Micromegas and the so called Saclay Drift Chambers) have special requirements with regards to its monitoring, and thus have a dedicated control system based on EPICS. This system publishes the most relevant data, which is read by a specially developed PVSS API (Application Programming Interface).\n\nThe Profibus protocol is used to transmit the data coming from the PLCs that monitor the detector gas systems to the PC that runs the Schneidar OPC server.\n\nMoreover, the magnetic field of the SM2 is measured with an NMRmeter that comes with a serial interface which, by use of the Profibus protocol, allows the communication with a standard PC, where a custom c program reads the information transmited, writes it in an ascii file and thereby makes it available for a PVSS API that collects the values and writes them into a datapoint.\n\n\n\\noindent\\section{\\uppercase{The Devices Layer}}\n\\label{Devices}\n\nMany different types of devices need to be controlled or simply monitored by the DCS, from high and low voltage crates and VME crates, to gas systems, sensors of temperature, humidity, pressure and magnetic fields.\n\nCOMPASS uses CAEN crates of different models to power most of its high voltage channels and for part of its low voltage channels. About 20 CAEN crates of older models (with CAENet interface) are in use, and six crates of newer models (with Ethernet interface).\nSeventeen Iseg high voltage modules are also in use and integrated in the DCS by use of their CAN interfaces. \nIn addition, fourteen Wiener low voltage power supplies are controlled, of which four are of type UEP6000, eight of type PL6021 and two of type PL508L. Nineteen VME crates are integrated in the DCS, both of older models (power supplies of type UEP5021) and newer models (power supplies of type UEP6021), the former being the majority. Both the power supplies and the VMEs are controlled by use of their CAN interface.\n\nIn subsystems where PLCs are used, the DCS only monitors the values that are published by them. This happens for the detector gas systems, the CEDAR detectors, and for systems that have dedicated control systems, namely the cryogenic systems of the polarized target, liquid hydrogen target and cold silicon detectors, see \\cite{Cesar, Anibus}.\n\nA wide range of devices are monitored by use of the ELMBs.\n\nHundreds of sensors are installed to monitor specific components of detectors or the experimental hall environment. For temperature monitoring, PT100 sensors in a 4-wire configuration are extensively used, whose output is read using the ELMBs.\n\nSome of the low voltage power supplies used in the experiment only have an interface for monitoring channel voltages or currents by means of voltage signals proportional to the values to be monitored, which is also read by ELMBs. In such cases, a calibration formula is used in the configuration of the CANopen OPC server, to provide the conversion to the real values delivered by the channels.\n\nMoreover, two of the most important magnets of the experiment, SM1 and Bend6, have their magnetic field monitored by Hall probes, whose output signals are read by ELMBs.\n\nIn the case of the second dipole magnet of the experiment, an NMRmeter is used. The NMRmeter has a serial interface, which, by use of the Profibus protocol, allows the communication with a PC.\n\nA custom power switch is also controlled, by use of the web server and driver provided with the equipment.\n\nThe DCS has an indirect monitoring of the powering system plus read-out chain of the four calorimeters of the experiment, based on the calibration signals of a laser system (for ECAL1) or a LED system (for the remaining three calorimeters). A component of the DAQ system calculates a spill-average amplitude of the signal read-out by each of the $\\sim$4500 channels, see \\cite{Konopka}, and saves this information in a mySQL conditions database, that is subsequently accessed by a PVSS script. In the DCS, a reference is chosen by the detector experts; afterwards, the DCS calculates, for each beam spill (using a synchronization scheme with the DAQ), the state of alert of each channel, based on the relative difference of the actual amplitude of the calibration amplitudes with respect to the reference values. The conditions to indicate alarms in the main panel are specified for the total of channels with a given alert state.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.4 \\textwidth]{ecal2.eps}\n\\caption{Panel for monitoring of the high voltage system of ECAL2, comprising around 3000 channels.}\n\\label{ECAL2}\n\\end{figure}\n\nThe main power supplies powering the electromagnetic calorimeters, their monitoring systems, and the subgroups distributor voltages are monitored by use of ELMBs.\n\nThe positions and movement of the electromagnetic calorimeters are controlled by CAMAC\\footnote{Computer Automated Measurement And Control.} modules, from whose readings the detector positions are calculated and read by the DCS.\n\nA recent integration in the main PVSS project is the monitoring of the most relevant parameters of the complex Polarized Target system. The communication with the devices required the usage of different protocols and front-end solutions: PLC S7, ModBus, DIP and ODBC (for MySQL and Oracle database connection). \n\nThe rates of the different triggers of the experiment, mo\\-ni\\-to\\-red online by the shift crew, are stored in a mySQL conditions database read by the DCS, which calculates rates normalised to the beam flux, and triggers alarms when those normalised rates fall outside predefined ranges.\n\nThe servers used in the DAQ run DIM servers to publish data related to internal temperatures, occupancy of their disks and status of important processes.\n\n\n\nThe beam line M2 belongs to CERN's infrastructure and thus is monitored and controlled by dedicated programs. The most important parameters, such as magnet currets, collimator positions, the primary target head or the beam file loaded are made available via DIP, thus providing alarms and and historical values of these parameters.\n\nMoreover, the CEDAR detectors (\\v{C}Erenkov Differential counters with Achromatic Ring focus), used in the hadron program of the experiment, are a responsability of CERN, and its relevant parameters are published using a DIP server. For the operation of these detectors, the density of the gas used must be within a predefined range. When this doesn't happen, the DCS displays a state of alarm, allowing the shift crew to start a procedure to refill the detectors. The high voltage system and the motors are also monitored.\n\n\\noindent\\section{\\uppercase{Conclusions}}\n\\label{Conclusions}\n\nThe DCS of the COMPASS experiment at CERN was presented in detail. This is a centralized system that displays to the end user in a homogeneised graphical user interface many different subsystems that use very different devices and thus require the use of a wide range of front-end solutions.\n\n\\section*{Acknowledgements}\n\nWe gratefully acknowledge the Controls group of CERN (IT\/CO and, later, EN\/ICE) and CERN's PhyDB for their constant and efficient support. This work was supported by FCT.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\begin{table}[t]\n\\centering\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{llll}\n\\multicolumn{4}{c}{\\textbf{List of Abbreviations}} \\\\ \\hline\n\\rowcolor[HTML]{EFEFEF} \n\\textbf{AMI} & Advanced Metering Infrastructure & \\textbf{HMS} & Head-End Management Server \\\\\n\\textbf{CA} & Certificate Authority & \\textbf{MAC} & Medium Access Control \\\\\n\\rowcolor[HTML]{EFEFEF} \n\\textbf{CRL} & Certificate Revocation List & \\textbf{OCSP} & Online Certificate Status Protocol \\\\\n\\textbf{DHT} & Distributed Hash Table & \\textbf{NIST} & National Institute of Standards and Technology \\\\\n\\rowcolor[HTML]{EFEFEF} \n\\textbf{DMZ} & Demilitarized Zone & \\textbf{PKI} & Public Key Infrastructure \\\\\n\\textbf{HES} & Head-End System & \\textbf{RSA} & Rivest\u2013Shamir\u2013Adleman \n\\\\ \\hline\n\\end{tabular}}\n\\end{table}\n\n\nThe existing power grid is currently going through a major transformation to enhance its reliability, resiliency, and efficiency by enabling networks of intelligent electronic devices, distributed generators, and dispersed loads \\citep{farhangi2010path}, which is referred to as \\textit{Smart(er) Grid}. Advanced Metering Infrastructure (AMI) network is one of the renewed components of Smart Grid that helps to collect smart meter data using a two-way communication\\citep{saputro2012}. Smart meters and integrated Internet-of-Things (IoT) devices are typically connected via a wireless mesh network with a gateway (or access point) serving as a relay between the meters and the utility company.\n\nThe security requirements for the AMI network are not different from the conventional networks as confidentiality, authentication, message integrity, access control, and non-repudiation are all needed to secure the AMI. Confidentiality is required to prevent exposure of customer's private data to unauthorized parties while integrity is necessary to ensure that power readings are not changed for billing fraud. Furthermore, authentication is crucial to prevent any compromised smart meters communicating with other smart meters. On the other hand, the National Institute of Standards and Technology (NIST) urges to use Public Key Infrastructure (PKI) for providing the security requirements of AMI \\citep{nist2014}. As an example, companies such as Landis\\&Gyr and Silver Spring Networks already use PKI to provide security for millions of smart meters in the US\\citep{landisgyr}. In such a PKI, the public-keys for smart meters and utilities are stored in \\textit{certificates} which are issued by Certificate Authorities (CAs). The employment of PKI in AMI requires management of certificates which include the creation, renewal, distribution and revocation. In particular, the certificate revocation and its association with smart meters are critical.\n\\vspace{10pt}\n\\begin{thisnote}\n\\subsection{Problem Description and Existing Solutions}\nSeveral reasons necessitate revoking certificates, such as key compromise, certificate compromise, excluding malicious meters, renewing devices, etc. Besides, if there is a vulnerability in the algorithms or libraries that are used in certificate generation, a massive number of revocations may additionally occur. For instance, a recent discovery of a chip deficiency on RSA key generation caused revocation of more than 700K certificates of devices that deployed this specific chip \\citep{2017ccsnemec} and renowned heartbleed vulnerability caused the revocation of millions of certificates, immediately \\citep{durumeric2014matter}. Thus, to establish secure communication, a smart meter should check the status of the other smart meter's certificate against a certificate revocation list (CRL) that keeps all revoked certificates. Considering the large number of smart meters in an AMI and the fact that the expiration period can be even lifelong in particular applications \\citep{landisgyr}, the CRL size will be huge. Consequently, revocation management becomes a burden for the AMI infrastructure which is typically restricted in terms of bandwidth. This overhead is particularly critical since the reliability and efficiency of AMI data communication are crucial for the functionality of the Smart Grid. Considering the potential impact on the performance of AMI applications \\citep{mahmoud2015investigating}, handling the overhead of revocation management is essential.\n\nCertificate revocation management is commonly handled by utilizing CRL that is stored in the smart meter. The status of a smart meter is determined by checking whether its certificate is listed in the CRL or not. An alternative method would be to store the CRL in a remote server as in the case of Online certificate status protocols (OCSPs) \\citep{galperin2013x}\\citep{pettersen2013transport}. In OCSP, an online and interactive certificate status server stores revocation information. Thus, each time a query is sent to the server to check the status of the certificate. While OCSP-like approaches can be advantageous on Internet communications, employing them for AMI is not attractive since it will require access to a remote server for each time. In this regard, another alternative would be to use OCSP \\textit{stapling} \\citep{pettersen2013transport} where the smart meters query the OCSP server at certain intervals and obtain a signed timestamped OCSP response which\nis included (\"stapled\") in the certificate. Again, this approach also needs frequent access to a remote server. Moreover, the 'stapled' certificates should be downloaded frequently by smart meters to ensure security, and this will create additional traffic overhead on the AMI which affects applications such as demand response or outage management. \n\n\\subsection{Our Approach and Contributions}\nIn this paper, we propose a communication-efficient revocation or CRL mananegment scheme for AMI networks by using RSA accumulators\\citep{camenisch2002dynamic}. RSA accumulator is a cryptographic tool which is able to represent a set of values with a single accumulator value (i.e., digest a set into a single value). Also, it provides a mechanism to check whether an element is in the set or not which implicitly means that cryptographic accumulators can be used for efficient membership testing. Due to the attractiveness of size, in this paper, we adapt RSA accumulators for our needs by introducing several novel elements as following:\n\n\\begin{itemize}\n\\item An accumulator manager is introduced within the utility company (UC) that is tasked with collection of CRLs from CAs and accumulating these CRLs (i.e., revoked certificates' serial numbers) to a single accumulator value which will then be distributed to the smart meters. \n\\item We also introduce a non-revoked proof tuple for allowing a smart meter to check whether another meter's certificate is revoked without referring to the CRL file.\n\\item We defined additional entities within AMI and assign functions to them to govern an accumulator based revocation management.\n\\item We introduced several security countermeasures against possible attacks to a accumulator-based scheme.\n\\end{itemize}\n\nThe computation and communication related aspects of the proposed approach is assessed via simulations in ns3 network. In addition, we built an actual testbed using in-house smart meters to assess the performance realistically. We compared our approach with the other methods that use conventional CRL schemes and Bloom-filters \\citep{akkaya2014efficient}. The results show that the proposed approach significantly outperforms the other existing methods in terms of reducing the communication overhead that is measured with the completion time. The overhead in terms of computation is not major and can be handled in advance within the utility that will not impact the smart meters.\n\nThis paper is organized as follows: In the next two sections, we summarize the related work and the background. Section IV introduces the threat model. Section V presents the proposed approach with its features. Section VI and VII are dedicated to evaluation criteria and experimental validation. Section VIII analyzes the security of the approach. Section IX discusses the benefits and limitations. The paper is concluded in Section X. \n\n\\end{thisnote}\n\n\\label{intro}\n\n\\section{Related Work} \\label{section2}\n\n\\begin{thisnote}\nDue to increasing threats towards Smart Grid, there has been a number of efforts to adapt PKI for Smart Grid communication infrastructure. For instance, Metke et al.\\citep{metke2010security} surveyed the existing key security technologies in Smart Grid domain by mainly focusing on PKI. On the other hand, the study \\citep{mahmoud2013efficient} stressed the importance of revocation overhead of PKI in Smart Grid. Beyond directly related studies on the PKI and Smart Grid relation, we also focus on studies about cryptographic accumulators and membership management. In this section, we examine the relation between this study and previous studies and highlight major differences.\n\n\\subsection{Revocation Management in AMIs}\nThe studies \\citep{mahmoud2015investigating} investigated different revocation management aspects such as short-lived-certificate scheme, tamper-proof device scheme, Online Certificate Status Protocol (OCSP), conventional CRL, and compressed CRL. However, this study just hypothetically analyzed the applicability of existent revocation solutions for AMI. The first offered approach that focused on reducing the revocation management overhead for AMI was based on Bloom Filters \\citep{rabieh2015scalable}. They provided a Bloom Filters based scheme particularly to reduce the size CRL. \\end{thisnote} However, since Bloom Filters suffer from false positives, the approach\nrequires accessing the CA to check the validity of a certificate. Our proposed scheme, on the other hand, never\nrequires accessing a remote server and provides a better reduction\non CRL size. The study in \\citep{cebe2017efficient} use distributed hash tables (DHT) to reduce the CRL size again. Although this study provides a reduction in CRL size, it suffers from additional inter-meter communication overhead for accessing the CRL information. \n\\begin{thisnote}\nWe would like to note that a very preliminary version of this work was published in \\citep{cebe2018efficient}. In this work, we improved the various aspects of the previous one. First, we improved computation performance utilizing Euler's Theorem. Second, we extended our threat\nmodel to new attack types that were not considered in the conference version. In this regard, we changed our approach in several ways: We proposed to use an initial secret during accumulation. We then introduced a non-revoked proof concept that was not used before in any of the revocation works. This required major changes to the accumulation process which was not in \\citep{cebe2018efficient}. We finally proposed an extensive certificate verification protocol as countermeasures to the new threats. This also required proposing a new secure multi-level AMI architecture as opposed to the monolithic architecture used in \\citep{cebe2018efficient}. In addition, we added several new experiments with accumulator computation overhead under various assumptions.\n\\end{thisnote}\n\\subsection{Cryptographic Accumulators}\nBenalog and DeMare \\citep{benaloh1993one} first introduced cryptographic accumulators. After their first appearance, there have been studies \\citep{camenisch2002dynamic,reyzin2016efficient,baldimtsi2017accumulators} offering to use them for membership testing. However, these studies solely focused on building the cryptographic fundamentals of accumulators, and thus, omit application-specific issues and security features when deploying them. Besides, these studies are offering to use accumulators for membership testing by accumulating a valid list. Considering AMI, accumulation of valid smart meter's certificates to provide a revocation mechanism would constitute a significant overhead due to the fact that revocation frequency is less than that of creating new certificates {(i.e., no need to update the accumulator each time when a new smart meter is added to AMI)}. Furthermore, since the number of revoked certificates is also less than the number of valid certificates which affects the required computation time significantly\\citep{durumeric2014matter}. \nOur approach mitigates these drawbacks by addressing security and application-specific issues and offering to use CRLs instead of valid certificates.\n\n\n\\section{Preliminaries}\n\\label{section3}\n\\begin{thisnote}\nBefore explaining our approach we provide some cryptographic background of accumulators and its particular form as RSA accumulators. In addition, to help the reader grasp a general idea of revocation management through CRLs, we explain the CRL and delta-CRL notions.\n\\end{thisnote}\n\\subsection{Background on Cryptographic Accumulators}\nBenaloh and De Mare\\citep{benaloh1993one} introduced the cryptographic accumulator concept which is a one-way hash function with a special property of being \\emph{quasi-commutative}. A quasi-commutative function is a special function $\\mathcal{F}$ such that $y_0,y_1,y_2 \\in \\mathbb{Y}:$\n\\begin{equation}\n\\mathcal{F}(\\mathcal{F}(y_0,y_1),y_2)=\\mathcal{F}(\\mathcal{F}(y_0,y_2),y_1) \\label{quasi}\n\\end{equation}\nThe properties of this function can be summarized as follows: \\textit{1)} it is a one-way function, i.e., hard to invert; \\textit{2)} it is a hash function for obtaining a secure digest $\\mathcal{A}$ (i.e., accumulator value) where $\\mathcal{A} = \\mathcal{F}(\\mathcal{F}(\\mathcal{F}(y_0,y_1),y_2),...,y_n)$ for a set of values $\\{y_0,y_1,y_2, . . . , y_n\\} \\in \\mathbb{Y}$; \\textit{3)} it is a \\emph{quasi-commutative} hash function which is different from other well-known hash functions such that the accumulator value $\\mathcal{A}$ does not depend on the order of $y_i$ accumulations. \n\nThese properties allow cryptographic accumulators to be used for a condensed representation of a set of elements. In addition, since the resulting accumulated hashes of $y_i$ ($\\mathbb{Y}=\\{y_i;~0 In fact, it is double the amount of time as shown in Table 1. The authors should not shy away from pointing this fact, while stressing on the other hand the space saving of their proposed approach}\n\n\\added{Response: We thank the reviewer for the comments.We updated the text accordingly}\n\\begin{itemize}\n \\item We observe that our approach has comparable results with the local CRL method which requires a simple text search over complete \\textit{full CRL} file, \\textcolor{red}{yet providing huge space saving benefit which effects both distribution and storage overhead.}\n\\end{itemize}\n\n\\end{itemize}\n\n\\item \\textit{Comment 7: Page 8, section 5. D.2: While the results in Table 1 show that the accumulator approach has higher revocation check time, it is hard to get a sense of the significance of the values provided in the table. It is not clear from the text whether the tests were done on the smart meters (and in this case, the technical characteristics of the smart meters are not provided), or on another device.}\n\n\\added{Response: We clarified the experiment is accomplished on the smart meter by adding following text to the revised manuscript}\n\n\\begin{itemize}\n \\item Finally, we looked at the computational time overhead for checking whether a certificate is revoked or not based on the three approaches compared. \\textcolor{red}{ This is an important experiment to understand the computation overhead our approach on the smart meter, which is shown in Figure 3.a, considering the fact that it has limited resources.}\n\\end{itemize}\n\n\\item \\textit{Comment 8: -A small typo in page 1: \"we focus efficient handling of this issue in this paper.\" -> focus \"on\" efficient\u2026\n}\n\\added{Response: The typo was relieved.}\n\n\\par\\bigskip\\hrul\n\n\\section{Responses to Reviewer 4}\n\\item \\textit{Comment 1: In section 4, it presented a high-level design of the approach but computation details of all algorithms are not given. \n}\n\n\\added{Response: We now give references to the equitations which are introduced in Background section}\n\n\\item \\textit{ Comment 2: In section 3.A, the introduction of one-way accumulator is slightly different from the paper they cited where $y_0$ and $y_1$, $y_2$ comes from different domains. Similarly, in section 3.B, domains of g and other variables are not clearly claimed.. \n}\n\n\\added{Response: This is the same issues raised by Reviewer #1 in Comment #2 and Reviewer #3 in Comment #3. We resolved these issues accordingly}\n\n\\item \\textit{ Comment 3: Since the details of the algorithms and variables are incomplete, the security of the approach cannot be evaluated.\n}\n\n\\added{Response: The revised manuscript now contains additional computational details. Moreover, it also contains Threat Model and Security Analysis sections. }\n\n\\item \\textit{Comment 4. To improve efficiency, CRL is used instead of valid list in the approach. However, there is still possibility that certificates presented by attackers which are not in CRL end up to be invalid ones. Further explanation is needed.}\n\n\\added{Response: We think Section V.D explains enough the authentication process. Nevertheless, I have added following text to help relieving confusions.}\n\n\\begin{itemize}\n\\item When two meters communicate by sending\/receiving signed messages, the signatures in these messages need to be verified. To be able to start the verification process, a receiving device needs to use the public key (for signature verification) presented in the certificate sent to itself. To ensure that this certificate is not revoked, then it needs to initiate a process which we call as certificate verification protocol. Figure~\\ref{fig:mutual} shows an overview of this process.\nBasically, the receiving device checks the corresponding $nr_{proof}$ tuple's signature to ensure that it is produced by the UC. Once the signature is verified, it then checks whether the the serial number within the tuple is same as the serial number of the provided certificate (i.e., either EndDevice\\#1.cer). For additional security, it also checks the length of the $nw_1\\&nw_2$ to see whether it is equal to the first accumulation setup parameter $k$. Finally, by performing $RevocationCheck()$ function, it checks whether the provided $nr_{proof}$ is correct. If all these steps are successful, the end-device has successfully complete the certificate verification protocol. \\textcolor{red}{Note that, without carrying the $nr_{proof}$ a smart meter can not be authenticated even if it has a valid certificate.}\n\\end{itemize}\n\n\\item \\textit{Comment 5: The implementation in simulation network consist of 81 and 196 meters which is far less than the scale in real smart grid, thus the results may not be credibly applied.\n}\n\n\\added{Response: Thanks to the reviewer for pointing out the fact that the produced simulation environment not exactly represent the real AMI size. However, to enable a larger AMI simulation, we should create our simulation environment by using Parallel Programming MPI + OpenMP and some sort of the cluster hardware. We believe that it will be out of scope of this study since the results clearly shows the scalibility of our approach by providing a revocation check mechanism which is independent from the size of CRL. In addition, the results shows that }\n\n\\item \\textit{Comment 6: There are some typos in section 2 and section 3. For example, \"collusion\" in line 21, section 3.A might be \"collision\".\n}\n\n\\added{Response: The typo is resolved.}\n\n\\par\\bigskip\\hrul\n\n\\section{Responses to Reviewer 5}\n\\item \\textit{Comment 1: About the use of CRLs, why ask the smart meter to store CRLs? I think the CRLs should be maintained by CAs. Whenever a smart meter MA needs to verify the validity of another smart meter MB's certificate, it just needs to download the latest certificate of the CA who signs MB's certificate. That is, the smart meter does not need to store the CRLs. If this is the case, then the motivation of this paper is not strong enough.}\n\n\\added{In light of the reviewer's comment, we think that could not clearly give the motivation of the problem in the Introduction section. Thus, we added following paragraph to the Introduction section. We hope this will help to unravel the motivation of the problem more clearly.}\n\n\\begin{itemize}\n \\item \\textcolor{red}{An alternative method would be to store the CRL in a remote server where an online and interactive certificate status server stores the revocation information as in the case of Online Certificate Status Protocol(OCSP) \\cite{galperin2013x}. By this way, a query can be sent to the server to check the status of a certificate whenever two meters would like to communicate. While OCSP can be advantageous for systems where Internet connections are always on, employing it for AMI is not an attractive idea since it will require access to a remote server for each interaction. In this regard, another alternative would be to use OCSP \\textit{stapling} \\cite{pettersen2013transport} where the smart meters query the OCSP server at certain intervals and obtains a timestamped OCSP response which is directly signed by the CA. This response is included (i.e., \"stapled\") in the certificate as a proof that it is not revoked. However, again, this approach requires frequent remote access. Furthermore, in order to perform authentication properly, the \"stapled\" smart meters' certificates should be download frequently by all the smart meters even in case there is no revocation incident, which will create enormous traffic on the AMI network. \n}\n\\end{itemize}\n\n\\item \\textit{Comment 2: In Section II.B, the authors claimed that their approach mitigated drawbacks of existing accumulator schemes by addressing security and application specific issues and offering to use CRLs instead of valid certificates in sense that the size of CRLs is much smaller than that of valid certificates. However, if the accumulator outputs a constant size of data, then it does not matter whether you accumulate a valid list or accumulate the CRLs.}\n\n\\added{Response: The reviewer is right about accumulating either valid or non-valid certificates would not be different in terms of accmulator size. However, the computation overhead and the number of times to update the accumulator value would be different. To clarify this issue, we modified the revised manuscript as follow:}\n\n\\begin{itemize}\n\\item Considering AMI, accumulation of valid smart meter's certificates to provide a revocation mechanism\nwould constitute a significant overhead due to the fact that revocation frequency is less than that of creating new certificates \\textcolor{red}{(i.e., no need to update accumulator for each new smart meter addition to AMI)} and number of revoked certificates is also less than the number of valid certificates \\textcolor{red}{ (i.e in terms of computation time)}\\cite{durumeric2014matter}.\n\\end{itemize}\n\n\\item \\textit{Comment 3: The proposed approach relies on the modification of RSA accumulator for the AMI case. However, the modifications inherit existing schemes [10] and the second part reducing the complexity of accumulator computation by using Euler's Theorem is actually common knowledge and thus not new. So what are the design challenges?}\n\n\\added{Response: As the reviewer pointed out, we are adapting existing cryptographic methods for revocation management. However, the adaptation of the existing cryptographic operations implicitly carry a number of different challenges which we resolved in our approach. The first challenge as pointed out is reducing the computation overhead by using Euler's theorem. The second is related to the adaption of certificate serial numbers to RSA accumulator. RSA accumulator requires special type of input to proper cryptographic operations. We implemented the method in [21] to produce a proper input to RSA accumulator from certificate serial numbers. Since the security of the accumulator is critical, we proposed a network topology and related additional components. Fourth, as we described in our Threat Model, to mitigate a possible attack regarding the \\textit{freshness of accumulator values}, we introduced a random secret $r_k$ accumulation. Fifth, we introduced $nr_{proof}$ concept and revocation check mechanism which is described in Figure 2 to mitigate any \\textit{stolen non-witness attack}. Without these modification, current accumulation approach can only mitigate a \\textit{comprised certificate attack}. In addition, naive application of the accumulator concept to the AMI for revocation management would be lack of many practical aspects as well who will compute accumulator value, how accumulator values should be updated to ensure the security of the mechanism through existing components, etc,.}\n\n\\item \\textit{Comment 4: The proposed algorithms are vaguely described, so it is not clear how these algorithms work.\n}\n\n\\added{Response: We have added additional description to Background section and its relation with our algorithm such as which equation used in which function in the Proposed appraoch section}\n\\item \\textit{Comment 5: The security analysis of the proposed protocol is missing.\n}\n\n\\added{Response: Now the revised text contains both Threat Model and Security Analysis.}\n\n\\section*{Acknowledgement}\nThis material is based upon work supported by the Department of Energy under Award Number DE-OE0000779.\n\\vspace{2\\baselineskip}\n\n\\begin{comment}\n\\begin{wrapfigure}[5]{l}{0.20\\textwidth}\n\\vspace{-9pt}\n\\centering\n\\includegraphics[width=0.20\\textwidth]{mumin.jpg}\n\\vspace{-34pt}\n\\end{wrapfigure}\n\\noindent Mumin Cebe is a PhD student in the Department of Electrical and Computer Engineering at Florida International University. He works at the Advanced Wireless and Security Lab (ADWISE) which is lead by Prof. Kemal Akkaya. He conducts research in the areas of wireless networking and security\/privacy that relates to Internet-of-Things (IoT) and Cyber-physical Systems (CPS), particularly in Smart Grids and Vehicular Networks.\n\n\\vspace{2\\baselineskip}\n\n\\begin{wrapfigure}[6]{l}{0.20\\textwidth}\n\\vspace{-9pt}\n\\centering\n\\includegraphics[width=0.20\\textwidth]{Akkaya.jpg}\n\\vspace{-34pt}\n\\end{wrapfigure}\n\\noindent Kemal Akkaya is a professor in the Department of Electrical and Computer Engineering at Florida International University. His current research interests include security and privacy, internet-of-things, and cyber-physical systems. He is the area editor of Elsevier Ad Hoc Network Journal and serves on the editorial board of IEEE Communication Surveys and Tutorials. He has published over 120 papers in peer reviewed journal and conferences. He has received ``Top Cited'' article award from Elsevier.\n\n\\end{comment}\n\n\\section*{References}\n\n\n\\section{Responses to Reviewer 1}\n\n\\begin{itemize}\n\n\\item \\textit{Comment 1: The paper is interesting, motivation is clear, the manuscript is well organized, and the authors able to convey their intended message. The paper explains the results in a clear manner.}\n\n\\added{\\textbf{Response}: We would like to thank the reviewer for his\/her appreciating comments.}\n\n\\item \\textit{Comment 2: It would be nice to list the limitations of current work precisely. If possible, discuss possible ways to attacks on the proposed certification revocation mechanism.}\n\n\\added{\\textbf{Response}: We thank the reviewer for this comment. In light of this comment, we further expanded our manuscript with a ``Benefits and Limitations'' Section to demonstrate the limitations of our approach where we discuss a possible side attack utilizing those limitations along with it precautions. Please refer to Subsection 9.2 for corresponding changes in the revised manuscript. (Page 32-33)}\n\n\n\\end{itemize}\n\n\\par\\bigskip\\hrul\n\n\n\\section{Responses to Reviewer 2}\n\n\\begin{itemize}\n\\item \\textit{The paper addresses the challenge of distributing and storing the certificate revocation list of smart metering systems under consideration of security protection. To avoid space limitations the paper proposes the use of cryptographic accumulators which shall allow reducing the size of needed revocation information. POINTS FOR IMPROVEMENT:}\n\\end{itemize}\n\\added{\\textbf{Response}: Thank you for the kind feedback. We addressed those issues in the revised manuscript as detailed below:}\n\\bigskip\n\n\\begin{itemize}\n\\item \\textit{Comment 1: FIU: Please introduce all abbreviations first. In the abstract try to avoid abbreviations at all. \nIn the abstract, the reader does not need to know that this has been applied at the FIU as this is obvious from the author affiliations and can be clarified in the detailed sections of the paper. Keep the Abstract short.}\n\n\\added{\\textbf{Response}: We thank the reviewer for this comment. We edited the abstract and removed all abbreviations and made abstract more succinct in the revised manuscript. In addition, we added a ``List of Abbreviations'' table at the end of the second page. \n}\n\n\n\\item \\textit{Comment 2: It does not become obvious in the introduction what the contribution will be. Do lines 73-80 introduce a second contribution? Make the contribution(s) of the paper clear, list them explicitly.\nHow does the proposed solutions relate to the problem introduced? This does not become clear. It might be advisable to restructure the introduction with clear subsections, i.e. background and motivation, problem statement, contribution, outline.}\n\n\\added{\\textbf{Response}: Thank you for the valuable suggestion. In light of the reviewer's comments, we clarified our position by restructuring the Introduction completely and precisely described the problem, motivation and our contribution in different subtitles. Please refer to pages 2-5 in the revised manuscript.} \n\n\n\\item \\textit{Comment 3: The related work is in need of a brief introduction. The reader does not get to know which related work is explained, why this is related and how this relates to the remainder of the paper.}\n\n\\added{\\textbf{Response}: In light of the reviewer's comment, related work is revised by adding a brief introduction and revising some portions to better describe the relation of our work with the previous ones. Please refer to pages 5-7 in the revised manuscript.}\n\n\n\\item \\textit{Comment 4: In line 115 it is discussed that this work provides 60 more content than in previous work. While this is an important information for the editor, it is not for the reader. The reader needs to know how this work relates to the previous work. Why have things been changed etc.}\n\n\\added{\\textbf{Response}: We thank the reviewer for this comment. We removed that part from the manuscript and re-structured the related work accordingly to resolve raised issues. Please see pages 5-6 in the revised manuscript.}\n\n\\item \\textit{Comment 5: Section 3 is also missing a brief introduction. Please explain to the reader what will happen in this section, why this section is needed and where these foundations are taken from. (Page 7)}\n\n\\added{\\textbf{Response}:\nTo help the reader, we provided a brief introduction to the Section 3 in the revised manuscript.}\n\n\n\\item \\textit{Comment 6: Also for Section 4, explain to the reader why this section is needed and how it relates to the contribution of the paper. How do the 4-points of the treat model relate to the approach. Are they already part of the approach? In this case integrate them with the next section.}\n\n\\added{\\textbf{Response}: We thank the reviewer for the comment. We are sorry to see that our explanations in the paper created a confusion about our threat model that is highly coupled with the approach. Actually, this was not our intention. To clarify this, we have completely modified the threat model section in the revised manuscript. Basically, in the threat model, we are now providing a more generic model where an adversary can both compromise the devices and communication. In addition, we simplified the system model so that it does not also get into approach description. Please refer to pages 10-12 for these changes. }\n\n\\item \\textit{Comment 7: For Section 5.2 please explain what this section is doing.}\n\\added{\\textbf{Response}: \nIn light of the reviewer's comments, we have updated the section and put an introduction to frame our bottom-up approach in Section 5.2 page 13 of the revised manuscript.}\n\n\\item \\textit{Comment 8: Between Section 5 and 6 a Section explaining the evaluation setup is needed. Please make clear what should be evaluated, why this shall be evaluated, and how this will show that the approach is feasible }\n\n\\added{\\textbf{Response}: We agreed that a section about evaluation would give a broad idea to reader on our evaluation criteria. Thus, we have added the requested section named \\textit{``Evaluation of the Approach and its Objective''} in the revised manuscript. (Pages 19-20)}\n\n\n\\item \\textit{Comment 9: Section 6 only lists loose proposals why the approach supports the threats from section 4. It is in need of deeper argumentation to explain or show why this is feasible. }\n\n\\added{\\textbf{Response}: Thanks for this valuable comment. This feedback help us to realize missing points related to the security analysis of our work. Thus, we now provided a deeper analysis and discussion for security analysis in the revised manuscript. Please refer to the \\textit{''Security Analysis''} for revised version. (Pages 28-31)}\n\n\n\\item \\textit{Comment 10: A discussion section is missing. Particularly, a discussion section should summarize the findings, discuss the limitations\/TTV, and deduce insights that go beyond the concrete technique and contribute to the state of the art in general. }\n\n\\added{\\textbf{Response}: In light of reviewer's comments, we added a new section to summarize our findings and also pointed out potential limitations of the proposed approach. Please refer to ``Benefits and Limitations'' section of the revised manuscript. (Pages 31-33)}\n\n\\item \\textit{Comment 11: Line 49: revoked certificates serial numbers -> revoked certificates' serial numbers\n}\n\n\\added{\\textbf{Response}: Corrected accordingly.}\n\n\\item \\textit{Comment 12: Line 89: remove one \"aspects such as\"\n}\n\n\\added{\\textbf{Response}: Corrected accordingly.}\n\n\\end{itemize}\n\n\\newpage\n\\section{Responses to Reviewer 5}\n\n\\begin{itemize}\n\n\\item \\textit{Comment 1: The objectives and potential impact of the research is not clearly presented. It should be included as a separate section in \"5 Proposed approach\"}\n\n\\added{\\textbf{Response}: Thanks for this valuable comment. This feedback help us to realize missing points related to the main objective and potential impact. As a result, we made some changes and now clearly state our objective and potential impacts in two different sections. The objective of our approach along with evaluation criteria are given in Section 6 after Section 5 as a separate section (page 19). The potential impacts of study are given in Section 9.1 on pages 31-32 of the revised manuscript.}\n\n\n\n\\item \\textit{Comment 2: The experimental simulation should be compared with a real world experiment and potential differences should be pointed out. Please also explain what is the meaning of \"realistic results\" in the paragraph: \"Finally, for more realistic results, we built an IEEE 802.11s-based mesh\"}\n\n\\added{\\textbf{Response}: The reviewer is right in pointing out what realistic refers to as we did not provide details about it. Basically, we built an actual mesh network at FIU by using Raspberry PIs and IEEE 802.11 antennas. Therefore, we updated manuscript accordingly to clarify the goal behind building such a testbed environment on pages 21-22 of the revised manuscript.}\n\n\\item \\textit{Comment 3: A security analysis is presented in section 6 but these aspects need to be addressed extensively in section 7 as well.}\n\n\\added{\\textbf{Response}: In light of reviewer's comment, we extended the security analysis section and moved it after experiments to be able emphasize obtained results and their affects on security. Please refer to Section 8 in the revised manuscript (Pages 28-31). }\n\n\\item \\textit{Comment 4: The added value of the research in comparison to other methods is not clearly addressed in the conclusion section. Some aspects are addressed in each of sub-sections of section 7. A suggestion is to address these aspects in a structured manner, pointing out the advantages and potential disadvantages in a separate section before conclusions.}\n\n\\added{\\textbf{Response}: Thank you the reviewer to stressing this important shortfall of the manuscript. As a result, we introduced a new ``Benefits and Limitations'' section. Please refer pages 31-34 for details in the revised manuscript.}\n\n\\end{itemize}\n\\end{document}\n\n\n\\section{Responses to Reviewer 1}\n\n\\begin{itemize}\n\n\\item \\textit{Comment 1: All the reviewer comments were addressed in the revised manuscript.}\n\n\\added{\\textbf{Response}: We thank the reviewer again for his\/her feedback which helped us to significantly improve the paper.}\n\n\\item \\textit{Comment 2: would recommend authors to add \"how the scalability or any other issues needs to handle when applied to the real testbeds such as EPIC [1] or real city-scale AMI metering infrastructure? \"\n}\n\n\\added{\\textbf{Response}: We thank the reviewer informing us a real testbed which is available to researchers. In light of this comment, we further expanded our future work where we discuss possibility of using a real testbed to assess our approach. Please refer to Section 10 for corresponding changes in the revised manuscript. (Page 34)}\n\n\n\\end{itemize}\n\n\\par\\bigskip\\hrul\n\n\n\\section{Responses to Reviewer 2}\n\n\\begin{itemize}\n\\item \\textit{Comment 1: The manuscript has considerably been revised. A great effort has been made to address all reviewer comments. The paper is in a state where it can be accepted as is. }\n\n\\added{\\textbf{Response}: We would like to thank the reviewer for his\/her appreciating comments.\n}\n\n\n\\item \\textit{Comment 2: There is one minor remark the authors might want to reconsider. The introduction section got rather long, which is not a bad thing in case of this paper. However, it might be advisable to use subsection headings within the introduction to guide the reader a bit more }\n\n\\added{\\textbf{Response}: Thank you for the valuable suggestion. In light of the reviewer's comments, we edited the introduction by adding a proper subsections introduction. Please refer to pages 2-5 in the revised manuscript.} \n\n\\end{itemize}\n\n\\newpage\n\\section{Responses to Reviewer 5}\n\n\\begin{itemize}\n\n\\item \\textit{Comment 1: The authors have addressed the comments\"}\n\n\\added{\\textbf{Response}: Thanks again for valuable comments during revision process to help us to realize missing points of our manuscript.}\n\n\n\n\\end{itemize}\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nTopological crystalline phases in non-interacting, clean structures have attracted a great deal of recent theoretical and experimental attention\\cite{Fu2011,Hsieh2012,ando2015topological,Hughes11,Turner2010,Turner2012}. \nFrom the discovery of helical edge states in $\\mathbb{Z}_2$ topological insulators\\cite{Kane04,bernevig2006quantum,konig2007quantum}, to surface Dirac cones protected by time-reversal or crystal symmetries\\cite{fukanemele,xia2009observation,Hsieh2012,Hourglass}, the experimental manifestations of band topology have come primarily through the exploration of surface states. \nThe recent theoretical prediction of higher-order topological insulators\\cite{hotis,Po2017,khalaf2018higher,benalcazar2017quantized} has triggered a wave of materials predictions\\cite{NaturePaper,bigmaterials,bigmaterials-china,ashvin-materials,xu2020high} and experimental efforts to observe their predicted gapped surfaces but gapless corners (in 2D) or hinges (in 3D). \nAt a theoretical level, topological band insulators can be classified by exploiting the constraints of symmetry, relating the topology of bands to the transformation properties of Bloch functions under crystal symmetries\\cite{Kruthoff2016,NaturePaper,Po2017,Fu2007,po2020symmetry,cano2020band,MTQC,watanabe2018structure,Zhang_TMCI_corep_RPB,Slager_magnetic_spacegroup_rep}. \nIn the simplest cases, the symmetry eigenvalues of occupied electronic wavefunctions at different crystal momenta in the Brillouin zone can be used to deduce the absence of an exponentially localized, position space description of the occupied states, and hence the presence of non-trivial topology. \nProgress along these lines has led to a full, predictive classification of topological band structures both with and without time-reversal symmetry. \nEssential to these efforts is the presence of discrete translation symmetry, which ensures that localized electronic functions are identical in each unit cell, and hence allows the symmetry properties of the system to be described as a function of momentum. \n\nAt the same time, the interplay between topological bands and symmetry-breaking order has started to attract a great deal of attention. \nIt has been argued theoretically that in topological systems with charge-density wave (CDW) order, the collective phason mode of the CDW may inherit topological properties from the Fermi sea, such as an induced axion coupling to electromagetic fields\\cite{wang2013chiral,you2016response,BurkovCDW,zyuzin2012weyl,maciejko2014weyl}. \nSignatures of this axion coupling have been recently experimentally detected in (TaSe$_4$)$_2$I\\cite{gooth2019evidence,shi2019charge}. \nAdditionally, the quantum anomalous Hall phase in the Dirac semimetal ZrTe$_5$ can be understood as originating from a magnetic-field induced CDW transition\\cite{tang2019three,qin2020theory,song2017instability,zhang2017transport}. \nBecause CDW order is in general incommensurate with the underlying lattice, a full understanding of the interplay between mean-field CDW order and band topology requires us to examine topology of incommensurately modulated electronic systems. \nSuch a study would also yield insights into topology in artificially modulated photonic\\cite{ozawa2018topological,ozawa2016synthetic}, metamaterial\\cite{grinberg2020robust}, and cold-atomic lattice systems\\cite{Ane1602685}, which have become a focus of recent research due to their tunability and experimental accessibility.\n\n\n\nNaively, the breaking of translational and point group symmetries implied by incommensurate modulation would seem to prohibit the application of symmetry-based tools which have been so successful in identifying and classifying topological crystalline systems. \nHowever, it is often possible to view the single-particle dynamics in an incommensurately modulated system as describing the behavior of a particle in a larger number of dimensions, the phase offsets of the incommensurate modulations playing the role of momenta in the extra ``synthetic'' dimensions. \nThe canonical example of this mapping is the 1D Harper (Aubry-Andr\\'{e}) model with incommensurate on-site potential. \nAs was shown some time ago\\cite{hofstadter-original}, the Hamiltonian for the Harper model is equivalent to the Hamiltonian for a 2D square-lattice system coupled to a background magnetic field\\cite{equivalence_Fibo_Harper_2012,Kraus_1D_QC_to_2D_QHE}. \nThe phase of the on-site modulation plays the role of the momentum in the second, synthetic dimension, while the wavevector of the modulation plays the role of the magnetic flux per plaquette in the 2D lattice. \nBands in the enhanced, 2D system can be characterized by a Chern number, which mandates the presence of gapless chiral modes at the edges of the system. \nReducing back to 1D, these two-dimensional edge states manifest as boundary states of a 1D wire which appear and disappear as a function of the phase of modulation, thus realizing a Thouless pump\\cite{Thouless_pump_original_paper,niu1984quantised,Topologically_quantized_current_PRR}. \n\\textcolor{black}{Recent studies also show that certain generalization of the 1D Harper model allows for the investigation of higher-order topological phases~\\cite{Zeng_generalized_AAH_model_HOTI_PRB}.} \n\n\nIn this work, we will extend the connection beyond 1D, to show how modulated systems in 2D and 3D can be related to topological crystalline phases in higher dimensions. \nWe will first review a general method for representing a modulated system as a higher dimensional system coupled to a background gauge field\\cite{RiceMele,Thouless_pump_original_paper,Kraus_1D_QC_to_2D_QHE,equivalence_Fibo_Harper_2012}.\nFor systems with negligible spin-orbit coupling and spin-independent modulation, the gauge field will be a $U(1)$ magnetic field; for spin-dependent modulations we will show that there can also be induced $SU(2)$ gauge fields. \nWe will exploit the fact that both $U(1)$ and $SU(2)$ gauge fields with constant field strength preserve inversion symmetry to show that 2D modulated systems can realize higher-order chiral ($U(1)$) and helical ($SU(2)$) topological phases in one extra synthetic dimension. \nWe show how the hinge states of these synthetic higher-order topological insulators (HOTIs) manifest as corner modes in 2D, with energies that can be tuned by changing the phase of the modulation. \nGoing further, we use the mapping to synthetic dimensions to bring order to the complex landscape of eigenstates of the modulated system, showing how the states can be interpreted as bulk and surface Landau level (LL) wavefunctions in synthetic dimensions. Finally, we also revisit a 3D minimal model for a Weyl semimetal (WSM) with (generally incommensurate) CDW order\\cite{dynamical_axion_insulator_BB}, and show how it realizes a 4D nodal line semimetal gapped into a phase with a non-trivial second Chern number. We will verify our conclusions with a combination of exact numerical results and approximate low-energy analytic calculations.\nWe will also exploit the fact that the phase of a (charge- or spin-) density wave (DW) order parameter can be shifted with an applied electromagnetic field, by exciting the (nominally gapless, but sometimes pinned) sliding mode\\cite{gruner1988dynamics}. \nThis will allow us to make predictions about topological pumping of boundary states in modulated structures, driven by the sliding mode of the DW.\nIn contrast to other recent proposals for topological pumping in synthetic dimensions, the coupling of the DW sliding mode to electromagnetic fields allows for tunability of synthetic dimensions in modulated structures. \nWe will comment on potential experimental realizations in condensed matter, photonic, and cold-atom systems throughout. \nThis work will enable new avenues for exploring higher-order topological phenomena which, with the exception of some promising results in Bismuth\\cite{hsu2019topology,nayak2019resolving,schindler2018higher}, have not been unambiguously identified in crystalline electronic systems.\n\n\nThe rest of the paper is organized as follows. \nIn Sec.~\\ref{sec_review_thouless_pump_Rice_Mele}, we review how the Thouless pump in a 1D Rice-Mele [Su-Schrieffer-Heeger (SSH)] chain is realized by the sliding of a CDW, and we review its connection to topology by promoting the model to a 2D $\\pi$-flux lattice. \nIn Sec.~\\ref{sec_Dimension_promotion} we next develop a general method to compute the $U(1)$ gauge fields that are coupled to a higher dimensional models promoted from a low-dimensional modulated system. \nIn Secs.~\\ref{sec_chiral_HOTI} and~\\ref{sec_helical_HOTI_sliding_modes}, we construct 2D modulated systems that can be promoted to 3D chiral and helical HOTIs coupled to $U(1)$ and $SU(2)$ gauge fields, respectively. \nWe demonstrate the pumping of corner modes by the sliding of DWs in these systems via numerical calculations of the energy spectra. \nWe examine the properties of wave functions in these 2D modulated systems by constructing low energy theories coupled to gauge fields in 3D. \nWe show how the evolution of bulk, edge, and corner states in 2D can be understood from the perspective of the low energy theory in 3D. \nIn Sec.~\\ref{sec:Weyl_CDW}, we turn to a model for a 3D WSM gapped by a CDW. We show that this model can be promoted to a 4D nodal line system gapped by a $U(1)$ gauge field. \nWe derive the corresponding low energy theory in 4D, and use it to explain both the existence of QAH surface states and the interpolation between topologically distinct QAH phases at the two inversion-symmetric values of the CDW phase in this 3D system.\nFinally, in Sec.~\\ref{sec:outlook}, we give an outlook as to how our work may extend the search of (higher-order) topological insulators in higher dimensions and enable simulations of $SU(2)$ gauge physics in higher dimensions. \nSome details of our models, further numerical results, and detailed derivations of the low energy theories are presented in the Supplementary Material (SM)\\cite{SM}. \n\nThroughout this paper, we use units where $\\hbar = c = |e| = 1$, and where the electron has charge $-|e| = -1$. \nFurthermore, the Einstein summation convention will not be used; whenever there is a summation over an index, we will write the summation explicitly.\n\n\n\n\\section{\\label{sec_review_thouless_pump_Rice_Mele}Review - Thouless Pump as Sliding Mode }\n\n\nIn this section, we review the CDW picture of the Rice-Mele (SSH) model\\cite{RiceMele,ssh1979}, and the interpretation of the Thouless pump\\cite{Thouless_pump_original_paper} as a CDW sliding mode. \nConsider the following Hamiltonian for a 1D chain\n\\begin{align}\n H_{\\text{Rice-Mele}} = \\sum_{n} & \\left( t + \\delta t (-1)^{n} \\cos{\\phi} \\right)c^{\\dagger}_{n+1}c_{n} + \\text{h.c.} \\nonumber\\\\\n & + \\sum_{n} (-1)^{n+1} \\Delta \\sin{\\phi} c^{\\dagger}_{n}c_{n}, \n \\label{eq:1DRiceMele}\n\\end{align}\nwhere $c^{\\dagger}_{n}$ is the creation operator for an electron at site $n$. \nThe nearest-neighbor hopping and on-site potential are modulated with periodicity $2$, and their relative strength is related to the phase $\\phi$ of the modulation. \nWe thus identify $\\phi$ as the phase of this CDW modulation. In this paper, we use the terms \"CDW sliding phase\" and \"phases of the mean-field CDW order parameter\" interchangeably to refer to $\\phi$. \nFor suitable choices of $t$, $\\delta t$ and $\\Delta$, the spectrum of this Hamiltonian is gapped for all $\\phi \\in [0,2\\pi)$. \nFocusing on the half-filled insulating ground state in this parameter regime, the occupied-band Wannier centers\\cite{Kohn59,Brouder2007,Marzari2012,NaturePaper,shockley1939surface} will be pumped by a length of one unit cell (two sites) as the phase $\\phi$ adiabatically slides from $0$ to $2\\pi$, leading to a quantized change of bulk polarization\\cite{xiao2010berry,Aris2014,RiceMele,ksv}. \nThis quantization has a topological origin: If we regard $\\phi$ as a crystal momentum along a second, synthetic dimension which we call $y$, Eq.~(\\ref{eq:1DRiceMele}) is equivalent to a 2D square lattice model with a $U(1)$ $\\pi$-flux (equivalent to half flux quantum $\\Phi_{0} = 2\\pi \\hbar \/ |e|$ where electron has charge $-|e|$) per plaquette, and with a fixed crystal momentum $k_{y}$ along $y$. \nThe quantized polarization change is then identified as the Chern number\\cite{tknn,niu1984quantised,niu1985quantized,Aris2014,bernevigbook} of the occupied bands in 2D.\nWe provide further details, including numerical verification of charge pumping, and the explicit construction of the dimensional promotion to 2D, in the SM\\cite{SM}.\n\n\nWe see from this example that promoting the dimension of a modulated system to a higher dimensional lattice coupled to gauge fields can help explain the topological origin of low-dimensional properties, including charge transport and boundary modes. \nA general method for dimensional promotion will thus be helpful in dealing with various topological modulated systems in more than 1D. \nIn what follows, we will show that the dimensional promotion approach can be extended to higher dimensions, and to cases where the modulation is incommensurate with the underlying lattice periodicity.\n\n\n\\section{\\label{sec_Dimension_promotion}Dimensional Promotion Procedure}\n\n\nIn this section, we will generalize the $1$D-to-$2$D dimensional promotion of the Rice-Mele chain to general dimensions. \nTo begin, let us consider a $d$-dimensional ($d$D) electronic model on a cubic lattice with $N$ mutually \nincommensurate on-site modulations\\cite{Earliest_dimension_promotion_superspace,Kraus_1D_QC_to_2D_QHE,equivalence_Fibo_Harper_2012,2D_QC_4D_QHE,time_periodic_1,time_periodic_2}, described by the Hamiltonian\n\\begin{equation}\n H_{\\text{low-dim}} = \\sum_{\\vec{n},\\vec{m}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m}} \\left[{H}_{\\vec{m}}\\right] {\\psi}_{\\vec{n}} + \\sum_{\\vec{n}} \\sum_{i=1}^{N}{\\psi}^{\\dagger}_{\\vec{n}} \\left[{V}^{(i)}_{\\vec{n}}\\right] {\\psi}_{\\vec{n}}.\\label{eq:h_low_dim}\n\\end{equation}\nHere both $\\vec{n} = (n_{1},\\cdots,n_{d})$ and $\\vec{m} = (m_{1},\\cdots,m_{d}) \\in \\mathbb{Z}^{d}$ are vectors in the $d$D cubic lattice, and ${\\psi}^{\\dagger}_{\\vec{n}}$ is the electron creation operator for an electron at position $\\vec{n}$ with a given set of spin and orbital degrees of freedom.\nWe denote by $\\left[{H}_{\\vec{m}}\\right]$ the hopping matrix connecting position $\\vec{n}$ to $\\vec{n}+\\vec{m}$, and by $\\left[{V}^{(i)}_{\\vec{n}}\\right]$ the matrix representing $i^{\\text{th}}$ modulated on-site energy at position $\\vec{n}$ ($i = 1 ,\\ldots, N$), with matrix indices encoding the spin and orbital dependence of the hopping\\footnote{throughout this work, we will use square brackets to denote matrices and matrix-valued functions}. \nNote that hermiticity of the Hamiltonian requires that $\\left[{H}_{\\vec{m}}\\right] = \\left[{H}_{-\\vec{m}} \\right]^\\dag$ {{and $\\left[{V}^{(i)}_{\\vec{n}}\\right]^{\\dagger}=\\left[{V}^{(i)}_{\\vec{n}}\\right]$}}. \nWe further assume that $\\left[{V}^{(i)}_{\\vec{n}}\\right] = \\left[f^{(i)}\\left(2\\pi\\vec{q}^{(i)}\\cdot\\vec{n} + \\phi^{(i)} \\right)\\right]$ with $\\left[f^{(i)}(x)\\right] = \\left[f^{(i)}(x+2\\pi)\\right]$, where $\\vec{q}^{(i)}$ is the $i^{\\text{th}}$ modulation wave vector and $\\phi^{(i)}$ is the sliding phase associated with the $i^{\\text{th}}$ modulation.\nFor the cubic system with unit lattice vectors we are discussing here, each component $q^{(i)}_{j}$, ($j = 1 ,\\ldots, d$) of $\\vec{q}^{(i)}$ is defined within $[0,1)$; that is, each $2\\pi \\vec{q}^{(i)}$ lies within the primitive Brillouin zone of the unmodulated system.\nSince each $\\left[{V}^{(i)}_{\\vec{n}}\\right]$ is a periodic function, they can be expanded in terms of Fourier series as\n\\begin{equation}\n \\left[{V}^{(i)}_{\\vec{n}}\\right] =\\sum_{p_{i}\\in \\mathbb{Z}} \\left[{V}^{(i)}_{p_{i}}\\right] e^{ip_{i}\\left( 2\\pi\\vec{q}^{(i)}\\cdot\\vec{n} + \\phi^{(i)} \\right)}, \\label{eq:expand_FT_V}\n\\end{equation}\nwhere $\\left[{V}^{(i)}_{p_{i}}\\right]$ is the matrix-valued ${p_{i}}^{\\text{th}}$ Fourier component of $\\left[{V}^{(i)}_{\\vec{n}} \\right]$. \nNote that $\\left[{V}^{(i)}_{p_{i}}\\right] = \\left[{V}^{(i)}_{-p_{i}} \\right]^{\\dagger}$ due to hermiticity of the Hamiltonian.\n\nTo perform the enhancement of dimensions, we first insert the expansion Eq.~($\\ref{eq:expand_FT_V}$) into the Hamiltonian Eq.~(\\ref{eq:h_low_dim}). \nWe then regard each $\\phi^{(i)}$ as the $i^{\\text{th}}$ crystal momentum $k^{i}$ along one of the additional $N$ synthetic dimensions. \nWe then promote the $d$D model to a $(d+N)$D space by summing over $\\vec{k} = (k^{1},\\cdots,k^{N}) \\in \\mathbb{T}^{N}$ (where $\\mathbb{T}^N$ denotes the $N$-dimensional torus), which yields the Hamiltonian in $(d+N)$D as\n\\begin{align}\n H_{\\text{high-dim}} &= \\sum_{\\vec{n},\\vec{m},\\vec{k}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{k}} \\left[{H}_{\\vec{m}}\\right] \\psi_{\\vec{n},\\vec{k}} \\nonumber \\\\\n & + \\sum_{\\vec{n},\\vec{k},i,p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{k}} \\left[{V}^{(i)}_{p_{i}}\\right] e^{ip_{i}k^{i}} e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\vec{n}}{\\psi}_{\\vec{n},\\vec{k}}. \\label{eq:d_n_Bloch}\n\\end{align}\nEach physically distinct configuration of $\\{\\phi^{(i)}\\}$ can be recovered by restricting the Hamiltonian Eq.~(\\ref{eq:d_n_Bloch}) to a single $\\vec{k}$-point. \nOnce we sum over $\\vec{k}$, however, we can reinterpret the Hamiltonian in a $(d+N)$D space. \nAs we will see below, adiabatic pumping of the phases $\\phi^{(i)}$ by an external field will allow us to explore dynamics in the full $d+N$ dimensional space. \n\nTo obtain the $(d+N)$D model in position-space, we perform an inverse Fourier transform of ${\\psi}^{\\dagger}_{\\vec{n},\\vec{k}}$, yielding\n\\begin{align}\n H_{\\text{high-dim}} &= \\sum_{\\vec{n},\\vec{m},\\vec{\\nu}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{\\nu}} \\left[{H}_{\\vec{m}}\\right] \\psi_{\\vec{n},\\vec{\\nu}} \\nonumber \\\\\n & + \\sum_{\\vec{n},\\vec{\\nu},i,p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}} \\left[{V}^{(i)}_{p_{i}}\\right] e^{i2\\pi p_{i}\\vec{q}^{(i)}\\cdot \\vec{n} }\\psi_{\\vec{n},\\vec{\\nu}}, \\label{eq:general_n_plus_d_model}\n\\end{align}\nwhere $\\vec{\\nu} = (\\nu_{1},\\cdots,\\nu_{N}) \\in \\mathbb{Z}^{N}$ and $\\hat{\\nu}_{i}$ is the unit vector along the $i^{\\text{th}}$ additional dimension, such that $\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}= (\\nu_{1},\\cdots,\\nu_{i}-p_{i},\\cdots,\\nu_{N})$. \nEq.~(\\ref{eq:general_n_plus_d_model}) can be viewed as the Hamiltonian for a system on a $(d+N)$D {{cubic lattice}} whose lattice sites are located at $(\\vec{n},\\vec{\\nu}) = (n_{1},\\cdots,n_{d},\\nu_{1},\\cdots,\\nu_{N}) \\in \\mathbb{Z}^{d+N}$. \nThe system is coupled to a continuous $U(1)$ gauge field\n\\begin{align}\n \\vec{A} = (\\underbrace{\\vec{0}}_\\text{$d$D},\\underbrace{2\\pi \\vec{q}^{(1)}\\cdot \\vec{r},\\cdots,2\\pi \\vec{q}^{(N)}\\cdot \\vec{r}}_\\text{$N$D}) \\label{eq:expression_A}\n\\end{align}\nthrough a Peierls substitution\\cite{Peierls_substitution}, explaining the appearance of the phase factors multiplying $\\left[{V}^{(i)}_{p_{i}}\\right]$ in Eq.~(\\ref{eq:general_n_plus_d_model}). \nNote $\\vec{r} \\in \\mathbb{R}^{d}$ is a vector in the original $d$D space.\n\n\nAs the vector potential in Eq.~(\\ref{eq:expression_A}) is linear in position $\\vec{r}$, the antisymmetric field strength $F_{\\mu \\nu} = \\partial_{\\mu}A_{\\nu} - \\partial_{\\nu}A_{\\mu}$ is constant in space. \nIn particular, Eq.~(\\ref{eq:expression_A}) implies that the nonzero components of $F_{\\mu \\nu}$ are given by\n\\begin{align}\n F_{i,j+d} = \\partial_{i}A_{j+d} - \\partial_{j+d}A_{i} = \\partial_{i}A_{j+d} = 2\\pi q^{(j)}_{i}, \\label{eq:F_i_j_plus_d}\n\\end{align}\nwhere $i = 1 ,\\ldots, d$ and $j = 1 ,\\ldots, N$. \nDue to the antisymmetry of the field strength, $F_{i+d,j}$ with $i = 1 ,\\ldots, N$, $j = 1 ,\\ldots, d$ is also nonzero and given by $F_{i+d,j} = -2\\pi q^{(i)}_{j}$. \nTherefore the (nonzero) constant field strength is proportional to the magnitude of the modulation wave vectors.\n\n\n\nThis shows that that a $d$D modulated system with phase offset $\\{\\phi^{(i)}\\}$ is equivalent to the Bloch Hamiltonian (see Eq.~(\\ref{eq:d_n_Bloch})) of the promoted $(d+N)$D lattice model with fixed crystal momenta $\\vec{k}$, once we identify $\\phi^{(i)}$ as $k^{i}$.\nIn practice, the modulation $\\left[{V}^{(i)}_{\\vec{n}}\\right]$ can be induced by a set of DW modulations. \nThe phase offset $\\{\\phi^{(i)}\\}$ is then regarded as the phason degrees of freedom, namely the phase of the $i^{\\text{th}}$ mean-field DW order parameter. \nBy applying electric fields that depin the DWs and make them slide\\cite{gooth2019evidence,Zakphase,gruner1988dynamics}, we may sample the whole spectrum of the $(d+N)$D model. \nIn particular, and as we will explore in subsequent sections, non-trivial topology in the $(d+N)$D lattice model--which may support localized boundary states--will manifest in the response of the $d$D model to adiabatic sliding of the DW phase mode(s). \n\\textcolor{black}{We emphasize here that in our dimensional promotion procedure for a DW system, there are no emergent electric fields in the promoted $(d+N)$D space. The electric fields mentioned here are external and serve as a way to depin the DW in order to vary $\\{\\phi^{(i)}\\}$ adiabatically. This allows for the sampling of the entire spectrum of the $(d+N)$D model as a function of $\\{\\phi^{(i)}\\}$, namely the additional crystal momenta.}\n\n\nBefore we move on to consider the band topology of promoted lattice models, let us make a few general comments about our dimensional promotion procedure. \nFirst, note that the dimensional promotion procedure places no constraints on the modulation vectors $\\vec{q}^{(i)}$; in particular, they need not be commensurate with the underlying lattice. \nIn the case of incommensurate modulation, the dimensional promotion procedure allows us to write the $d$D incommensurate model in terms of a periodic $(d+N)$D model with an irrational $U(1)$ flux per plaquette. \nWe will see below how we can use this to explore the topology of systems with incommensurate modulation. \n\\textcolor{black}{We emphasize that the dimensional promotion procedure is independent of whether in the original $d$D space the system is infinite or finite. When we promote the dimension of a $d$D system to $(d+N)$D space, the $(d+N)$D system is inherently infinite along the additional $N$ dimensions, as it allows a Fourier transformation to obtain the Bloch Hamiltonian with fixed $N$ additional crystal momenta. From this viewpoint there are two ways to utilize the dimensional promotion procedure. If we promote the dimension of an infinite $d$D system, we will obtain an infinite $(d+N)$D system that allows us to discuss the non-trivial bulk topology in the promoted $(d+N)$D space. If we instead promote the dimension of a finite $d$D system, we will obtain a $(d+N)$D system which is finite along the original $d$ dimensions and infinite in the additional $N$ dimensions. This allows us to compute the energy spectrum to examine whether there are boundary states protected by the non-trivial bulk topology in $(d+N)$D space.}\n\n\nSecond, although here we consider only dimensional promotion of a $d$D cubic lattice model with only on-site modulations \\textcolor{black}{and all orbitals located at the lattice points labelled by $\\vec{n} \\in \\mathbb{Z}^{d}$} to a $(d+N)$D cubic lattice model, we may generalize our method to $d$D models with modulations in both on-site and hopping matrix elements, together with non-orthogonal lattice vectors \\textcolor{black}{and arbitrary orbital positions}. \nWe show how to systematically promote the dimensions of such $d$D models to $(d+N)$D and compute the corresponding $U(1)$ gauge fields in the SM\\cite{SM}. \nWe also give several examples in the SM\\cite{SM}, including the dimensional promotion of: (1) the 1D Rice-Mele chain in Sec.~\\ref{sec_review_thouless_pump_Rice_Mele} to a 2D square lattice with $\\pi$-flux, (2) 1D lattices with modulation in both on-site energies and hopping terms to 2D hexagonal lattices under a perpendicular magnetic field, and (3) 2D modulated systems with hexagonal lattice to 3D systems also with hexagonal lattices coupled to a $U(1)$ gauge field. \nThe $U(1)$ gauge fields will take a slightly different form from Eq.~(\\ref{eq:expression_A}) when we consider a system with non-orthogonal lattice vectors. \nHowever, the vector potentials will still be linear in $\\vec{r} \\in \\mathbb{R}^{d+N}$, and hence will still produce constant field strengths $F_{\\mu \\nu}$. \nFurthermore, note that although we considered for simplicity models where the electrons were localized to the origin of each unit cell, this is not essential for the application of our formalism.\n\n\nThird, we emphasize that no additional parameters are used in the above derivation. \nThe hopping matrices connecting $(\\vec{n},\\vec{\\nu})$ to $(\\vec{n}+\\vec{m},\\vec{\\nu})$ and $(\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i})$ are given by $\\left[{H}_{\\vec{m}}\\right]$ and $\\left[{V}^{(i)}_{p_{i}}\\right]$, respectively, in the $(d+N)$D model. \nThe phase $\\phi^{(i)}$ corresponds to the $i^{\\text{th}}$ crystal momentum along the $i^{\\text{th}}$ additional dimensions. \nFurther, the modulation wave vectors $\\vec{q}^{(i)}$ specify the strength of the $U(1)$ gauge field in $(d+N)$D, see Eq.~(\\ref{eq:expression_A}). \nNotice that the on-site modulations $\\left[{V}^{(i)}_{\\vec{n}}\\right]$ only lead to hopping parallel to $\\hat{\\nu}_{i}$ in $(d+N)$D. \nIf we also consider modulated hopping matrices in $d$D, upon dimensional promotion we will get hopping along $\\vec{m} + p_{i}\\hat{\\nu}_{i}$ in $(d+N)$D\\cite{equivalence_Fibo_Harper_2012,Oded_4D_CI_to_2D_HOTI}, which we show in the SM\\cite{SM}. \nNotice that the index $i$ is not summed over in $p_{i}\\hat{\\nu}_{i}$. \nRecall also that $\\vec{m}$ and $\\hat{\\nu}_{i}$ are vectors in the original $d$D and additional $N$D space, respectively. \nAn example that demonstrates this is the 1D Rice-Mele model\\cite{RiceMele} in Sec.~\\ref{sec_review_thouless_pump_Rice_Mele}. \nIn the SM\\cite{SM} we promote Eq.~(\\ref{eq:1DRiceMele}) to a 2D lattice with $\\pi$-flux per plaquette in which the electrons can hop along $\\hat{x}+\\hat{y}$ (where $\\hat{x}$ and $\\hat{y}$ are in the original 1D and additional 1D space, respectively).\n \nNext, our construction provides a way to compute the promoted $(d+N)$D model and the $U(1)$ gauge field to which it is coupled. \nAs a $U(1)$ gauge field breaks time-reversal-symmetry (TRS), this dimensional promotion procedure is suitable to investigate non-trivial topological phases in $(d+N)$D space without TRS. \nBelow we will also consider a dimensional promotion to $(d+N)$D space with an $SU(2)$ gauge field, which preserves TRS and allows us to explore non-trivial topological phases protected by TRS\\cite{Kane04,bernevig2006quantum,ryu2010topological,KitaevClassify}.\nIn order to construct a low dimensional modulated model equivalent to a higher dimensional lattice coupled to an $SU(2)$ gauge field, we adopt a top-down approach. \nWe will in Sec.~\\ref{sec_helical_HOTI_sliding_modes} present a 2D modulated model which is obtained from a 3D model coupled to one $SU(2)$ gauge field with a fixed crystal momentum.\n\n\nIn the following sections, we explore various 2D and 3D modulated systems that admit a dimensional promotion to a higher dimensional topological phases coupled to either $U(1)$ or $SU(2)$ gauge fields. \nWe will show how an analysis of the higher-dimensional models can shed light on the eigenstates and boundary state dynamics of incommensurate DWs.\n\n\n\n\\section{\\label{sec_chiral_HOTI}Chiral Higher-Order Topological Sliding Modes}\nIn this section, we will show how the dimensional promotion procedure can be used to realize 3D chiral HOTIs in 2D density wave (DW) materials. \nWe will first construct a Hamiltonian for an insulating 2D modulated system that is inversion-symmetric for special values of the DW sliding phase $\\phi$. \nThen, we will show how, after dimensional promotion, the Hamiltonian corresponds to a 3D inversion-symmetric chiral HOTI coupled to a $U(1)$ gauge field. \nWe will explore the connection between hinge states of the 3D system and corner states of the 2D system using a combination of numerical diagonalization and a 3D low-energy $\\vec{k}\\cdot\\vec{p}$ theory.\n\n\\subsection{Dimensionally Promoted Chiral Model}\n\nConsider the following 2D Hamiltonian for electrons on a square-lattice, with one modulated on-site potential $[V(\\vec{q},\\vec{n},\\phi)]$:\n\\begin{equation}\n\\begin{aligned}\nH = {} & \\sum_{\\vec{n}} \\psi^{\\dagger}_{\\vec{n}+\\hat{x}} [H_{+\\hat{x}}]\\psi_{\\vec{n}} + \\psi^{\\dagger}_{\\vec{n}+\\hat{y}} [H_{+\\hat{y}}]\\psi_{\\vec{n}} + \\text{h.c.}\\\\\n&+ \\sum_{\\vec{n}} \\psi^{\\dagger}_{\\vec{n}}\\left( [H_{\\text{on-site}}] + [V(\\vec{q},\\vec{n},\\phi)] \\right)\\psi_{\\vec{n}},\n\\end{aligned}\\label{chiral_modulated_2}\n\\end{equation}\nwhere the unmodulated hoppings and on-site energies are\n\\begin{align}\n & [H_{+\\hat{e}_{i}}] = \\frac{J_{i}}{2}\\tau_{z}\\sigma_{0} - \\frac{\\lambda_{i}}{2i}\\tau_{x}\\sigma_{i}, \\label{eq:chiral_xy_hopping}\\\\\n & [H_{\\text{on-site}}] = M \\tau_{z}\\sigma_{0}+\\tau_{0} \\vec{B_0} \\cdot \\vec{\\sigma} . \\label{eq:chiral_on_site}\n\\end{align}\nWe use $\\hat{e}_{i}$ to denote the unit vector along the $i^{\\text{th}}$ ($i = 1, 2$) direction. \nThe Pauli matrices $\\vec{\\tau} = (\\tau_{x},\\tau_{y},\\tau_{z})$ and $\\vec{\\sigma} = (\\sigma_{x},\\sigma_{y},\\sigma_{z})$ denote respectively orbital (for example $s$ and $p$ orbitals) and spin degrees of freedom. \nThis Hamiltonian is inversion-symmetric, with inversion symmetry represented by $\\tau_z$. \nFurthermore, when $\\vec{B}_0=0$ the model is also time-reversal (TR) symmetric, with the TR operator represented as $i\\sigma_y\\mathcal{K}$ (where $\\mathcal{K}$ is the complex conjugation operator).\n\nWe assume that both orbital degrees of freedom are located at the lattice sites.\nThe hopping matrices $[H_{+\\hat{e}_{i}}]$, and $M \\tau_{z}\\sigma_{0}$ give rise to, at low energy, four-component massive Dirac fermions, allowing us to access various topological phases\\cite{ryu2010topological,haldanemodel,bernevigbook}.\nPhysically, we can interpret $M \\tau_{z}\\sigma_{0}$ as the on-site energy difference for different orbitals, and $\\tau_{0}\\vec{B_0} \\cdot \\vec{\\sigma}$ as a ferromagnetic potential which splits the spin degeneracy of bands\\cite{wieder2018axion}. \nThe modulated on-site potential, which can arise from a density wave modulation, is\n\\begin{align}\n [V(\\vec{q},\\vec{n},\\phi)] =& J_{z}\\cos\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{z}\\sigma_{0} + \\lambda_{z}\\sin\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{x}\\sigma_{z}, \\label{eq:chiral_modulated_V}\n\\end{align}\nwhere $\\theta_{\\vec{q},\\vec{n},\\phi} = 2\\pi \\vec{q}\\cdot \\vec{n} + \\phi$, $\\vec{q} = (q_{x},q_{y})$ is the modulation wave vector in 2D, $\\vec{n} \\in \\mathbb{Z}^{2}$ is the lattice position, and $\\phi$ is the sliding phase. \nThe first term in Eq.~(\\ref{eq:chiral_modulated_V}) modulates the mass $M \\tau_{z}\\sigma_{0}$ in Eq.~(\\ref{eq:chiral_on_site}), while the second modulation denotes an on-site spin-orbit coupling between $s$ and $p$ orbitals. \nNote that the modulation $J_{z}\\cos\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{z}\\sigma_{0}$ is a TR-even charge ordering, while $\\lambda_{z}\\sin\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{x}\\sigma_{z}$ is a TR-odd spin ordering. \nTo see this, note that TR maps $(\\tau_{0},\\tau_{x},\\tau_{y},\\tau_{z}) \\to (\\tau_{0},\\tau_{x},-\\tau_{y},\\tau_{z})$ and $(\\sigma_{0},\\sigma_{x},\\sigma_{y},\\sigma_{z}) \\to (\\sigma_{0},-\\sigma_{x},-\\sigma_{y},-\\sigma_{z})$. \nIn addition, the modulations $J_{z}\\cos\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{z}\\sigma_{0}$ and $\\lambda_{z}\\sin\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{x}\\sigma_{z}$ are both inversion-symmetric when $\\phi=0$, $\\pi$.\n\n\nDenoting the third, synthetic dimension as $z$ and identifying $\\phi$ as the corresponding crystal momentum $k_{z}$, we may use our general procedure in Sec.~\\ref{sec_Dimension_promotion} to promote this 2D modulated system to a 3D lattice model. \nWe first expand the modulations in terms of Fourier series as\n\\begin{align}\n & J_{z}\\cos\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{z}\\sigma_{0} = \\frac{J_{z}}{2}\\left(e^{i\\theta_{\\vec{q},\\vec{n},\\phi}} + e^{-i\\theta_{\\vec{q},\\vec{n},\\phi}} \\right) \\tau_{z}\\sigma_{0}, \\label{eq:J_expand} \\\\\n & \\lambda_{z}\\sin\\theta_{\\vec{q},\\vec{n},\\phi}\\tau_{x}\\sigma_{z} = \\frac{\\lambda_{z}}{2i}\\left( e^{i\\theta_{\\vec{q},\\vec{n},\\phi}} - e^{-i \\theta_{\\vec{q},\\vec{n},\\phi}} \\right)\\tau_{x}\\sigma_{z}. \\label{eq:lambda_z_expand}\n\\end{align}\nAccording to Eqs.~(\\ref{eq:expand_FT_V}) and (\\ref{eq:general_n_plus_d_model}), the hopping along $+\\hat{z}$ can be identified with the terms associated with $e^{-i \\theta_{\\vec{q},\\vec{n},\\phi}}$ in Eqs.~(\\ref{eq:J_expand}) (\\ref{eq:lambda_z_expand}). \nTherefore, the hopping along $+\\hat{z}$ in the promoted 3D space reads\n\\begin{equation}\n[H_{+\\hat{z}}]=\\frac{J_{z}}{2}\\tau_{z}\\sigma_{0} -\\frac{\\lambda_{z}}{2i}\\tau_{x}\\sigma_{z}.\n\\end{equation} \nFrom Eq.~(\\ref{eq:expression_A}) we can also identify the vector potential in the promoted 3D space as\n\\begin{align}\n \\vec{A} = (0,0,2\\pi \\vec{q} \\cdot \\vec{r}) = (0,0,2\\pi q_{x} x + 2\\pi q_{y}y), \\label{A_U1}\n\\end{align}\nwhere $\\vec{r} = (x,y) \\in \\mathbb{R}^{2}$. \nTherefore, we have that the lattice Hamiltonian in the promoted 3D space is given by\n\\begin{widetext}\n\\begin{align}\n H = \\sum_{\\vec{n}} & \\left[ \\left( \\psi^{\\dagger}_{\\vec{n}+\\hat{x}} [H_{+\\hat{x}}]\\psi_{\\vec{n}} +\\psi^{\\dagger}_{\\vec{n}+\\hat{y}} [H_{+\\hat{y}}]\\psi_{\\vec{n}} + {\\psi}^{\\dagger}_{\\vec{n}+\\hat{z}} [H_{+\\hat{z}}] e^{-i2\\pi (q_{x}n_{x}+q_{y}n_{y})} {\\psi}_{\\vec{n}} + \\text{h.c.} \\right)+ \\psi^{\\dagger}_{\\vec{n}} [H_{\\text{on-site}}] \\psi_{\\vec{n}} \\right], \\label{eq:lattice_model_chiral_sliding}\n\\end{align}\n\\end{widetext}\nwhere the vector potential Eq.~(\\ref{eq:expression_A}) is coupled to the system through a Peierls substitution\\cite{Peierls_substitution}, and\n$[H_{+\\hat{x}}]$, $[H_{+\\hat{y}}]$, $[H_{+\\hat{z}}]$ and $[H_{\\text{on-site}}]$ are given by Eqs.~(\\ref{eq:chiral_xy_hopping}) and (\\ref{eq:chiral_on_site}), respectively. \nHereafter, we will set $J_{x} = J_{y} = J_{z} = J$ for simplicity. \nIf we Fourier transform Eq.~(\\ref{eq:lattice_model_chiral_sliding}) along $z$ and regard $k_{z}$ (the wavenumber along $z$) as the sliding phase $\\phi$, we can obtain the 2D modulated system in Eq.~(\\ref{chiral_modulated_2}). \n\nWe will now use Eq.~(\\ref{eq:lattice_model_chiral_sliding}) to analyze the topological properties of the higher-dimensional model, in order to infer the properties of the low-dimensional modulated system.\nThis approach can also be employed in other low-dimensional modulated systems provided the corresponding higher-dimensional models are constructed. \nFor $q_{x}=q_{y}=0$ and $\\vec{B_0}=0$, Eq.~(\\ref{eq:lattice_model_chiral_sliding}) describes a TR and inversion-symmetric insulator whose inversion operation is represented by $\\tau_{z}$ (note that inversion symmetry acts to flip the sign on the synthetic momentum $k_z$). \nWe can employ the theory of symmetry-based indicators of band topology~\\cite{khalaf,Po2017,song2017,xu2020high,MTQC,Wieder_spin_decoupled_helical_HOTI,dynamical_axion_insulator_BB,Kruthoff2016} to compute the $\\mathbb{Z}_4$ indicator\n\\begin{align}\n z_{4} = \\frac{1}{4}\\sum_{\\vec{k}_{a} \\in \\text{TRIMs}} \\left( n^{a}_{+} - n^{a}_{-} \\right) \\text{ mod }4,\n\\end{align}\nwhere $n^{a}_{+}$[$n^{a}_{-}$] is the number of positive[negative] parity eigenvalues in the valence band at the time-reversal invariant momentum (TRIM) $\\vec{k}_{a} $. \nWe find that for $|M\/J|>3$, $|M\/J|<1$, $13$, $|M\/J|<1$, $1<|M\/J| < 3$ we have $\\tilde{z}_{4} = 0$, $0$ and $2$. \nThe corresponding weak indices are all necessarily trivial.\nTherefore, for $1 < |M\/J| <3$ with $q_{x}=q_{y}=0$, the system described by Eq.~(\\ref{eq:lattice_model_chiral_sliding}) gives a strong TI with $\\vec{B_0}=0$ and a chiral HOTI (axion insulator)\\cite{MTQC} with $\\vec{B_0}\\ne0$, where the gapless surface states of the strong TI are gapped by the inversion-preserving ferromagnetic potential $\\tau_{0} \\vec{B_0} \\cdot \\vec{\\sigma}$. \nTherefore, Eq.~(\\ref{eq:lattice_model_chiral_sliding}) with $\\vec{q} \\ne 0$ describes an inversion-symmetric chiral HOTI\\cite{pozo2019quantization} coupled via a Peierls substitution to a 3D $U(1)$ gauge field given by the $\\vec{A}$ in Eq.~(\\ref{eq:expression_A}).\nThis $\\vec{A}$ produces a constant $U(1)$ magnetic field \n\\begin{align}\n \\nabla \\cross \\vec{A} = (2\\pi q_{y},-2\\pi q_{x},0), \\label{B_U1}\n\\end{align} \nwhich preserves the inversion symmetry represented by $\\tau_{z}$ in 3D, up to a gauge transformation (see SM\\cite{SM}). \nTherefore, for a suitable choice of parameters, as long as the $U(1)$ gauge field does not close the bulk gap in 3D, the insulating ground state will be in the same inversion symmetry-protected non-trivial chiral HOTI phase. \nThis implies that our model should exhibit the characteristic boundary modes of a chiral HOTI in 3D. \nIn particular, our promoted model will support odd numbers of sample-encircling chiral hinge modes in rod geometries which respect inversion symmetry~\\cite{pozo2019quantization,wieder2018axion,dynamical_axion_insulator_BB}.\n\n\\begin{figure}[t]\n\\hspace{-0.5cm}\n\\includegraphics[width=\\columnwidth]{fig_main_text_chiral_sliding_change_ticking_freq.pdf}\n\\caption{(a) $\\phi$-sliding spectrum of the chiral 2D model in Eq.~(\\ref{chiral_modulated_2}) with parameters given in the text. \n(b) Probability distribution of corner modes in the gap-crossing bands at $\\phi = 0.5\\pi$ and $E=-0.1368$. \n(c) $\\&$ (d) Probability distribution of edge and bulk modes at $\\phi = 0.9\\pi$ and $E = -0.2508$ and $E = 0.278$, respectively. \nThe darker (black) color in (b)--(d) implies higher probability density. \\textcolor{black}{(b), (c) and (d) correspond to the corner mode, edge-confined mode and bulk-confined mode discussed in Sec.~\\ref{sec:chiral_sliding_corner_modes}, \\ref{sec:chiral_sliding_confined_edge_modes} and \\ref{sec:chiral_sliding_confined_bulk_modes}, respectively.}\nIn (b), (c) and (d), the $x$- and $y$-coordinate both range from $-15 ,\\ldots, +15$.}\n\\label{fig:chiral_sliding_main_text}\n\\end{figure}\n\n\\subsection{\\label{sec:chiral_sliding_corner_modes}Corner states}\n\nRecalling that in our case the $\\hat{z}$ direction is conjugate to the phase $\\phi$ of the sliding mode (regarded as the crystal momentum $k_z$), it is natural for us to consider inversion-symmetric rod geometries which are finite in the $\\hat{x}$ and $\\hat{y}$ directions, and infinite in the $\\hat{z}$ direction.\nIn our 2D system, this corresponds to considering the properties of a finite system as a function of the phase $\\phi$.\nWe can thus compute the energy spectrum of our 2D system in an open geometry with size $L_{x} \\times L_{y}$ as a function of $\\phi$ to obtain the energy dispersion along $k_{z}$ in the promoted model. \nIn the following, we call this kind of calculation the {\\it $\\phi$-sliding spectrum}, since the variation of $\\phi$ can be obtained by electromagnetically exciting the sliding mode of the underlying DW.\nFig.~\\ref{fig:chiral_sliding_main_text} (a) shows the $\\phi$-sliding spectrum of Eq.~(\\ref{chiral_modulated_2}) with parameters $J = 1$, $M =2 $, $\\lambda_{i}= 1$, $(\\vec{B}_{0})_{i}= 0.5\/\\sqrt{3}$\\cite{pozo2019quantization}, and $\\vec{q} = (0,q_{y})$, where $q_{y} = 0.11957$ is comparable with the experimental CDW wave vectors in (TaSe$_4$)$_2$I\\cite{shi2019charge} and is incommensurate with the underlying 2D square lattice in Eq.~(\\ref{chiral_modulated_2}).\nThe system size is $31 \\times 31$. \nAs we can see the spectrum contains modes which, as a function of $\\phi$, traverse the bulk spectral gap. \nExamining the wave functions of these ``gap-crossing modes,'' we see that they are localized to the corners of our $2$D sample, as shown in Fig.~\\ref{fig:chiral_sliding_main_text} (b). \nThe gap-crossing modes with opposite slopes correspond to states at inversion-related corners; in our example one mode is localized at the corner $(x_{\\text{corner}},y_{\\text{corner}})=(L\/2,-L\/2)$ (Fig.~\\ref{fig:chiral_sliding_main_text} (b)) and the other at $(x_{\\text{corner}},y_{\\text{corner}})=(-L\/2,L\/2)$ where $L = 30$.\nIf we start in a half-filled insulating ground state (with Fermi level $E_{F}=0$), then as $\\phi$ slides from $0$ to $2\\pi$, we realize charge pumping as one corner mode merges into the occupied-state subspace while the inversion-related counterpart flows into the unoccupied state subspace. \nThe ground states at the two inversion-symmetric values $\\phi=0,\\pi$ differ in electron number by $1$, demonstrating a ''filling anomaly''\\cite{benalcazar2018quantization,wieder2020strong}. \\tabularnewline\nBecause these corner modes originate as hinge modes in the $3$D dimensionally promoted system (where, recall, $\\phi$ is the momentum $k_z$), their existence is mandated by the non-trivial higher-order topology of the model Eq.~(\\ref{eq:lattice_model_chiral_sliding}).\n\nBy analyzing the low energy theory of the 3D hinge modes, we will now derive the dynamics of the 2D corner modes as a function of $\\phi$. \nIn 3D, the corresponding low energy 1D hinge Hamiltonian\\cite{hasan2010colloquium,khalaf,hotis} with a chiral mode as a function of $k_{z}$ is given by\n\\begin{align}\n H_{\\text{hinge}} = \\xi v_{F} \\left( k_{z} + 2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{y}y_{\\text{hinge}} \\right) \\right). \\label{eq:chiral_low_energy_hinge_H_1}\n\\end{align}\nWe have assumed that for the hinge along $z$ at position $(x_{\\text{hinge}},y_{\\text{hinge}})$ there is only one chiral mode with Fermi velocity $\\xi v_{F}$ where $v_{F} > 0$. \nWe have introduced $\\xi = \\pm 1$ to denote whether the chiral mode has positive or negative velocity. \nFollowing our dimensional promotion procedure, Eq.~(\\ref{eq:chiral_low_energy_hinge_H_1}) is then minimally coupled to a $U(1)$ gauge field in Eq.~(\\ref{A_U1}) through the Peierls substitution $k_{z} \\to k_{z} + 2\\pi (q_{x}x+q_{y}y)$, where $x = x_{\\text{hinge}}$ and $y = y_{\\text{hinge}}$ are fixed. \n\nTo map Eq.~(\\ref{eq:chiral_low_energy_hinge_H_1}) in 3D to the corner mode dispersion in 2D, it is helpful to first compute the $\\phi$-sliding spectrum for Eq.~(\\ref{chiral_modulated_2}) with $\\vec{q} = (0,0)$, as shown in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (a). \nIf we identify $\\phi$ as $k_{z}$ in the hinge theory (modulo a constant offset that we will fix later), Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (a) is the $\\hat{z}$-directed rod band structure for Eq.~(\\ref{eq:lattice_model_chiral_sliding}) without coupling to any vector potential. \nAs we can see, there are linear dispersing hinge modes spanning the bulk gap, which cross each other at $k_{z} = \\pi$. \nThis will be used below in Eq.~(\\ref{eq:low_E_corner_chiral}) to complete the mapping from Eq.~(\\ref{eq:chiral_low_energy_hinge_H_1}) to 2D. \nFig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (a) will also serve as a reference calculation when we examine the response of the $\\phi$-sliding spectrum as we increase the magnitude of $\\vec{q}$, which will confirm our low energy analysis.\n\n\nWe now use Eq.~(\\ref{eq:chiral_low_energy_hinge_H_1}) to construct a low energy description of the corner modes in Fig.~\\ref{fig:chiral_sliding_main_text} (a) for Eq.~(\\ref{chiral_modulated_2}). \nUpon projecting from 3D to 2D, the fixed hinge mode position $(x_{\\text{hinge}},y_{\\text{hinge}})$ becomes the fixed corner mode position $(x_{\\text{corner}},y_{\\text{corner}})$, and the hinge modes become corner modes. \nSince the gap-crossing modes in the $q=0$ system shown in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (a) intersect at $\\phi = \\pi$, we replace $k_{z}$ in the hinge theory by $\\Delta \\phi = \\phi - \\pi$. \nThus, we obtain an effective low energy description of the corner modes as\n\\begin{align}\n H_{\\text{corner}} = \\xi v_{F} \\cdot \\left(\\Delta \\phi+2\\pi \\left(q_{x}x_{\\text{corner}}+q_{y}y_{\\text{corner}}\\right)\\right).\n \\label{eq:low_E_corner_chiral} \n\\end{align}\nWe now verify Eq.~(\\ref{eq:low_E_corner_chiral}) by numerically computing the $\\phi$-sliding spectrum shown in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (b) with same parameters as Fig.~\\ref{fig:chiral_sliding_main_text} (a) but with $q_{y}$ changed to $0.02$. This small value of $q_y$ gives a smooth modulation--and hence a low flux per plaquette in the dimensionally-promoted model--and is thus a suitable platform to examine the low energy theory with minimal coupling. \nWe observe gap-crossing modes with negative and positive slopes corresponding to corner modes at $(-L\/2,L\/2)$ and $(L\/2,-L\/2)$ where $L =30$, respectively. \nThese are shown in Figs.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (c) and (d) at $\\phi = 0.4\\pi$ and $1.6\\pi$, respectively. \nUsing Eq.~(\\ref{eq:low_E_corner_chiral}), we have the low energy descriptions for these two corner modes governed by the Hamiltonians\n\\begin{align}\n & H_{\\text{corner 1}} = -v_{F}\\left( \\Delta \\phi + \\pi q_{y}L \\right), \\label{eq:eff_corner_chiral_1} \\\\\n & H_{\\text{corner 2}} = +v_{F}\\left( \\Delta \\phi - \\pi q_{y}L \\right), \\label{eq:eff_corner_chiral_2}\n\\end{align}\nwhere we have used $q_{x} = 0$. \nThus, if we ramp up $q_{y}$ from $0$ to some non-zero value, we expect to see the corner mode dispersion shift along the $\\phi$-axis. \nThis is demonstrated in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (b), which is to be compared with Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (a). \nIn fact, a careful examination of Figs.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (a) and (b) shows that the dispersions of the two corner modes shift in opposite directions as a function of $\\Delta\\phi$, as indicated in Eq.~(\\ref{eq:eff_corner_chiral_1}) and Eq.~(\\ref{eq:eff_corner_chiral_2}), with the shift given by $\\pi q_{y}L \\approx 0.6 \\pi$ for $q_{y} = 0.02$ and $L = 30$. \nWe thus see that the corner mode dispersion in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (b) can be explained by Eq.~(\\ref{eq:low_E_corner_chiral}). \nThis demonstrates the origin of the corner modes in the 2D modulated system as higher dimensional hinge modes minimally coupled to a $U(1)$ gauge field. \nIf we consider larger $q_{y}$, such as in Fig.~\\ref{fig:chiral_sliding_main_text} where we have $q_{y} = 0.11957$, then the shift of the corner mode dispersion is predicted to be $\\pi q_{y} L \\approx 3.5871 \\pi$. This lies outside the first Brillouin zone and needs to be folded back into the range $\\phi = [0,2\\pi)$. \nThis occurs because, in passing from low energy continuum theory to a lattice model, the periodicity of $\\phi$--which in the promoted dimension is the continuous wavenumber $k_{z}$--is restored.\nAdditionally, note that Eq.~(\\ref{eq:low_E_corner_chiral}) implies that we may tune the range of $\\phi$ where the corner mode energies emerge from the bulk continuum by varying the periodicity of the modulation $\\sim 1\/|\\vec{q}|$. \nAs shown in Eq.~(\\ref{A_U1}) and Eq.~(\\ref{B_U1}), tuning $\\vec{q}$ is equivalent to changing the direction and strength of the $U(1)$ gauge field and the corresponding magnetic field in 3D.\n\n\n\\subsection{\\label{sec:chiral_sliding_confined_edge_modes}Edge states}\n Having accounted for the low energy description of the corner modes, we observe that in Fig.~\\ref{fig:chiral_sliding_main_text} (a), there are additional modes with {\\it flat dispersion}. \n These non-dispersing modes describe states confined either to the bulk or edge of the system, as shown in Figs.~\\ref{fig:chiral_sliding_main_text} (c) and (d). \n We now use low energy theories to demonstrate that these states originate from the $U(1)$ Landau quantization of the surface and bulk electrons in the promoted 3D chiral HOTI. \n We will revisit Figs.~\\ref{fig:chiral_sliding_main_text} (c) and (d) after we complete the low energy theory analysis using relatively small $q_{y}$.\n\n\n\nWe start with the edge-confined modes. \nSince a chiral HOTI can be obtained by gapping out the surface of a 3D inversion and TR-symmetric TI with a TR-breaking mass term, the generic surface theory reads\\cite{khalaf,wieder2018axion,vanderbiltaxion}\n\\begin{align}\n H_{\\text{surf}} = \\left( \\vec{p} \\times \\vec{\\sigma}' \\right)\\cdot \\hat{n} + m \\hat{n} \\cdot \\vec{\\sigma}',\n\\end{align}\nwhere $\\vec{\\sigma}'$ are Pauli matrices that act in the basis of low-energy surface states and which capture their spin and orbital texture, $\\vec{p}$ is the momentum operator, and $\\hat{n}$ is the surface normal vector. \nThe time-reversal operator in this surface theory is given by $\\mathcal{T} = i\\sigma_{y}'\\mathcal{K}$ such that $\\mathcal{T} \\vec{\\sigma}' \\mathcal{T}^{-1} = -\\vec{\\sigma}'$. \nThe momentum dependent term $\\left( \\vec{p} \\times \\vec{\\sigma}' \\right)\\cdot \\hat{n}$ describes a helical surface Dirac cone, while $m \\hat{n} \\cdot \\vec{\\sigma}'$ is the TR-breaking mass term. \nAs shown in Eq.~(\\ref{B_U1}), if $q_{x} = 0$, which is the case we consider in Fig.~\\ref{fig:chiral_sliding_main_text} and Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp}, we have that $\\nabla \\cross \\vec{A}$ is parallel to $\\hat{x}$. \nWe then consider a surface theory on the $yz$-plane coupled to a perpendicular magnetic field $B \\hat{x}$ generated by a Landau-gauge $U(1)$ gauge field $\\vec{A} = (0,0,By)$.\nThe corresponding surface Hamiltonian with the $U(1)$ gauge field reads ${{H_{\\text{surf}} = p_{y}\\sigma_{z}' - \\left(p_{z}+By \\right)\\sigma_{y}' + m\\sigma_{x}',}}$ \nwhere we have made a Peierls substitution such that $p_{z} \\to p_{z} + By$, and we have assumed that both $B$ and $m$ are positive. \nTo facilitate the derivation, we perform a basis transformation through a $-2\\pi\/3$ radian spin rotation $U$ along the $[1,1,1]$ axis such that $U^{\\dagger} (\\sigma_{x}',\\sigma_{y}',\\sigma_{z}') U = (\\sigma_{z}',\\sigma_{x}',\\sigma_{y}')$. \nThe transformed Hamiltonian then reads\n\\begin{align}\n H_{\\text{surf}} = p_{y}\\sigma_{y}' - \\left(p_{z}+By \\right)\\sigma_{x}' + m\\sigma_{z}'. \\label{eq:surf_H_chiral_HOTI_transformed}\n\\end{align}\nFourier transforming Eq.~(\\ref{eq:surf_H_chiral_HOTI_transformed}) to replace {{$p_{z}$}} by the wavenumber $k_{z}$, and defining a $k_{z}$-dependent ladder operator\n\\begin{align}\n & {{a^{\\dagger}_{k_{z}} = \\frac{1}{\\sqrt{2B}}\\left(\\left(k_{z}+By \\right)-ip_{y} \\right),}} \\label{eq:U1_ladder}\n\\end{align}\nwhere $[a_{k_{z}},a^{\\dagger}_{k_{z}}]=1$, we can rewrite Eq.~(\\ref{eq:surf_H_chiral_HOTI_transformed}) as\n\\begin{align}\n {{H_{\\text{surf}}(k_{z}) = \\begin{bmatrix}\n m & -\\sqrt{2B} a_{k_{z}} \\\\\n -\\sqrt{2B} a^{\\dagger}_{k_{z}} & -m\n \\end{bmatrix}.}}\n \\label{eq:chiral_reexpress_H_surf}\n\\end{align}\nWe can solve for the eigenstates and energy eigenvalues of Eq.~(\\ref{eq:chiral_reexpress_H_surf}) to find\n\\begin{widetext}\n\\begin{align}\n & {{\\psi^{-}_{k_{z},n=0} = e^{ik_{z}z} \\begin{bmatrix}\n 0 \\\\ \\ket{0,k_{z}}\n \\end{bmatrix},\\ E^{-}_{k_{z},n=0} = -m,}} \\nonumber \\\\\n & {{\\psi^{-}_{k_{z},n>0} = e^{ik_{z}z} \\begin{bmatrix}\n \\alpha_{-}(n) \\ket{n-1,k_{z}}\\\\ \\ket{n,k_{z}}\n \\end{bmatrix},\\ E^{-}_{k_{z},n>0} = - \\sqrt{m^{2} + 2Bn},}} \\nonumber \\\\\n & {{\\psi^{+}_{k_{z},n>0} = e^{ik_{z}z} \\begin{bmatrix}\n \\alpha_{+}(n) \\ket{n-1,k_{z}}\\\\ \\ket{n,k_{z}}\n \\end{bmatrix},\\ E^{+}_{k_{z},n>0} = + \\sqrt{m^{2} + 2Bn},}} \\nonumber \\\\\n & {{\\text{where } \\alpha_{\\pm}(n) = \\frac{-1}{\\sqrt{2Bn}}\\left(\\pm\\sqrt{m^{2}+2Bn}+m \\right).}} \\label{eq:U1_surface_EE_EV}\n\\end{align}\n\\end{widetext}\nHere $n$ is a non-negative integer labelling the $U(1)$ Landau levels (LLs), and $\\ket{n,k_{z}}$ is the $n^{\\text{th}}$ simple harmonic oscillator (SHO) eigenstate localized along $y$ defined by the $a^{\\dagger}_{k_{z}}$ in Eq.~(\\ref{eq:U1_ladder}). \nNotice that the energies $E^{-}_{k_{z},n=0}$, $E^{-}_{k_{z},n>0}$ and $E^{+}_{k_{z},n>0}$ of these LLs shown in Eq.~(\\ref{eq:U1_surface_EE_EV}) are all independent of {{$k_{z}$}}. \nAs before, we now construct the low energy description of the edge-confined modes in the 2D modulated system from the above low energy surface theory in Eq.~(\\ref{eq:surf_H_chiral_HOTI_transformed}). \nWe identify $k_{z}$ in the surface theory as $\\Delta \\phi = \\phi - \\pi$, since we have flat bands as a function of $\\phi$ in our 2D modulated system. \nWe also identify $B$ with $2\\pi q_{y}$ since in our examples of Fig.~\\ref{fig:chiral_sliding_main_text} (a) and Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (b), we have $q_{x} = 0$ and the corresponding vector potential is $\\vec{A} = (0,0,2\\pi q_{y}y)$. \nWhen we project down to the 2D model, the surface electrons correspond to states confined in the left and right edges, as shown in Fig.~\\ref{fig:chiral_sliding_main_text} (c) and Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (e)--(h). \nWe again use $q_{y} = 0.02$ to demonstrate the low energy theory.\n\n\n\n\\begin{figure*}[t]\n\\hspace{-0.5cm}\n\\includegraphics[width=\\linewidth]{main_text_chiral_sliding_low_energy_theory.pdf}\n\\caption{(a) $\\&$ (b) $\\phi$-sliding spectrum of the chiral 2D model with the same parameters as Fig.~\\ref{fig:chiral_sliding_main_text} (a) but with $q_{y} = 0$ and $0.02$, respectively. \n(c) $\\&$ (d) Probability distribution of the corner modes in (b) at $\\phi = 0.4 \\pi$ and $1.6 \\pi$ and both with $E = 0.0365$. \n(e)--(g) Average of the probability distribution for the doubly degenerate edge-confined modes in the flat bands in (b) at $\\phi=\\pi$. \nThe double degeneracy is due to the pair of opposite edges related by inversion symmetry. \n(e)--(g) are edge-confined modes at $\\phi = \\pi$ with energies $E=-0.4705$, $-0.2235$ and $0.3811$, which are marked orange, green and red respectively in (b). \nThe corresponding energy eigenvalues are $E^{-}_{k_{z}=0,n=1}$, $E^{-}_{k_{z}=0,n=0}$ and $E^{+}_{k_{z}=0,n=1}$ in Eq.~(\\ref{eq:U1_surface_EE_EV}). \n(h) Edge-confined mode at $\\phi = 0.9 \\pi$ with energy $E = -0.2235$ corresponding to $E^{-}_{k_{z}=-0.1\\pi,n=0}$ in Eq.~(\\ref{eq:U1_surface_EE_EV}). \nThe darker (black) color in (c)--(h) implies higher probability density. \nThe inset in (e)--(h) is the probability distribution integrated over all $x$ coordinates. \nIn (c)--(h), the $x$- and $y$-coordinate both range from $-15 ,\\ldots, +15$.}\n\\label{fig:chiral_sliding_low_energy_main_text_temp}\n\\end{figure*}\n\n\n\nWe now remark on the implications of our low-energy analysis. \nFirst, the spectrum in Eq.~(\\ref{eq:U1_surface_EE_EV}) breaks particle-hole symmetry as there is a $-m$ energy eigenvalue but no $+m$ energy eigenvalue. \nThis can be observed in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (b), where there are no flat bands of edge-confined modes around $E \\approx +0.2$, which corresponds to $E = +m$. \nWe thus identify the flat bands in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (b) marked by red, green and orange as $E^{+}_{k_{z},n=1}$, $E^{-}_{k_{z},n=0}$ and $E^{-}_{k_{z},n=1}$ in Eq.~(\\ref{eq:U1_surface_EE_EV}). \n\n\nSecond, from Eq.~(\\ref{eq:U1_surface_EE_EV}), the probability distributions for the states $\\psi^{-}_{k_{z},n=0}$ and $\\psi^{\\pm}_{k_{z},n=1}$ are given by\n\\begin{align}\n & {{|\\psi^{-}_{k_{z},n=0}|^2 \\propto\\left|\\varphi_{0,B}(y+k_{z}\/B) \\right|^{2}}} \\\\\n & {{|\\psi^{\\pm}_{k_{z},n=1}|^2 \\propto\\left| \\alpha_{\\pm}(1) \\right|^{2}\\left|\\varphi_{0,B}(y+k_{z}\/B) \\right|^{2} + \\left|\\varphi_{1,B}(y+k_{z}\/B) \\right|^{2}}}\n\\end{align}\nup to a normalization factor, where $\\varphi_{n,B}(y)$ is the $n^{\\text{th}}$ eigenstate of an SHO localized along {{$y$}}. \nNotice that we have indicated the explicit $B$-dependence on $\\varphi_{n,B}(y)$ since the cyclotron frequency and the localization of the wave function depend on the strength of magnetic field. \nThis implies that $\\psi^{-}_{k_{z},n=0}$ has a pure Gaussian distribution. Furthermore, we expect that $\\psi^{-}_{k_{z},n=1}$ is more characteristic of an SHO first excited state than $\\psi^{+}_{k_{z},n=1}$ since $\\left| \\alpha_{-}(1) \\right|^{2} = \\left( -\\sqrt{m^{2} + 2B} + m \\right)^{2}\/(2B) < \\left( \\sqrt{m^{2} + 2B} + m \\right)^{2}\/(2B) = \\left| \\alpha_{+}(1) \\right|^{2}$,\nas we have assumed both $B$ and $m$ are positive. \nFigs.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (e)--(g) show the 2D wave function probability distributions at $\\phi = \\pi$ for edge confined modes in different LLs in our lattice model, together with the insets showing the integrated wave function probability over all $x$-coordinates. \nWhile both Figs.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (e) and (g) corresponds to $n = 1$ LL, the former is at the negative energy branch and the latter is at the positive energy branch.\nTherefore Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (e) shows split peaks characteristic of the SHO first excited state, more so than Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (g). \nIn contrast, Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (f), corresponding to the $n=0$ LL wave function, shows the Gaussian probability distribution characteristic of the SHO ground state. \nWe see that the qualitative properties of the wave functions are all consistent with the low energy surface theory.\n\nThird, the definition of the ladder operator in Eq.~(\\ref{eq:U1_ladder}) implies that the center of the wave functions will be shifted by $-k_{z}\/B$ from $y = 0$. \nIdentifying $k_{z}$ in the low energy theory as $\\Delta \\phi = \\phi - \\pi$ and $B$ as $2\\pi q_{y}$, we deduce that the distance $l$ that the edge-confined mode gets shifted along $y$ in the lattice model will be $l = -\\Delta \\phi\/({2\\pi q_{y}})$. \nNotice that the edge-confined mode in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (h) at $\\phi = 0.9\\pi$ ($\\Delta \\phi = -0.1\\pi$) is shifted by $\\approx +2.5$ lattice constants along $y$ comparing with Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (f), which is at $\\phi = \\pi$ ($\\Delta \\phi = 0$). \nThis is consistent with our prediction, as $l$ will be $+2.5$ when $\\Delta \\phi = -0.1\\pi$ and $q_{y} = 0.02$.\n\nFourth, although Eq.~(\\ref{eq:U1_surface_EE_EV}) predicts non-degenerate energy levels for a single surface with a perpendicular $U(1)$ magnetic field, in Fig.~\\ref{fig:chiral_sliding_main_text} (a) the flat band corresponding to the $E^{-}_{k_{z},n=0}$ level is highly degenerate. \nThis is due to zone-folding effects, similar to what we observed for the corner mode dispersion in Fig.~\\ref{fig:chiral_sliding_main_text} (a). \nAs the gap-crossing modes are shifted outside $\\phi = [0,2\\pi)$, they get folded back to $\\phi = [0,2\\pi)$ together with the flat bands connected to them. \nUp to the degeneracy due to zone folding, the universal feature is that the edge-confined modes appearing in our 2D chiral DW system originate from the projection of surface electrons in a chiral HOTI with $U(1)$ Landau quantization.\n\n\nBefore moving on, let us remark on the robustness of our low-energy predictions to perturbations of the model. \nIf we consider a more complicated modulated system with, for example, long-range and anisotropic hopping terms together with other on-site potentials, as long as the promoted 3D system still preserves inversion symmetry and the gap is not closed, the 3D system will still be in the same chiral HOTI phase. \nHowever, the low energy theories that we have constructed might be modified. \nFor example, the low energy theory at the surfaces, which we model with Eq.~(\\ref{eq:surf_H_chiral_HOTI_transformed}), may be modified as \n\\begin{align}\n H_{\\text{surf}} =& \\alpha_{y}p_{y}\\sigma_{y}' - \\alpha_{z}\\left(p_{z}+By \\right)\\sigma_{x}' + m\\sigma_{z}' + \\Delta \\sigma_{0}' \\nonumber \\\\\n & + \\mathcal{O}(p_{y}^{2},p_{z}^{2},p_{y}p_{z}).\\label{eq:higher_order_terms}\n\\end{align}\nDifferences between $\\alpha_{x}$ and $\\alpha_{y}$ can lead to an anisotropic gapped Dirac cone. \nA nonzero $\\Delta$ induces unequal masses in different subspace of $\\vec{\\sigma}'$ which can shift the entire energy spectrum. \n$\\mathcal{O}(p_{y}^{2},p_{z}^{2},p_{y}p_{z})$ represents higher-order terms in the low energy theory which might cause nonlinearity in the band dispersion in Eq.~(\\ref{eq:higher_order_terms}) without minimal coupling.\nBy the same reasoning, we might also have non-linear hinge mode energies with a quadratic momentum correction in Eq.~(\\ref{eq:chiral_low_energy_hinge_H_1}).\nAll of these additional terms will change the energetic feature of the system, such as energy spectra, Fermi velocities, together with the detailed form of the wave functions, which will be inevitably different from Eq.~(\\ref{eq:U1_surface_EE_EV}). \nNevertheless, the following features are universal: (1) There will be electrons confined to the surface that undergo $U(1)$ Landau quantization, and therefore there will be states that are confined along some directions. \nUpon projecting down to the 2D modulated system, we will still obtain edge-confined modes. \n(2) There will be (non-)linear hinge mode dispersion that will be shifted along $k_{z}$ due to the minimal coupling. \nTherefore the statement that we can tune the range of $\\phi$ where the gap-crossing corner modes appear by tuning the magnitude of the modulation wave vectors, will still hold. \nWe use the low energy theories Eq.~(\\ref{eq:chiral_low_energy_hinge_H_1}) and Eq.~(\\ref{eq:surf_H_chiral_HOTI_transformed}) since these allow us to uncover the relation between the states in the promoted dimension and those in the original low dimensional modulated system in an analytically tractable way.\n\n\n\\subsection{\\label{sec:chiral_sliding_confined_bulk_modes}Bulk states}\n\nThe above analysis on corner- and edge-confined modes shows that the corresponding higher dimensional description of our modulated system is a 3D chiral HOTI minimally coupled to a $U(1)$ gauge field. \nTo complete our analysis, we will now focus on the bulk states.\nAs expected, the low energy description of the bulk-confined modes, shown in Fig.~\\ref{fig:chiral_sliding_main_text} (d), will correspond to the low energy theory of bulk electrons in a 3D chiral HOTI minimally coupled to a $U(1)$ gauge field. \nWe start with the Bloch Hamiltonian of the promoted 3D chiral HOTI (Eq.~(\\ref{eq:lattice_model_chiral_sliding}) with $q_{x} = q_{y} = 0$) expanded around the $\\Gamma$ point\\cite{pozo2019quantization}, \n\\begin{align}\n H_{\\text{bulk}} = m_{\\text{bulk}} \\tau_{z}\\sigma_{0} + \\tau_{x}\\vec{p} \\cdot \\vec{\\sigma} + \\tau_{0} \\vec{M} \\cdot \\vec{\\sigma}. \\label{eq:3D_chiral_HOTO_bulk}\n\\end{align}\nWe have defined several parameters to make Eq.~(\\ref{eq:3D_chiral_HOTO_bulk}) compact for later convenience, and introduced $\\tau_{0} \\vec{M} \\cdot \\vec{\\sigma}$ where $\\vec{M} = (M,M,M)$ corresponding to the ferromagnetic potential in Eq.~(\\ref{eq:chiral_on_site}). \nWe now couple this $H_{\\text{bulk}}$ to $\\vec{A} = By \\hat{z}$, which is equivalent to Eq.~(\\ref{A_U1}) with $q_{x} = 0$. \nThis can be done via the minimal substitution $p_{z} \\to p_{z} + By$. \nFourier transforming along $x$ and $z$ to replace $p_{x}$ and $p_{z}$ by wavenumbers $k_{x}$ and $k_{z}$, and defining the $k_{z}$-dependent ladder operator as\n\\begin{align}\n a^{\\dagger}_{k_{z}} = \\frac{1}{\\sqrt{2B}}\\left( k_{z} + By - ip_{y} \\right), \\label{eq:3D_chiral_bulk_U1_ladder}\n\\end{align}\nwe can rewrite Eq.~(\\ref{eq:3D_chiral_HOTO_bulk}) coupled to $\\vec{A} = By \\hat{z}$ in terms of $a_{k_{z}}$ and $a^{\\dagger}_{k_{z}}$ as\n\\begin{widetext}\n\\begin{align}\n H_{\\text{bulk}}(k_{x},k_{z}) = m_{\\text{bulk}} \\tau_{z}\\sigma_{0} + \\tau_{x} \\begin{bmatrix}\n \\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}+a^{\\dagger}_{k_{z}} \\right) & k_{x} -\\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}-a^{\\dagger}_{k_{z}} \\right) \\\\\n k_{x} +\\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}-a^{\\dagger}_{k_{z}} \\right) & -\\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}+a^{\\dagger}_{k_{z}} \\right)\n \\end{bmatrix} + \\tau_{0} \\vec{M} \\cdot \\vec{\\sigma}. \\label{eq:3D_bulk_LL_Hamiltonian}\n\\end{align}\n\\end{widetext}\nWe have numerically shown in SM\\cite{SM} that the effective theory in Eq.~(\\ref{eq:3D_bulk_LL_Hamiltonian}) captures several properties of the flat bulk bands in Fig.~\\ref{fig:chiral_sliding_low_energy_main_text_temp} (b) with relatively small $q_{y} = 0.02$, such as energy asymmetry with respect to $E = 0$ and the confinement direction of the bulk states due to $U(1)$ Landau quantization.\n\nFrom the above analysis on corner-, edge-, and bulk-confined modes, we conclude that we can characterize this topological 2D modulated system with chiral sliding modes in terms of a 3D chiral HOTI coupled to a $U(1)$ gauge field. \nIn addition, such 2D modulated systems provide a platform to examine the properties of a 3D chiral HOTI, by sliding the DW order parameter $\\phi$.\n\n\n\n\\section{\\label{sec_helical_HOTI_sliding_modes}Helical Higher-Order Sliding Modes and $SU(2)$ Gauge Fields }\n\n\n\n\nNext, we will generalize our formalism to time-reversal invariant spinful systems. \nIn doing so, we will see that incommensurate modulations induce coupling to $SU(2)$ gauge fields in the dimensionally promoted models.\n$SU(2)$ gauge fields can be used to represent spin-orbit coupling\\cite{YiLi_SU2_Hofstadter}, which is ubiquitous in topological states of matter. \nFor example, $SU(2)$ gauge fields in 3D and 4D generates $SU(2)$ LLs that give rise to 3D TIs and 4D QHEs\\cite{YiLi_TI_SU2_LL,zhang2001four}. \nA non-Abelian $SU(2)$ Peierls phase in 2D and 3D lattices can also lead to 2D and 3D TIs\\cite{SU2_gauge_in_2D_Goldman,YiLi_SU2_Hofstadter}. \nIn addition, in response to a bulk $SU(2)$ gauge flux insertion, a 2D TI can bind various quasi-particle excitations such as spinons, holons and chargeons\\cite{qispincharge}. \nIn this section, we present a 2D modulated system that allows us to simulate a 3D helical HOTI coupled to an $SU(2)$ gauge field.\n\n\\begin{figure}[t]\n\\hspace{-0.5cm}\n\\includegraphics[width=\\columnwidth]{fig_main_text_helical_sliding_revised_change_ticking_freq.pdf}\n\\caption{(a) $\\phi$-sliding spectrum of the 2D helical model in Eq.~(\\ref{eq:H_helical_sliding}) with parameters given in the text. \n(b) Summation of the probability density of the doubly-degenerate corners modes at $\\phi = 0.4\\pi$ and $E=0.0146$. \nThe two corner modes at the same $\\phi$ are related to each other by the $\\mathcal{I}\\mathcal{T}$-symmetry, and hence they are localized at inversion-related corners and have opposite spins. \n(c) $\\&$ (d) Probability distribution of edge and bulk modes at $\\phi = 0.9\\pi$ and $E = -0.1459$ and $E = 0.5227$, respectively. \nThe darker (black) color in (b)--(d) implies higher probability density. \nIn (b), (c) and (d), the $x$- and $z$-coordinate both range from $-15 ,\\ldots, +15$.}\n\\label{fig:helical_sliding_main_text}\n\\end{figure}\n\n\\subsection{Dimensionally Promoted Helical Model}\n\nWe start by considering the following 2D Hamiltonian on a square lattice with one modulated on-site potential $[V(\\vec{q},\\vec{n},\\phi)]$:\n\\begin{equation}\n\\begin{aligned}\nH = {} & \\sum_{\\vec{n}} \\psi^{\\dagger}_{\\vec{n}+\\hat{x}} [H_{+\\hat{x}}]\\psi_{\\vec{n}} + \\psi^{\\dagger}_{\\vec{n}+\\hat{z}} [H_{+\\hat{z}}]\\psi_{\\vec{n}} + \\text{h.c.}\\\\\n&+ \\sum_{\\vec{n}} \\psi^{\\dagger}_{\\vec{n}}\\left( [H_{\\text{on-site}}] + [V(\\vec{q},\\vec{n},\\phi)] \\right)\\psi_{\\vec{n}}, \\label{eq:H_helical_sliding}\n\\end{aligned}\n\\end{equation}\nwhere the unmodulated couplings are are\n\\begin{align}\n & [H_{+\\hat{x}}] = \\frac{v_{x}}{2}\\tau_{z}\\mu_{0}\\sigma_{0} - \\frac{u_{x}}{2i}\\tau_{y}\\mu_{y}\\sigma_{0}, \\\\\n & [H_{+\\hat{z}}] = \\frac{v_{z}}{2}\\tau_{z}\\mu_{0}\\sigma_{0} - \\frac{u_{z}}{2i}\\tau_{x}\\mu_{0}\\sigma_{0}, \\\\\n & [H_{\\text{on-site}}] = m_{1}\\tau_{z}\\mu_{0}\\sigma_{0}+m_{2}\\tau_{z}\\mu_{x}\\sigma_{0} + m_{3}\\tau_{z}\\mu_{z}\\sigma_{0} \\nonumber \\\\ \n & +m_{v_{1}}\\tau_{0}\\mu_{z}\\sigma_{0} + m_{v_{2}}\\tau_{0}\\mu_{x}\\sigma_{0}. \\label{eq:helical_H_on_site}\n\\end{align}\nThe matrices $\\vec{\\tau}$, $\\vec{\\mu}$ and $\\vec{\\sigma}$, are Pauli matrices and denote the orbital, sub-lattice and spin degrees of\nfreedom, respectively. \nThe hopping matrices $[H_{+\\hat{x}}]$ and $[H_{+\\hat{z}}]$, together with the on-site potential $[H_{\\text{on-site}}]$ respect both inversion and TR symmetries. \nThe inversion and TR operations are represented by $\\tau_{z}$ and $i\\tau_{z}\\sigma_{y}\\mathcal{K}$, respectively\\cite{Wieder_spin_decoupled_helical_HOTI}. \nThese hoppings give rise to low energy four-component Dirac fermions in each spin subspace, realizing a topological critical point. \nThe modulated on-site energy is given by\n\\begin{eqnarray}\\nonumber\n[V(\\vec{q},\\vec{n},\\phi)]&=& v_{y}\\tau_{z}\\mu_{0}\\begin{bmatrix}\n \\cos\\theta^{+}_{\\vec{q},\\vec{n},\\phi} & 0 \\\\ 0 & \\cos\\theta^{-}_{\\vec{q},\\vec{n},\\phi}\n \\end{bmatrix}\\\\\n&& + v_{H} \\tau_{y}\\mu_{z}\\begin{bmatrix}\n \\sin\\theta^{+}_{\\vec{q},\\vec{n},\\phi} & 0 \\\\ 0 & \\sin\\theta^{-}_{\\vec{q},\\vec{n},\\phi}\n \\end{bmatrix},\n \\label{eq:helical_sliding_modulation_H}\n\\end{eqnarray}\nwhere $\\theta^{\\pm}_{\\vec{q},\\vec{n},\\phi} = 2\\pi \\vec{q}\\cdot \\vec{n}\\pm\\phi$, $\\vec{q} = (q_{x},q_{z})$ is the modulation wave vector in 2D, $\\vec{n} \\in \\mathbb{Z}^{2}$ is the lattice position, and $\\phi$ is the sliding phase.\nThe first term in Eq.~(\\ref{eq:helical_sliding_modulation_H}) modulates the mass $m_{1}\\tau_{z}\\mu_{0}\\sigma_{0}$ in Eq.~(\\ref{eq:helical_H_on_site}), which may represent unequal on-site energy for $s$ and $p$ orbitals, with forward ($-\\phi$) and backward ($+\\phi$) sliding phase in each spin subspace\\cite{SU2_gauge_in_2D_Goldman,SU2_gauge_2D_to_1D_Goldman}. \nThe second term in Eq.~(\\ref{eq:helical_sliding_modulation_H}) describes a modulation of the on-site energy which mixes $s$ and $p$ orbitals with unequal strength for different sublattices. \nSimilarly, we have forward and backward sliding phases in different spin subspaces for the second term. \nSince the modulation in Eq.~(\\ref{eq:helical_sliding_modulation_H}) has opposite phase offsets in each spin subspace, it may be induced from spin-orbit coupled spin ordering. \nThis modulation is TR- and inversion-symmetric only when $\\phi=0$, $\\pi$. \nNote, however, that the product of inversion and TR symmetry, which we will denote $\\mathcal{I}\\mathcal{T}$-symmetry, is preserved for all values of $\\phi$.\nIf we denote the third, synthetic dimension as $y$, this 2D model is equivalent to the inversion and TR symmetric 3D helical HOTI model of Ref.~\\onlinecite{Wieder_spin_decoupled_helical_HOTI}, coupled to an $SU(2)$ gauge field given by\n\\begin{align}\n \\vec{A} = (0,2\\pi (q_{x}x+q_{z}z)\\sigma_{z},0). \\label{SU2_A}\n\\end{align}\nThis matrix-valued $\\vec{A}$ produces a constant $SU(2)$ magnetic field\\cite{Estienne_2011} $\\vec{B} = \\vec{\\nabla} \\cross \\vec{A} - i \\vec{A} \\cross \\vec{A}$, determined from the field strength\\cite{eguchi1980gravitation} $F_{\\mu \\nu} = \\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu} - i \\left[ A_{\\mu},A_{\\nu}\\right]$, and given by\n\\begin{align}\n \\vec{B} = (-2\\pi q_{z}\\sigma_{z},0,2\\pi q_{x} \\sigma_{z} ). \\label{SU2_B}\n\\end{align}\nThis constant $SU(2)$ field strength preserves both inversion and TR symmetry in 3D, up to a spin-dependent gauge transformation (see SM\\cite{SM}). \nNotice that Eq.~(\\ref{SU2_B}) implies that the $SU(2)$ magnetic field in this example can be interpreted as a $U(1)$ magnetic field with opposite sign for spin-up and spin-down electrons\\cite{SU2_gauge_in_2D_Goldman,SU2_gauge_2D_to_1D_Goldman}. \nWe then expect that, for a suitable choice of parameters such that the $SU(2)$ gauge field does not close the bulk gap in 3D, the insulating ground-state will be in the same non-trivial helical HOTI phase as the model with $\\vec{q}=0$\\cite{khalaf,hotis,Po2017}.\nTherefore, in 3D, our promoted model will support an odd number of pairs of sample-encircling helical hinge modes respecting inversion and TR symmetries\\cite{khalaf,Wieder_spin_decoupled_helical_HOTI}.\nUpon projected back to 2D, the helical hinge modes in 3D become $\\mathcal{I}\\mathcal{T}$-related pairs of corner modes at the same $\\phi$ in the 2D modulated system. \nIn the SM\\cite{SM}, we give the form of the 3D dimensionally-promoted model in position-space. \n\n\\subsection{Calculation of the Spectrum}\n\nLet us now numerically verify these conclusions. \nFig.~\\ref{fig:helical_sliding_main_text} (a) shows the $\\phi$-sliding spectrum of Eq.~(\\ref{eq:H_helical_sliding}) with parameters $m_{1} = -3$, $m_{2} = 0.3$, $m_{3} = 0.2$, $m_{v_{1}} = -0.4$, $m_{v_{2}} = 0.2$, $v_{x}=v_{z}=u_{x}=u_{z} = 1$, $v_{y} = 2$, $v_{H} = 1.2$\\cite{Wieder_spin_decoupled_helical_HOTI}, and $\\vec{q} = (0,q_{z})$ where $q_{z} = 0.11957$ \\cite{shi2019charge}. \nThe system size is $31 \\times 31$. \nThere are doubly-degenerate pairs of states which cross the gap as a function of $\\phi$, where the degeneracy is protected by $\\mathcal{IT}$-symmetry. \nWe see from the wave functions that these are corner modes related by $\\mathcal{IT}$-symmetry, as shown in Fig.~\\ref{fig:helical_sliding_main_text} (b) for the branch with negative slope around $\\phi \\approx 0.4\\pi$. \nIn the other branch of doubly-degenerate gap-crossing states with positive slope, the corner modes are the inversion-symmetric counterpart (where recall that inversion symmetry leaves spin invariant) to those in Fig.~\\ref{fig:helical_sliding_main_text} (b). \nTherefore, as $\\phi$ slides from $0$ to $2\\pi$, this model realizes a $\\mathbb{Z}_{2}$ pump\\cite{fu2006time,teo2010topological} as one of the pairs of corner states will merge into the occupied state subspace (with Fermi level $E_{F}=0$) while the other pair will flow out. \nIn our specific examples, the two states in each $\\mathcal{I}\\mathcal{T}$-related pair at the same $\\phi$ are spin eigenstates and therefore in this case the $\\mathbb{Z}_2$ pump is a spin pump; our conclusions, however, hold even when spin is not conserved.\n\n\nAs mentioned earlier, and in analogy with our chiral HOTI model, the corner modes here are equivalent to hinge modes along $y$ in 3D. \nThe corresponding low energy theory for these corner modes is \n\\begin{align}\n H_{\\text{corner}}= v_{F} \\left( \\phi \\sigma_{z}' + 2\\pi \\left( q_{x}x_{\\text{corner}} + q_{z}z_{\\text{corner}} \\right) \\sigma_{0}' \\right), \\label{eq:H_helical_hinge_1}\n\\end{align}\nwhere $v_{F}$ is the group velocity of the hinge modes in 3D. \nWe use the Pauli matrices $\\vec{\\sigma}'$ to denote the effective basis where in each subspace the states have opposite spin together with some orbital and sub-lattice textures. \nWe have assumed without loss of generality that there is only one pair of helical hinge modes at the hinge along $y$ in the promoted 3D system. \nBy denoting $\\phi$ as $k_{y}$, which is the crystal momentum along $y$, we recognize Eq.~(\\ref{eq:H_helical_hinge_1}) as the hinge mode dispersion $H(k_{y}) = v_{F}k_{y}\\sigma_{z}'$ in 3D minimally coupled to an $SU(2)$ gauge field described by Eq.~(\\ref{SU2_A}).\nSimilar to Sec.~\\ref{sec_chiral_HOTI}, as we vary $\\vec{q}$--which is equivalent to changing the strength and (spatial) direction of the $SU(2)$ gauge field in Eqs.~(\\ref{SU2_A}) and (\\ref{SU2_B})--the dispersion of the spin-polarized corner modes will shift along the $\\phi$-axis. \nIn the SM\\cite{SM} we present a complete low energy theory analysis for the corner modes with the same structure as Sec.~\\ref{sec_chiral_HOTI}.\n\nIn addition, we show in Figs.~\\ref{fig:helical_sliding_main_text} (c) and (d) the probability density for the edge- and bulk-confined modes in the flat bands of Fig.~\\ref{fig:helical_sliding_main_text} (a). \nSimilar to the corner modes, these can be respectively understood in terms of 3D low energy surface and bulk theories minimally coupled to an $SU(2)$ gauge field, leading to an $SU(2)$ Landau quantization\\cite{YiLi_TI_SU2_LL,YiLi_SU2_Hofstadter}. \nThe relevant surface theory describes a time-reversed pair of Chern insulators. \nThe relevant bulk theory is the $\\vec{k}\\cdot\\vec{p}$ expansion around $\\Gamma$ of the promoted 3D helical HOTI Hamiltonian\\cite{Wieder_spin_decoupled_helical_HOTI}. \nWe provide further details in the SM\\cite{SM}.\nTogether with the corner mode analysis, we see that this topological 2D modulated system with helical sliding modes can be characterized by a 3D lattice model coupled to an $SU(2)$ gauge field. \nIn addition, we have shown how 2D modulated systems can provide a platform to examine $SU(2)$ gauge physics in higher dimensions, by sliding the phase $\\phi$ of the DW order parameter.\n\n\n\\section{\\label{sec:Weyl_CDW}Weyl-CDWs and 4D topological modes}\n\n\nAs a final demonstration of our dimensional promotion formalism and its utility to investigating physics in more than 3D, we consider the mean-field state of a correlated inversion-symmetric 3D Weyl semimetal with CDW distortion (Weyl-CDW)\\cite{dynamical_axion_insulator_BB,gooth2019evidence,shi2019charge,wang2013chiral,CDW_Weyl_Sehayek,CDW_in_Q1D_Cohn,Monopole_CDW_in_Weyl_Yi_Li,yu2020dynamical}. \nIt has been shown that such a system can realize various topological phases. \nDepending on the phase $\\phi$ of the CDW order parameter, the system can interpolate between quantum anomalous Hall (QAH) and {\\it obstructed} QAH (oQAH) phase\\cite{dynamical_axion_insulator_BB}. \nThis is due to the $\\pi$ mod $2\\pi$ axion angle difference $\\delta \\theta_{\\phi} = \\theta(\\phi = \\pi) - \\theta(\\phi = 0)$ for the system with $\\phi = 0$ and $\\phi = \\pi$, in the thermodynamic limit. \nPhysically, this leads to a Hall conductance difference\n\\begin{align}\n \\left|G_{xy}(\\phi = \\pi) - G_{xy}(\\phi = 0) \\right| = e^{2}\/h \\text{ mod } 2e^{2}\/h \\label{eq:main_G_xy_eqn}\n\\end{align}\nfor a semi-infinite slab [see also Eq.~(\\ref{eq:agnostic}) below, as well as Refs.~\\onlinecite{olsen2020gapless,2020_Axion_coupling_Vanderbilt}]. \nIn this section, we analyze a minimal model of a 3D inversion-symmetric magnetic Weyl-CDW system, which admits a dimensional promotion to 4D with a $U(1)$ gauge field. \nWe will explain the origin of the background QAH response and the interpolation between QAH and oQAH phases using the corresponding 4D theory. \nIn the following, we will denote a sample infinite along $x$ and $y$ with finite thickness $L_z$ along the $z$ direction as an $xy$-slab. \nSimilarly, we will use the term $y$-rod to denote a sample infinite along $y$ and finite along $x$ and $z$ with size $L_{x}\\times L_{z}$.\n\n\n\\subsection{3D Weyl-CDW Model and Dimensional Promotion}\n\nTo begin, we consider electrons on a 3D cubic-lattice with Hamiltonian $H = H_{0} + H_{CDW}(\\phi) $. \nHere $H_0$ is a periodic tight-binding Hamiltonian given by\n\\begin{align}\n H_0&=\\left(\\sum_{\\vec{n}}\\left[-it_x\\psi^\\dag_{\\vec{n}+\\hat{x}}\\sigma_x \\psi_{\\vec{n}}-it_y \\psi^\\dag_{\\vec{n}+\\hat{y}}\\sigma_y \\psi_{\\vec{n}}+t_z\\psi^\\dag_{\\vec{n}+\\hat{z}}\\sigma_z \\psi_{\\vec{n}}\\right]\\right. \\nonumber \\\\\n &+\\sum_{\\vec{n}}\\frac{m}{2}\\left(\\psi^\\dag_{\\vec{n}+\\hat{x}}\\sigma_z \\psi_{\\vec{n}} + \\psi^\\dag_{\\vec{n}+\\hat{y}}\\sigma_z \\psi_{\\vec{n}} -2 \\psi^\\dag_{\\vec{n}}\\sigma_z \\psi_{\\vec{n}} \\right) \\nonumber \\\\\n &\\left.-\\sum_{\\vec{n}} t_z \\left(\\cos (\\pi q)\\right) \\psi^\\dag_{\\vec{n}}\\sigma_z c_{\\vec{n}}\\right) +\\mathrm{h.c.}\n\\end{align}\nin position space. \nThe corresponding Bloch Hamiltonian is\n\\begin{align}\n &H_0(\\vec{k})= -2[t_x\\sin (k_x)\\sigma_x +t_y\\sin (k_y)\\sigma_y] \\nonumber \\\\\n & -m[2-\\cos (k_x) - \\cos (k_y)]\\sigma_{z} +2t_z[\\cos (k_z) -\\cos (\\pi q)]\\sigma_{z} \\label{eq:Bloch_H_0} , \n\\end{align}\nwith $m\/2 \\ge t_{x},-t_{y},t_{z} >0$. \nWe take for the on-site modulation\n\\begin{align}\n H_{CDW}(\\phi)&=2|\\Delta|\\sum_{\\vec{n}}\\cos(2\\pi q n_{z}+\\phi)\\psi^\\dag_{\\vec{n}}\\sigma_z \\psi_{\\vec{n}}.\n\\end{align}\nHere $2|\\Delta|$ is the strength of the CDW modulation, $2\\pi q$ is the magnitude of the modulation wave vector $2\\pi \\vec{q}=(0,0,2\\pi q)$ and $\\phi$ is the phase of CDW order parameter. \nWe again use $\\vec{\\sigma}$ to denote the Pauli matrices, which here index an orbital degree of freedom. \nThe inversion and TR operation are represented by $\\sigma_{z}$ and $\\mathcal{K}$, respectively (note that this is a model of spinless electrons).\nThe Hamiltonian $H_{0}$ then describes a TR-breaking, inversion-symmetric magnetic Weyl semimetal (WSM) with Weyl nodes at $\\vec{k} = (0,0,\\pm \\pi q)$, see Eq.~(\\ref{eq:Bloch_H_0})\\cite{mccormick2017minimal}.\nThe perturbation $H_{CDW}(\\phi)$ is the CDW modulation that couples these two Weyl nodes and opens a gap in the bulk spectrum~\\cite{dynamical_axion_insulator_BB}. \nNote that in this simple model, we have chosen the modulation wavevector to be exactly equal to the Weyl node separation vector for simplicity of analysis. \nEven though the bulk is gapped, the surface of this 3D Weyl-CDW is gapless, due to the presence of QAH surface states. \nIn Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (a) we show the $\\phi$-sliding spectrum for a $y$-rod of $H_{0}+H_{CDW}(\\phi)$ at $k_{y} = 0$ with size $L_{x} \\times L_{z} = 25 \\times 25$, $t_{x} = - t_{y} = t_{z} = 1$, $m = 2$, $2|\\Delta|=0.75$ and $q = 1\/5$. \nThis corresponds to a commensurate Weyl-CDW system. \nThe mid-gap zero modes in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (a) correspond to the QAH surface states. \nIn Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (b) we show the probability distribution of the 10 zero modes at $\\phi = 0$. \nTogether with Wilson loop and Berry curvature calculation in the SM\\cite{SM}, we verify that the corresponding $xy$-slab with $L_{z} = 25$ carries a slab Hall conductance $ G_{xy}(\\phi = 0) = -5 e^{2}\/h$. \nWe can then identify the weak Chern number\\cite{qi2008topological,kohmoto1992diophantine,halperin1987possible,fukanemele} of the 3D periodic system with $5=25\/5$ unit cells (since $q= 1\/5$) as $\\nu_{z} = -1$.\n\n\n\\begin{figure*}[t]\n\\hspace{-0.5cm}\n\\includegraphics[width=\\linewidth]{Q_2pi_over_5_tx_1_ty_-1_tz_1_m_2_D_0_75_Lz_25.pdf}\n\\caption{(a) $\\phi$-sliding spectrum of the Weyl-CDW model in a $y$-rod geometry at $k_{y} = 0$ with size $L_{x}\\times L_{z} = 25 \\times 25$, $t_{x} = -t_{y} = t_{z} = 1$, $m = 2$, $2|\\Delta| = 0.75$ and $q = 1\/5$. \n(b) The average probability distribution of the 10 zero modes at $\\phi = 0$ in (a). \nThese zero modes correspond to QAH surface states. \n(c) The average probability distribution of the 5 non-trivial states at $\\vec{k} = \\Gamma$ of the $xy$-slab at $\\phi = 0$, which in total lead to $G_{xy}(\\phi = 0) = -5 e^{2}\/h$. \n(d) The average probability distribution of the 8 zero modes at $\\phi = \\pi$ in (a). \nThese zero modes correspond to QAH surface states. \n(e) The average probability distribution of the 4 non-trivial states at $\\vec{k} = \\Gamma$ of the $xy$-slab at $\\phi = \\pi$, which in total lead to $G_{xy}(\\phi = \\pi) = -4 e^{2}\/h$. \nThe darker (black) color in (b)--(e) implies higher probability density. \nIn (b) and (d), the $x$- and $z$-coordinate both range from $-12 ,\\ldots, +12$. \nIn (c) and (e), the $z$-coordinate ranges from $-12 ,\\ldots, +12$.}\n\\label{fig:temporary_3d_weyl_cdw_fig}\n\\end{figure*}\n\n\n\nAs in Sec.~\\ref{sec_chiral_HOTI} and~\\ref{sec_helical_HOTI_sliding_modes}, we identify $\\phi$ with the crystal momentum $k_{w}$ along a fourth, synthetic direction denoted by $w$. \nUsing the dimensional promotion procedure in Sec.~\\ref{sec_Dimension_promotion}, we can promote $H_{0} + H_{CDW}(\\phi)$ to a 4D nodal line system coupled to a $U(1)$ gauge field. \nIn the SM\\cite{SM} we give the explicit form of the promoted model in 4D position space. \nThe corresponding 4D nodal line system (with $q=0$) has a Bloch Hamiltonian\n\\begin{align}\n &H(\\vec{k})= -2[t_x\\sin (k_x)\\sigma_x +t_y\\sin (k_y)\\sigma_y] + 2|\\Delta|\\cos{(k_{w})}\\sigma_{z} \\nonumber \\\\\n & -m[2-\\cos (k_x) - \\cos (k_y)]\\sigma_{z} +2t_z[\\cos (k_z) -\\cos (\\pi q)]\\sigma_{z}. \n\\label{eq:ham0}\n\\end{align}\nThe spectrum of this Hamiltonian features nodal lines at $k_{x}=k_{y} = 0$ defined by the implicit equation\n\\begin{align}\n t_{z}\\cos{(k_{z})} + |\\Delta| \\cos{(k_{w})} = t_{z} \\cos{(\\pi q)}.\n\\end{align}\nAccording to Eq.~(\\ref{eq:expression_A}), we then couple this Hamiltonian to a 4D $U(1)$ gauge field given by\n\\begin{align}\n \\vec{A} = (0,0,0,2\\pi q z), \\label{eq:U1_4D}\n\\end{align}\nsince $2\\pi \\vec{q} = 2\\pi q \\hat{z}$ in this system. \nThis $\\vec{A}$ only produces non-zero field strength threading the $zw$ plane,\n\\begin{align}\n F_{zw} = -F_{wz}= \\partial_{z}A_{w} - \\partial_{w}A_{z}=2 \\pi q, \\label{eq:Fzw}\n\\end{align}\nwhere all other components of $F_{\\mu \\nu} = \\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu}$ are zero. \nWe are now in a position to reinterpret the existence of a background QAH response and QAH surface states when the bulk gap is opened due to the CDW. \nWe will see how these features emerge from the low energy approximation for this 4D system minimally coupled to Eq.~(\\ref{eq:U1_4D}).\n\n\\subsection{Low Energy Theory Analysis}\n\n\\begin{widetext}\n\nWe start from the 4D Bloch Hamiltonian in Eq.~(\\ref{eq:ham0}). \nExpanding around $\\vec{k} = \\vec{0}$, we have\n\\begin{align}\n H(\\vec{k}) \\approx -2[t_x k_{x} \\sigma_{x} +t_y k_{y} \\sigma_{y}] + 2t_{z} \\left( 1 - \\frac{k_{z}^{2}}{2} - \\cos (\\pi q) \\right)\\sigma_{z} + 2|\\Delta| \\left( 1 - \\frac{k_{w}^{2}}{2}\\right) \\sigma_{z}. \\label{eq:ham0_low_energy}\n\\end{align}\nThe nodal line in this low energy theory is an ellipse in the $k_{z}$-$k_{w}$ plane with $k_{x}=k_{y}=0$, defined by\n\\begin{align}\n t_{z}k_{z}^{2} + |\\Delta|k_{w}^{2} = 2t_{z}\\left[ 1 - \\cos{ (\\pi q)} \\right] + 2|\\Delta| > 0.\n\\end{align}\nReplacing the 4D wave vector $\\vec{k}=(k_{x},k_{y},k_{z},k_{w})$ by the 4D momentum operator $\\vec{p}=(p_{x},p_{y},p_{z},p_{w})$ using the so-called envelope function approximation\\cite{Envelope_function_approximation_1,Envelope_function_approximation_2,Envelope_function_approximation_3,Envelope_function_approximation_4,Envelope_function_approximation_5,Qi_Zhang_RMP,hasan2010colloquium,bernevig2006quantum}, the Hamiltonian governing the low energy dynamics reads\n\\begin{align}\n H = -2[t_x p_{x} \\sigma_{x} +t_y p_{y} \\sigma_{y}] + 2t_{z} \\left( 1 - \\frac{p_{z}^{2}}{2} - \\cos (\\pi q)\\right) \\sigma_{z} + 2|\\Delta| \\left( 1 - \\frac{p_{w}^{2}}{2}\\right) \\sigma_{z}. \\label{eq:ham0_low_energy_p}\n\\end{align}\nNext, let us minimally couple Eq.~(\\ref{eq:ham0_low_energy_p}) to a {{4D}} $U(1)$ gauge field $\\vec{A} = (0,0,0,2\\pi qz)$ via a Peierls substitution such that $p_{w} \\to p_{w} + 2\\pi q z$. \nEq.~(\\ref{eq:ham0_low_energy_p}) then becomes\n\\begin{align}\n H = -2[t_x p_{x} \\sigma_{x} +t_y p_{y} \\sigma_{y}] + 2 \\left( t_{z}\\left[ 1 - \\cos{(\\pi q)} \\right] + |\\Delta| \\right) \\sigma_{z} - \\left( t_{z}p_{z}^{2} +|\\Delta| \\left( p_{w} + 2 \\pi q z \\right)^{2} \\right) \\sigma_{z}, \\label{eq:after_4D_U1_couple}\n\\end{align}\nwhere we have assumed that the particle carries $-1$ charge. \nFourier transforming along $x$, $y$ and $w$, we may replace $p_{x}$, $p_{y}$ and $p_{w}$ by the corresponding wavenumbers $k_{x}$, $k_{y}$, $k_{w}$, such that\n\\begin{align}\n H(k_{x},k_{y},k_{w}) = -2[t_x k_{x} \\sigma_{x} +t_y k_{y} \\sigma_{y}] +2 \\left( t_{z}\\left[ 1 - \\cos{(\\pi q)} \\right] + |\\Delta| \\right) \\sigma_{z} - \\left( t_{z}p_{z}^{2} + |\\Delta| \\left( k_{w} + 2\\pi q z \\right)^{2} \\right) \\sigma_{z}. \\label{eq:FT1_4D_U1_LL}\n\\end{align}\n\\end{widetext}\n\nNotice that the coefficient of $\\sigma_z$ in the final term in the Hamiltonian,\n\\begin{align}\n t_{z}p_{z}^{2} + |\\Delta| \\left( k_{w} + 2 \\pi q z \\right)^{2}, \\label{eq:4D_U1_SHO_H}\n\\end{align}\nis an SHO Hamiltonian along $z$ which can be diagonalized as\n\\begin{align}\n 4 \\pi q \\sqrt{t_{z}|\\Delta|} \\left( n + \\frac{1}{2} \\right).\n\\end{align}\nHere $n$ is a non-negative integer and the eigenvalue of the number operator $ a^{\\dagger}_{k_{w},q} a_{k_{w},q}$ with\n\\begin{align}\n a^{\\dag}_{k_{w},q} = \\frac{1}{\\sqrt{ 4\\pi q}} \\left( \\frac{t_{z}}{|\\Delta|} \\right)^{\\frac{1}{4}}\\left[ \\left({\\frac{|\\Delta|}{t_{z}}}\\right)^{\\frac{1}{2}} \\left(k_{w}+ 2\\pi q z \\right) - ip_{z} \\right]. \\label{eq:4D_U1_LL_ladder}\n\\end{align}\nThe quantum number $n$ is the 4D $U(1)$ LL index. \nBy restricting to a subspace of the full Hilbert space with fixed $n$ and $k_{w}$, we see that the 4D low energy Hamiltonian Eq.~(\\ref{eq:after_4D_U1_couple}) may be decomposed into a direct sum of 2D low energy Chern insulators (CIs) in $xy$-plane parameterized by $n$ and $k_{w}$.\nThe Hamiltonian for these 2D CIs is given by\n\\begin{align}\n H_{\\text{2D CI}}(n,k_{w}) = -2[t_x p_{x} \\sigma_{x} +t_y p_{y} \\sigma_{y}] + 2 m \\sigma_{z}, \\label{eq:2D_CI_subspace}\n\\end{align}\nwhere\n\\begin{align}\n m = t_{z}\\left(1 - \\cos{(\\pi q)} \\right)+ |\\Delta| - 2\\pi q \\sqrt{t_{z}|\\Delta|}\\left( n + \\frac{1}{2} \\right). \\label{eq:mass_term_emerge_CIs}\n\\end{align}\nSince we have restricted to the subspace with fixed $n$ and $k_{w}$ in Eq.~(\\ref{eq:2D_CI_subspace}), according to Eq.~(\\ref{eq:4D_U1_SHO_H}) the wave function along $z$ and $w$ will be SHO eigenstates centered at $z = -k_{w} \/ (2\\pi q)$ multiplied by a plane wave $e^{ik_{w} w}$. \nNotice that the $k_{w}$-dependence of Eq.~(\\ref{eq:2D_CI_subspace}) is due to the integer $n$ in Eq.~(\\ref{eq:mass_term_emerge_CIs}) which is an eigenvalue of the number operator $a_{k_{w},q}^{\\dagger}a_{k_{w},q}$. \nTherefore, the eigenstates in the low energy approximation take the form of plane waves in $w$, and Chern insulator eigenstates as a function of $(x,y)$ localized at different constant-$z$ planes for different $k_{w}$.\nThis provides a four-dimensional interpretation of the layer construction of the Weyl-CDW presented in Refs.~\\onlinecite{dynamical_axion_insulator_BB,CDW_Weyl_Sehayek}.\n\nAs in a 3D nodal ring system with a perpendicular magnetic field~\\cite{Nodal_ring_perpendicular_B}, Eq.~(\\ref{eq:2D_CI_subspace}) can yield a gapped 4D bulk spectrum provided that $m \\ne 0$ $\\forall n \\ge 0$. \nThis insulating ground state will then carry non-trivial topology inherited from the nodal line system, since in Eq.~(\\ref{eq:2D_CI_subspace}) we found that the gapped 4D continuum theory is composed of low energy 2D CIs. \nWe then expect that there will be CI layers in the $xy$-plane of the corresponding 4D lattice model (see SM\\cite{SM}). \nThe CI layers will also be separated along $z$ by $2\\pi \/ (2\\pi q) = 1\/q$ for a fixed $k_{w}$, due to the $2\\pi$ periodicity of $k_{w}$ in the lattice model. \nIn our current example this separation is $5$ since $q =1\/5$. \nNotice that $k_{w}$ is now interpreted as the crystal momentum along the $4^{\\text{th}}$ dimension. \nTo connect these observation in 4D back to the physical 3D Weyl-CDW system with Hamiltonian $H_{0} + H_{CDW}(\\phi)$, we notice that each 3D Weyl-CDW system with a fixed $\\phi$ corresponds to the 4D theory with a fixed $k_{w}$. \nFocusing on the $xy$-slab with $\\phi = 0$ and thickness $L_{z} = 25$, in which $ G_{xy}(\\phi = 0) = -5 e^{2}\/h$, we show wavefunctions corresponding to the only $5$ layers of non-trivial CIs separated from each other by $5$ lattice constants along $z$ in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (c). \nEach of these CI layers carries Chern number $C = -1$ and contributes one chiral edge mode along $y$ in the ${y}$-rod, shown in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (b). \nIn the SM\\cite{SM} we provide technical details on identifying the non-trivial CI layers using hybrid Wannier function, Berry phases and Berry curvature calculations for an $xy$-slab. \nWe can thus regard the CDW-induced gap opening and the existence of background QAH response as the results of $U(1)$ Landau quantization in the 4D nodal line system. \n\n\nNext, we address the interpolation between the QAH phase at $\\phi = k_w=0$ and the oQAH phase at $\\phi = k_w=\\pi$ using the 4D theory. \nBefore we turn to the 4D low energy theory, we begin with the observation that in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (a), the number of mid-gap zero modes corresponding to QAH surface states decreases by $2$ as $\\phi$ slides from $0$ to $\\pi$; one state is lowered into the valence band, while one state is elevated to the conduction band.\nThis is consistent with the change in Hall conductance Eq.~(\\ref{eq:main_G_xy_eqn}), which is derived in the thermodynamic limit where the 2D slab thickness $L_{z} \\to \\infty$ with infinitesimal but non-zero $2|\\Delta|$~\\cite{dynamical_axion_insulator_BB}. \nThe ambiguity modulo $2e^{2}\/h$ in the change of Hall conductance is due to the axion angle $\\theta$, which is only well-defined mod $2\\pi$, as shown below in Eq.~(\\ref{eq:agnostic}). \nTaking $L_{z} \\to \\infty$ with infinitesimal $2|\\Delta|$ ensures that the only effect the CDW modulation has is to open the gap at the Weyl points without inverting bands at other high-symmetry points in the 3D Brillouin zone. \nWe have also verified that our choice for the parameters in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (a) is adiabatically connected to this condition by increasing $L_{z}$ and decreasing $2|\\Delta|$. \nThe slab Hall conductance $G_{xy}$ of the $xy$-slab contains both an extensive contribution from the bulk QAH phase through the weak Chern number $\\nu_{z}$, and an intensive contribution from axion angle $\\theta$, which collectively gives\\cite{vanderbiltaxion,2020_Axion_coupling_Vanderbilt}\n\\begin{align}\n G_{xy} = \\frac{e^2}{h}(\\nu_{z} l_{z} + \\theta \/ \\pi), \\label{eq:agnostic}\n\\end{align}\nwhere $l_{z}$ is the number of unit cells in the slab. \nIn our examples for $q = 1\/5$, $l_{z}$ will be given by $L_{z}\/5$. \nRecall also that as we slide $\\phi$ from $0$ to $\\pi$, the bulk gap of the 3D Weyl-CDW system never closes, hence the $\\nu_{z}$ is unchanged during the process. \nPutting this all together, we see that Eq.~(\\ref{eq:main_G_xy_eqn}) implies that there is a $\\pi$ mod $2\\pi$ change in the axion angle between $\\phi = 0$ and $\\phi = \\pi$. \nTo be more specific, in our current examples we have $G_{xy}(\\phi = 0) = -5 e^{2}\/h$ and $G_{xy}(\\phi = \\pi) = -4 e^{2}\/h$. \nThis quantized change of $G_{xy}$ or $\\theta$ can be explained again using the 4D low energy theory, as we now show. \n\n\nGoing back to the 4D low energy theory, Eq.~(\\ref{eq:4D_U1_SHO_H}) predicts that if we shift $k_{w}$ to $k_{w} + \\Delta k_{w}$, the corresponding CI layers described by the Hamiltonian in Eq.~(\\ref{eq:2D_CI_subspace})--which are localized around $z = -k_{w}\/(2\\pi q)$--will be shifted in the $z$ direction by $\\Delta z = -\\Delta k_{w} \/ (2\\pi q)$. \nConnecting this observation back to the physical 3D Weyl-CDW system, it implies that as we slide $\\phi$ from $0$ to $\\pi$, all the CI layers will be shifted by $\\Delta z = -\\pi \/ (2\\pi q) = -1\/(2q)$; for our choice of $q= 1\/5$ this gives a shift of $\\Delta z = -2.5$. \nWe demonstrate this numerically in Figs.~\\ref{fig:temporary_3d_weyl_cdw_fig} (d) and (e) which show the probability distribution of the $8$ QAH zero modes and the corresponding 4 non-trivial CI layers (with Chern number $C = -1$) for $\\phi = \\pi$. \nThe physical interpretation of Eq.~(\\ref{eq:main_G_xy_eqn}) is now clear: As we slide $\\phi$ from $0$ to $\\pi$, the non-trivial CI layers will be shifted by $\\Delta z = -2.5$ unit cells, all in the same direction. \nTherefore, the bottom non-trivial CI layer at $\\phi = 0$ and $z = -10$ depicted in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} (c) will be shifted outside the finite sample and hence will not appear when $\\phi=\\pi$. \nAt $\\phi = \\pi$, there will be only $4$ non-trivial CI layers remaining. \nThis leads to a change in the Hall conductance by $e^2\/h$, as indicated by Eq.~(\\ref{eq:main_G_xy_eqn}). \nSimultaneously, the number of QAH zero modes in the $y$-rod decreases by $2$ when we slide $\\phi$ from $0$ to $\\pi$. \nPhysically, these two QAH zero modes are pushed toward the boundary of the system, due to the shifting of the bottom non-trivial CI layer. \nTherefore, their energies will be pushed toward the bulk continuum, leading to the inevitable appearance of gap-crossing bands as shown in Fig~\\ref{fig:temporary_3d_weyl_cdw_fig} (a). \nNumerically, we have observed that in all of our examples (Figs.~\\ref{fig:temporary_3d_weyl_cdw_fig} and~\\ref{fig:temporary_3d_weyl_cdw_fig_2}), the zero modes in the band structure of the $y$-rod only appear at $k_{y} = 0$. \nTherefore, as far as the zero modes are concerned, we can focus on the energy spectrum of the $y$-rod at $k_{y}=0$, as in Figs.~\\ref{fig:temporary_3d_weyl_cdw_fig} (a) and \\ref{fig:temporary_3d_weyl_cdw_fig_2} (a). \nAnalytically, this can be understood from the Hamiltonian of the low energy Chern insulator Eq.~(\\ref{eq:2D_CI_subspace}) for each $n$ and $k_{w}$, which has zero energy edge modes only at $k_{y} = 0$\\cite{hasan2010colloquium,Qi_Zhang_RMP,bernevigbook,Jackiw_Rebbi}.\n\n\n\nTo summarize, we have shown that the identity Eq.~(\\ref{eq:main_G_xy_eqn}) can be regarded as a consequence of the $U(1)$ Landau quantization of a 4D nodal line system in which the localization centers along $z$ of the states are directly related to $k_{w}$.\nWe then identified $k_w$ with the sliding phase $\\phi$ through our dimensional promotion formalism in Sec.~\\ref{sec_Dimension_promotion}. \nThe change in conductance as a function of $\\phi$ can thus be regarded as a physical manifestation of the {\\it Chern number polarization}, which can alternatively be computed in terms of $z$-localized hybrid Wannier centers~\\cite{2020_Axion_coupling_Vanderbilt,zilberberg2018photonic,dynamical_axion_insulator_BB,yu2020dynamical,olsen2020gapless}.\n\n\\begin{figure*}[t]\n\\hspace{-0.5cm}\n\\includegraphics[width=\\linewidth]{Q_taupi_over_2_tx_1_ty_-1_tz_1_m_2_D_2_Lz_21.pdf}\n\\caption{(a) $\\phi$-sliding spectrum of the Weyl-CDW model in a $y$-rod geometry at $k_{y} = 0$ with size $L_{x}\\times L_{z} = 21 \\times 21$, $t_{x} = -t_{y} = t_{z} = 1$, $m = 2$, $2|\\Delta| = 2$ and $q = \\tau \/4$ where $\\tau = (1+\\sqrt{5})\/2$. \n(b) The average probability distribution of the 18 zero modes at $\\phi = 0$ in (a). \nThese zero modes correspond to QAH surface states. \n(c) The average probability distribution of the 9 non-trivial states at $\\vec{k} = \\Gamma$ of the $xy$-slab at $\\phi = 0$, which in total lead to $G_{xy}(\\phi = 0) = -9 e^{2}\/h$. \n(d) The average probability distribution of the 16 zero modes at $\\phi = \\pi$ in (a). \nThese zero modes correspond to QAH surface states. \n(e) The average probability distribution of the 8 non-trivial states at $\\vec{k} = \\Gamma$ of the $xy$-slab at $\\phi = \\pi$, which in total lead to $G_{xy}(\\phi = \\pi) = -8 e^{2}\/h$. \nThe darker (black) color in (b)--(e) implies higher probability density. \nIn (b) and (d), the $x$- and $z$-coordinate both range from $-10 ,\\ldots, +10$. \nIn (c) and (e), the $z$-coordinate ranges from $-10 ,\\ldots, +10$.}\n\\label{fig:temporary_3d_weyl_cdw_fig_2}\n\\end{figure*}\n\nHaving demonstrated the utility of our dimensional promotion formalism for a 3D Weyl-CDW system coupled to a commensurate CDW with $q = 1\/5$, we next explore the case of incommensurate modulations which are prevalent in nature\\cite{gruner1988dynamics}. \nIn particular, the experimentally intriguing Weyl-CDW (TaSe$_4$)$_2$I is incommensurate\\cite{shi2019charge,tasei_original,heeger_tasei,tournier2013electronic,zhang2020first,shi2019charge}. \nWe still consider $H_{0}+H_{CDW}(\\phi)$ with $t_{x}=-t_{y}=t_{z}=1$, $m=2$, $2|\\Delta|=2$. \nHowever, we now choose the modulation $q = \\tau \/4$ where $ \\tau = (1+\\sqrt{5})\/2$ is the golden ratio. \nFor an $xy$-slab we choose $L_{z} = 21$ and for $y$-rod we choose $L_{x} \\times L_{z} = 21\\times 21$. \nSince $q=\\tau\/4$ is an irrational number, the modulation $H_{CDW}(\\phi)$ is incommensurate with $H_{0}$. \nCrucially though, we can use our dimensional promotion procedure regardless of whether or not the modulation is commensurate with the underlying lattice. \nThe $U(1)$ gauge field to which the 4D nodal line system is coupled still takes the form in Eq.~(\\ref{eq:U1_4D}).\nThe main difference is that now, the 4D system has an irrational flux $2\\pi q = \\pi\\tau\/2$ per plaquette in the $zw$ plane. \nWe have verified that for the $xy$-slab we have $G_{xy}(\\phi = 0) = -9 e^{2}\/h$ and $G_{xy}(\\phi = \\pi) = -8 e^{2}\/h$, consistent with Eq.~(\\ref{eq:agnostic}). \nWe also show in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig_2} (a) the $\\phi$-sliding spectrum of the $y$-rod at $k_{y}=0$. \nWe see that there are $2$ fewer QAH zero modes at $\\phi = \\pi$ than at $\\phi = 0$. \nThe 18 and 16 QAH zero modes for $\\phi = 0$ and $\\phi = \\pi$ are shown in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig_2} (c) and (e), respectively. \nWe again identify 9 and 8 non-trivial states in the $xy$-slab at $\\vec{k} = \\Gamma$ for $\\phi = 0$ and $\\phi = \\pi$, and show their probability distributions in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig_2} (b) and (d), respectively. \nIn the SM\\cite{SM} we present the details of the numerical methods for identifying non-trivial states in the $xy$-slab. \nThe existence of non-zero QAH response and QAH zero modes can again be attributed to 4D $U(1)$ Landau quantization which gaps the 4D nodal line system, yielding a topologically non-trivial insulating ground state. \nIn particular, we also have $\\left| G_{xy}(\\phi = \\pi) - G_{xy}(\\phi = 0) \\right| = e^{2}\/h$ mod $2e^{2}\/h$. \nThis can again be understood from the shifting of non-trivial CI layers. \nIn this case, as $\\phi$ slides from $0$ to $\\pi$, all the non-trivial CI layers will be shifted downward by $\\Delta z = -\\pi \/ (2\\pi q) = -2 \/ \\tau \\approx -1.236$ lattice constants. \nThe non-trivial CI layer at the bottom ($z = -10$) of Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig_2} (b) will be shifted outside the finite size system and thus the absolute value of slab Hall conductance will be changed by $-1$. \nConsequently, the number of QAH zero modes in the $y$-rod at $k_{y}=0$ will be decreased by $2$. \nTherefore, together with the examples in Sec.~\\ref{sec_chiral_HOTI} and Sec.~\\ref{sec_helical_HOTI_sliding_modes}, we see that our dimensional promotion procedure provides a way to understand topological properties of systems with incommensurate modulations.\n\n\n\\subsection{Weyl-CDW and 4D Chern Number}\n\nWe can also understand the topological properties of the Weyl-CDW model from the perspective of 4D response theory. \nCombining the field strength in Eq.~(\\ref{eq:Fzw}) with our analysis of the Hall conductance above allows us to formulate a $(4+1)$D field-theoretical description of the QAH response in a 3D Weyl-CDW system. \nThe corresponding action is that of the $(4+1)$D Chern-Simon theory\\cite{qi2008topological,zhang2001four}\n\\begin{align}\n S = \\frac{C_{2}}{24 \\pi^{2}} \\sum_{\\mu\\nu\\lambda\\rho\\sigma}\\int d^{5}x \\epsilon^{\\mu \\nu \\lambda \\rho \\sigma} A_{\\mu} \\partial_{\\nu} A_{\\lambda} \\partial_{\\rho} A_{\\sigma}, \\label{eq:action}\n\\end{align}\nwhere $C_{2}$ is the second Chern number, $A_{\\mu}$ is the electromagnetic gauge potential and $\\epsilon^{\\mu \\nu \\lambda \\rho \\sigma}$ is the Levi-Civita symbol in $(4+1)$D.\nThe Greek indices here are taken to run over all $4+1$ dimensions. \nEq.~(\\ref{eq:action}) gives the electromagnetic response through\n\\begin{align}\n J^{\\mu} = \\frac{\\delta S}{\\delta A_{\\mu}} =\\frac{C_{2}}{32 \\pi^{2}} \\sum_{\\nu\\lambda\\rho\\sigma}\\epsilon^{\\mu \\nu \\lambda \\rho \\sigma} F_{\\nu \\lambda} F_{\\rho \\sigma},\n\\end{align}\nwhere $J^{\\mu}$ is the current density along the $\\mu$ direction. \nSince we have $F_{zw} = 2\\pi q$, an electric field $E^{y}$ along $y$ (implying $F_{ty} = E_{y}$) \nwill induce a Hall current density along $x$ through\n\\begin{align}\n J^{x} = q \\frac{C_{2}}{2\\pi} E^{y}. \\label{eq:Hall_current_x}\n\\end{align}\nIntegrating this along the $z$ direction, we find then that, with non-zero $C_{2}$, the Hall conductance $G_{xy}$ is proportional to $qL_z$. \nThis is consistent with Eq.~(\\ref{eq:agnostic}) and the the recent calculation\\cite{dynamical_axion_insulator_BB} showing that the Hall conductance $G_{xy}$ of a 3D Weyl-CDW system is given by \n\\begin{align}\n G_{xy} = \\left( |\\vec{Q}|L_{z}+2\\theta \\right) \\cdot e^{2} \/ (2\\pi h), \\label{eq:Eq_from_dynamical_axion_insulator_preprint}\n\\end{align}\nwhere $L_{z}$ is the thickness of the $xy$-slab, $\\vec{Q}$ is the CDW wave vector along $z$, which in our specific model system is $\\vec{Q} = 2\\pi q \\hat{z}$, and $\\theta$ is the bulk axion angle computed from the inversion-symmetric unit cell. \nAs we take the thermodynamic limit $L_{z} \\to \\infty$, the axion angle contribution to $G_{xy}$ becomes negligible and thus $G_{xy}$ can also be regarded as proportional to the magnitude of CDW wave vector, which is consistent with Eq.~(\\ref{eq:Hall_current_x}).\nTherefore, the field strength in Eq.~(\\ref{eq:Fzw}) indeed allows a sensible construction of higher dimensional continuum theory.\n\nTo see concretely that the 3D Weyl-CDW system indeed emulates a 4D system with non-zero $C_{2}$, we notice that for both examples in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} and Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig_2}, the system can be deformed into a limit where we have layers of decoupled Chern insulators localized along $z$. \nIn the decoupled-layer limit, for the commensurate case, for example, $q = 1\/5$, where we consider the single nontrivial band in each unit cell, this implies that $C_{2}$, which is defined through\\cite{2D_QC_4D_QHE,zilberberg2018photonic,ozawa2016synthetic,4D_QHE_ultracold_atom,qi2008topological}\n\\begin{align}\n C_{2} = \\frac{1}{4\\pi^{2}} \\int_{\\mathbb{T}^{4}} d^{4}k \\left( \\Omega_{xy}\\Omega_{zw} + \\Omega_{wx}\\Omega_{zy} + \\Omega_{zx} \\Omega_{yw} \\right)\n\\end{align}\nbecomes\n\\begin{align}\n C_{2} = \\frac{1}{4\\pi^{2}} \\left(\\int_{\\mathbb{T}^{2}} dk_{x}dk_{y} \\Omega_{xy}\\right)\\left(\\int_{\\mathbb{T}^{2}} dk_{z}dk_{w}\\Omega_{zw}\\right)\n\\end{align}\nin this limit, where $\\Omega_{\\mu \\nu}$ is the Abelian Berry curvature in the $k_{\\mu}$-$k_{\\nu}$ plane. \nFor both examples in Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} and Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig_2}, we have identified the weak Chern number $\\nu_{z} = -1$, implying that both systems have $\\frac{1}{2 \\pi} \\int_{\\mathbb{T}^{2}} dk_{x}dk_{y} \\Omega_{xy} = -1$. \nIn fact, for 3D Weyl-CDW system it have been shown that there will always be background QAH response in the $xy$ plane\\cite{dynamical_axion_insulator_BB}, implying that in the limit of decoupled Chern insulators we have $\\frac{1}{2 \\pi} \\int_{\\mathbb{T}^{2}} dk_{x}dk_{y} \\Omega_{xy} \\ne 0$. \nFurthermore, as we shift the CDW sliding phase $\\phi$, which is equivalent to shifting the momentum $k_{w}$, by $2\\pi$, all the Chern insulating layers will be shifted by $\\Delta z = -\\Delta k_{w}\/(2\\pi q) = -2\\pi \/(2\\pi q) = -1\/q$, implying a non-trivial Thouless charge pump along $z$. \nSpecifically, for Fig.~\\ref{fig:temporary_3d_weyl_cdw_fig} with $q = 1\/5$, all the Chern insulating layers will be shifted by $\\Delta z = -5$, which is equal to the unit cell length along $z$, implying $\\left| \\frac{1}{2 \\pi} \\int_{\\mathbb{T}^{2}} dk_{z}dk_{w}\\Omega_{zw} \\right|=1$. \nThe fact that the 3D Weyl-CDW system can be viewed as layers of Chern insulators\\cite{dynamical_axion_insulator_BB} and the expression $\\Delta z = -1\/q$ governing the charge pumping along $z$ as we vary $k_{w}$ by $2\\pi$ collectively predict a non-zero $\\frac{1}{2 \\pi} \\int_{\\mathbb{T}^{2}} dk_{z}dk_{w}\\Omega_{zw}$. \nTherefore, for a 3D Weyl-CDW system with QAH surface states\\cite{dynamical_axion_insulator_BB}, the corresponding 4D theory is described by a (4+1)D Chern-Simon theory in Eq.~(\\ref{eq:action}) with non-zero $C_{2}$. \nFurthermore, this result holds even as we deform away from the decoupled-layer limit, provided no energy gaps close. \nThus the 3D Weyl-CDW system serves as a platform to study higher-dimensional topological field theories.\n\n\nLet us conclude with two remarks. \nFirst, from the above analysis, we see that a 3D Weyl-CDW system with QAH surface states provides a platform to examine a 4D nodal line system gapped by a $U(1)$ gauge field and carries non-zero second Chern number $C_{2}$. \nSecondly, as opposed to Secs.~\\ref{sec_chiral_HOTI} and~\\ref{sec_helical_HOTI_sliding_modes} where we have in higher dimensions a gapped topological phase coupled to $U(1)$ or $SU(2)$ gauge fields, in the 4D model promoted from a 3D Weyl-CDW system it is precisely the coupling to a $U(1)$ gauge field that opens up a bulk gap, inducing emergent CI layers, QAH surface states and non-zero $C_{2}$.\n\n\n\\section{\\label{sec:outlook}Outlook}\n\n\nTo conclude, we have shown in Secs.~\\ref{sec_chiral_HOTI} and \\ref{sec_helical_HOTI_sliding_modes} that higher-order topology in 3D can be probed in 2D DW systems. \nFurthermore, we showed in Sec.~\\ref{sec:Weyl_CDW} how 3D Weyl-CDW systems with background QAH response can be used to study 4D topology. \nThe next and natural step is to identify 3D systems with modulations coexisting with hinge or corner modes. \nThis will be a platform for studying 4--or even higher--dimensional higher-order topology. \nOur dimensional promotion procedure in Sec.~\\ref{sec_Dimension_promotion} can also be used together with the topological classification based on crystalline symmetries\\cite{Po2017,khalaf2018higher,Chiu2016,NaturePaper} in the promoted dimensions, in order to explore topological crystalline phases in higher dimensions. \nWith suitably chosen modulated systems, we may either study (1) how topological crystalline insulators diagnosed by symmetry-based indicators\\cite{Po2017,comment,khalaf,ashvin-materials,po2020symmetry,watanabe2018structure} in the promoted dimensions respond to a background $U(1)$ or $SU(2)$ gauge fields, or (2) how topological semimetals\\cite{armitage2018weyl} in the promoted dimensions can be gapped by background $U(1)$ or $SU(2)$ gauge fields. \nWith the dimensional promotion procedure, we may also extend our studies of topological materials to those with space groups beyond 3D, known as superspace groups\\cite{superspace1,superspace2,superspace3,superspace4,superspace5}. \nTo extract the full information in higher dimension, a way to control the phase offset $\\{\\phi^{(i)} \\}$ experimentally is needed, and currently applying electromagnetic fields to depin the (charge- or spin-)density waves is one practical approach\\cite{gooth2019evidence,gruner1988dynamics}. \nIn addition, since we can systematically compute the background continuous gauge field coupled to the dimensionally-promoted model, we can again use low dimensional modulated systems to study the low energy dynamics in higher dimensions, by minimally coupling the low energy theory to the known continuous gauge fields as in Sec.~\\ref{sec:Weyl_CDW}. \nAs our dimensional promotion procedure can be carried out for both commensurate and incommensurate modulations, this approach can be used to study topological properties of system with quasi-periodic potentials~\\cite{Earliest_dimension_promotion_superspace,2D_QC_4D_QHE,zilberberg2018photonic,equivalence_Fibo_Harper_2012,Kraus_1D_QC_to_2D_QHE} where conventional band theory is not applicable. \nThe general procedure will be to promote the dimension of these quasi-periodic systems and examine the response of possible topological phase in higher dimensions to a gauge field producing an irrational flux per plaquette. \nThese techniques can be applied to analyze the DW phases in material systems of interest such as (TaSe$_4$)$_2$I\\cite{gooth2019evidence,shi2019charge,zhang2020first,tournier2013electronic} and ZrTe$_5$\\cite{tang2019three,qin2020theory,song2017instability,zhang2017transport}. \nThis can also lead to interesting studies on the higher-dimensional Hofstadter butterfly, complementing the recent studies of Refs.~\\onlinecite{Bernevig_fragile_LL,Herzog_Hof_topo}. \n\\textcolor{black}{Another interesting direction is to introduce dynamics to the DW modulation. This can happen, for example, when the phase offsets $\\{ \\phi^{(i)}\\}$ acquire non-adiabatic time-dependence and become $\\{ \\phi^{(i)}(t)\\}$. Previous studies have focused on promoting the dimension of a periodically-driven system to a Floquet lattice, which under certain conditions can lead to topologically-protected quantized energy pump~\\cite{time_periodic_1,time_periodic_2,Refael_Topo_freq_conversion_cavity_PRB}. We expect that richer phenomena in higher-dimensional space can be investigated when the DW modulations are not only periodic in real-space but also (1) periodic in time or (2) have general time-dependence.} Finally, we have shown in Sec.~\\ref{sec_helical_HOTI_sliding_modes} the simplest case of how $SU(2)$ gauge field physics may be studied through a 2D modulated system. \nRecently, the spin-orbit-coupled Hofstadter models induced by non-Abelian $SU(2)$ gauge fields have also been studied both in 2D\\cite{2D_SU2_butterfly_Joannopoulos} and 3D\\cite{3D_SU2_butterfly_Joannopoulos}, where Dirac points with up to 16-fold degeneracy and various topological insulating states were found. \nWe expect that 3D DW materials with different types of spin-orbit coupled modulations may enable simulation of various aspects of the physics of $SU(2)$ gauge fields in 4D or higher dimensions, including topological states and $SU(2)$ Hofstadter butterflies~\\cite{YiLi_SU2_Hofstadter,2D_SU2_butterfly_Joannopoulos,3D_SU2_butterfly_Joannopoulos}. \nWe hope that this work will lay the groundwork for the exciting future investigations mentioned above, and extend the search for exotic topological phases beyond 3D. \nIn particular, there are many possible defects that one can imagine in a spin-orbit coupled density wave order parameter, each of which may correspond to a non-trivial response to $SU(2)$ gauge field defects in the higher-dimensional system.\n\n\n\n\n\\begin{acknowledgments}\nThe authors would like to thank Y. Li and B. Wieder for fruitful discussions. \nThis work was supported by the Alfred P. Sloan Foundation, and by the National Science Foundation under grant DMR-1945058. \nNumerical computations made use of the Illinois Campus Cluster, a computing resource that is operated by the Illinois Campus Cluster Program (ICCP) in conjunction with the National Center for Supercomputing Applications (NCSA) and which is supported by funds from the University of Illinois at Urbana-Champaign. \nNumerical calculations in this work employed the open-source PythTB package\\cite{PythTB}.\n\\end{acknowledgments}\n\n\n\n\\section{\\label{sec:General_dimension_promotion}General dimensional promotion procedure}\n\nIn this section, we develop a general procedure to promote a $d$-dimensional ($d$D) modulated system to a $(d+N)$D lattice model coupled to a $U(1)$ gauge field. \nGeneralizing Sec.~III in the main text, here we allow for both on-site energy and hopping modulations, general (potentially non-cubic) lattice structures in the higher-dimensional model, as well as arbitrary orbital positions. \nSec.~\\ref{subsec:formalism} is devoted to the development of the method. \nSpecific examples are given in Sec.~\\ref{subsec:examples_general_dim_promotion}.\n\n\n\\subsection{\\label{subsec:formalism}Formalism}\n\n\\subsubsection{\\label{subsubsec:__dimensional_promotion}Dimensional promotion}\nSuppose we have a $d$D periodic electronic system with lattice vectors $\\{\\vec{a}_{1},\\cdots,\\vec{a}_{d} \\}$, which form a general $d$D Bravais lattice. \nNote that the $\\vec{a}_i$ need not be mutually orthogonal. \nWe add $N$ mutually incommensurate on-site and hopping modulations with modulation wave vectors $\\{\\vec{q}^{(1)},\\cdots,\\vec{q}^{(N)}\\}$, such that the Hamiltonian is given by\n\\begin{equation}\n H_{\\text{low-dim}} = \\sum_{\\vec{n},\\vec{m}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m}} \\left[H_{\\vec{m}}\\right] {\\psi}_{\\vec{n}} + {\\sum_{\\vec{n},\\vec{l}}} \\sum_{i=1}^{N} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l}} \\left[H_{\\vec{l},\\vec{n}}^{(i)} \\right] {\\psi}_{\\vec{n}} + \\sum_{\\vec{n}} \\sum_{i=1}^{N}{\\psi}^{\\dagger}_{\\vec{n}} \\left[ V^{(i)}_{\\vec{n}} \\right] {\\psi}_{\\vec{n}}. \\label{eq:H_low_general}\n\\end{equation}\nHere $\\vec{n} = (n_{1},\\cdots,n_{d})$, $\\vec{m} = (m_{1},\\cdots,m_{d})$ and $\\vec{l} = (l_{1},\\cdots,l_{d})\\in \\mathbb{Z}^{d}$ index lattice positions in reduced coordinates. \nFor example, the vector $\\vec{n}$ uniquely denotes the lattice site $\\sum_{j=1}^d n_j\\vec{a}_j$. \nThe multi-component operator ${\\psi}^{\\dagger}_{\\vec{n}}$ creates an electron at site $\\vec{n}$ with a given set of spin and orbital degrees of freedom.\nThe matrix $\\left[H_{\\vec{m}}\\right]$ is the unmodulated hopping matrix connecting lattice position $\\vec{n}$ to $\\vec{n}+\\vec{m}$, with matrix indices encoding the spin and orbital dependence of the hopping. \nSimilarly, $\\left[H_{\\vec{l},\\vec{n}}^{(i)} \\right]$ is the $i^{\\text{th}}$ ($i = 1 ,\\ldots, N$) modulated hopping matrix connecting lattice position $\\vec{n}$ to $\\vec{n}$+$\\vec{l}$, and $\\left[ V^{(i)}_{\\vec{n}} \\right]$ is the $i^{\\text{th}}$ ($i = 1 ,\\ldots, N$) modulated on-site energy at lattice position $\\vec{n}$. \n\nEach $\\vec{q}^{(i)}$ is associated with a set of modulated couplings $\\left[V^{(i)}_{\\vec{n}} \\right]$ and $\\left[ H^{(i)}_{\\vec{l},\\vec{n}} \\right]$. \nNote it is possible that for a given $\\vec{q}^{(i)}$, $\\left[ H^{(i)}_{\\vec{l},\\vec{n}} \\right]$ is only non-zero for some $\\vec{l}$.\nHere we sum over all $\\vec{l}$, and set $\\left[ H^{(i)}_{\\vec{l},\\vec{n}} \\right] =0$ if for a given $\\vec{q}^{(i)}$ there is no modulated hopping matrix with the given $\\vec{l}$. \nIn most practical cases of interest, $\\vec{l}$ will be restricted to a nearest-neighbor hopping. \nNote that in the situations considered in the main text we took $\\left[ H^{(i)}_{\\vec{l},\\vec{n}} \\right]=0$ for all $\\vec{l}$. \nAnalogous considerations let us analyze systems with modulated hoppings but no modulated on-site potentials by taking $[V_{\\vec{n}}^{(i)}]=0$.\n\nWe require $H_{\\text{low-dim}}$ to be Hermitian, such that\n\\begin{align}\n & \\left[ H_{-\\vec{m}} \\right]^{\\dagger} = \\left[H_{\\vec{m}} \\right], \\label{eq:Hermitian_1} \\\\\n & \\left[ H^{(i)}_{-\\vec{l},\\vec{n}+\\vec{l}} \\right]^{\\dagger} = \\left[ H^{(i)}_{\\vec{l},\\vec{n}} \\right], \\label{eq:Hermitian_2} \\\\\n & \\left[ V^{(i)}_{\\vec{n}} \\right]^{\\dagger} = \\left[ V^{(i)}_{\\vec{n}} \\right].\\label{eq:Hermitian_3} \n\\end{align}\nEq.~(\\ref{eq:Hermitian_2}) is another statement that the hopping process from $\\vec{n}$ to $\\vec{n}+\\vec{l}$ is the conjugate of the hopping process from $\\vec{n}+\\vec{l}$ to $\\vec{n}$. \nWe assume that the modulations are periodic with functional dependence\n\\begin{align}\n & \\left[ H_{\\vec{l},\\vec{n}}^{(i)} \\right] = \\left[ g^{(i)}_{\\vec{l}} \\left( 2\\pi \\vec{q}^{(i)} \\cdot \\left( n_{1}\\vec{a}_{1}+\\cdots+n_{d}\\vec{a}_{d} \\right) + \\phi^{(i)} \\right)\\right], \\label{eq:H_form_1} \\\\\n & \\left[ V^{(i)}_{\\vec{n}} \\right] = \\left[f^{(i)} \\left( 2\\pi \\vec{q}^{(i)} \\cdot \\left( n_{1}\\vec{a}_{1}+\\cdots+n_{d}\\vec{a}_{d} \\right) + \\phi^{(i)} \\right)\\right], \\label{eq:H_form_2}\n\\end{align}\nsuch that\n\\begin{align}\n & \\left[ g^{(i)}_{\\vec{l}} (x)\\right] = \\left[ g^{(i)}_{\\vec{l}} (x + 2\\pi)\\right], \\\\\n & \\left[f^{(i)} (x)\\right] = \\left[f^{(i)} (x+2\\pi)\\right].\n\\end{align}\nTherefore, we can expand $\\left[ H_{\\vec{l},\\vec{n}}^{(i)} \\right]$ and $\\left[ V^{(i)}_{\\vec{n}} \\right]$ in terms of Fourier components,\n\\begin{align}\n & \\left[ H_{\\vec{l},\\vec{n}}^{(i)} \\right] = \\sum_{p } \\left[ H_{\\vec{l},p}^{(i)} \\right] e^{i p \\left( 2\\pi \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) + \\phi^{(i)} \\right) }, \\label{eq:general_FT_1} \\\\\n & \\left[ V^{(i)}_{\\vec{n}} \\right] = \\sum_{p } \\left[ V^{(i)}_{p} \\right] e^{i p \\left( 2\\pi \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) + \\phi^{(i)} \\right) }, \\label{eq:general_FT_2}\n\\end{align}\nwhere $p \\in \\mathbb{Z}$. The matrices $\\left[ H_{\\vec{l},p}^{(i)} \\right]$ and $\\left[ V^{(i)}_{p} \\right]$ are the $p^{\\text{th}}$ (matrix-valued) Fourier components of $\\left[ H_{\\vec{l},\\vec{n}}^{(i)} \\right]$ and $\\left[ V^{(i)}_{\\vec{n}} \\right]$, respectively. \nEqs.~(\\ref{eq:Hermitian_2}) and (\\ref{eq:Hermitian_3}) also imply\n\\begin{align}\n & \\left[ H^{(i)}_{-\\vec{l},-p} \\right]^{\\dagger} e^{ip 2 \\pi \\vec{q}^{(i)} \\cdot \\sum_{j=1}^{d}l_{j}\\vec{a}_{j}} = \\left[ H^{(i)}_{\\vec{l},p} \\right], \\label{eq:Hermitian_4}\\\\\n & \\left[ V^{(i)}_{-p} \\right]^{\\dagger} = \\left[ V^{(i)}_{p} \\right]. \\label{eq:Hermitian_5}\n\\end{align}\n\nAlthough Eq.~(\\ref{eq:Hermitian_4}) involves several different indices, it will be crucial in forming our dimensionally-promoted Hamiltonian. \nAs such, it is illuminating to prove it as follows. \nStarting from Eq.~(\\ref{eq:general_FT_1}), we can obtain\n\\begin{align}\n \\left[ H_{-\\vec{l},\\vec{n}}^{(i)} \\right] = \\sum_{p } \\left[ H_{-\\vec{l},p}^{(i)} \\right] e^{i p \\left( 2\\pi \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) + \\phi^{(i)} \\right) }\n\\end{align}\nby changing $\\vec{l} \\to -\\vec{l}$. \nWe next shift $\\vec{n} \\to \\vec{n} + \\vec{l}$ such that\n\\begin{align}\n \\left[ H_{-\\vec{l},\\vec{n}+\\vec{l}}^{(i)} \\right] = \\sum_{p } \\left[ H_{-\\vec{l},p}^{(i)} \\right] e^{i p \\left( 2\\pi \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) + \\phi^{(i)} \\right) } e^{ip 2\\pi \\vec{q}^{(i)} \\cdot \\sum_{j=1}^{d}l_{j}\\vec{a}_{j}}. \\label{eq:general_before_Hermitian_1}\n\\end{align}\nTaking Hermitian conjugate of Eq.~(\\ref{eq:general_before_Hermitian_1}) and setting $p \\to -p$, we have\n\\begin{align}\n \\left[ H_{-\\vec{l},\\vec{n}+\\vec{l}}^{(i)} \\right]^{\\dagger} = \\sum_{p } \\left[ H_{-\\vec{l},-p}^{(i)} \\right]^{\\dagger} e^{i p \\left( 2\\pi \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) + \\phi^{(i)} \\right) } e^{ip 2\\pi \\vec{q}^{(i)} \\cdot \\sum_{j=1}^{d}l_{j}\\vec{a}_{j}}. \\label{eq:pf_1}\n\\end{align}\nIn order for Eq.~(\\ref{eq:pf_1}) being consistent with the Hermiticity constraint Eq.~(\\ref{eq:Hermitian_2}), we deduce that Eq.~(\\ref{eq:Hermitian_4}) must hold.\n\nTo continue with our development of the dimensional promotion procedure, we plug Eq.~(\\ref{eq:general_FT_1}) and Eq.~(\\ref{eq:general_FT_2}) into Eq.~(\\ref{eq:H_low_general}) such that Hamiltonian of the $d$D modulated system can be written as\n\\begin{align}\n H_{\\text{low-dim}} & = \\sum_{\\vec{n},\\vec{m}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m}} \\left[ H_{\\vec{m}}\\right] {\\psi}_{\\vec{n}} \\nonumber \\\\\n & + {\\sum_{\\vec{n},\\vec{l}}} \\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l}} \\left[H_{\\vec{l},p_{i}}^{(i)}\\right] e^{i p_{i} \\left( 2\\pi \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) + \\phi^{(i)} \\right) } {\\psi}_{\\vec{n}} \\nonumber \\\\\n & + \\sum_{\\vec{n}} \\sum_{i=1}^{N} \\sum_{p_{i}}{\\psi}^{\\dagger}_{\\vec{n}} \\left[ V^{(i)}_{p_{i}} \\right] e^{i p_{i} \\left( 2\\pi \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) + \\phi^{(i)} \\right) } {\\psi}_{\\vec{n}}. \\label{eq:H_low_general_2}\n\\end{align}\nNotice that for each $i$ (which indexes the modulation wave vectors) we sum over all $p_{i} \\in \\mathbb{Z}$. \nTo promote Eq.~(\\ref{eq:H_low_general_2}) to a $(d+N)$D space, let us introduce a set of additional lattice vectors $\\{\\vec{c}_{1},\\cdots,\\vec{c}_{N} \\}$ such that, together with $\\{\\vec{a}_{1},\\cdots,\\vec{a}_{d} \\}$, we have a linearly independent basis spanning a $(d+N)$D lattice. \nNotice that this $(d+N)$D lattice is not necessarily orthorhombic, since we do not require the lattice vectors $\\{\\vec{a}_{1},\\cdots,\\vec{a}_{d} \\}$ and $\\{\\vec{c}_{1},\\cdots,\\vec{c}_{N} \\}$ be pairwise orthogonal. \nNext, we introduce the corresponding reciprocal lattice vectors $\\{\\vec{g}_{1},\\cdots,\\vec{g}_{d}\\}$ and $\\{\\vec{G}_{1},\\cdots,\\vec{G}_{N}\\}$ in the $(d+N)$D reciprocal space such that\n\\begin{align}\n & \\vec{g}_{i} \\cdot \\vec{a}_{j} = 2 \\pi \\delta_{ij},\\ i = 1 ,\\ldots, d,\\ {\\text{and }}j = 1 ,\\ldots, d, \\label{eq:reciprocal_1} \\\\\n & \\vec{g}_{i} \\cdot \\vec{c}_{j} = 0,\\ i = 1 ,\\ldots, d,\\ {\\text{and }}j = 1 ,\\ldots, N, \\label{eq:reciprocal_2} \\\\\n & \\vec{G}_{i} \\cdot \\vec{a}_{j} = 0,\\ i = 1 ,\\ldots, N,\\ {\\text{and }}j = 1 ,\\ldots, d, \\label{eq:reciprocal_3} \\\\\n & \\vec{G}_{i} \\cdot \\vec{c}_{j} = 2\\pi \\delta_{ij},\\ i = 1 ,\\ldots, N,\\ {\\text{and }}j = 1 ,\\ldots, N. \\label{eq:reciprocal_4}\n\\end{align}\nWe can now identify $\\phi^{(i)} \\in [0,2\\pi)$ with $\\vec{k} \\cdot \\vec{c}_{i}$, which is $2\\pi$ times the coefficient of $\\vec{k}$ along $\\vec{G}_{i}$ (see Eq.~(\\ref{eq:reciprocal_4})); the periodicity of $\\phi^{(i)}$ is reflected in the periodicity of the $(d+N)$D Brillouin zone. \nWe then promote the $d$D model to a $(d+N)$D space by summing over $\\{ \\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N} \\} \\in \\mathbb{T}^{N}$, where $\\mathbb{T}^N$ denotes the $N$-torus represented as a parallelepiped with boundary spanned by $\\{\\vec{G}_{1},\\cdots,\\vec{G}_{N} \\}$, and opposite edges identified. \nWe further label the original creation and annihilation operators by additional symbols $\\{ \\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N} \\} \\in \\mathbb{T}^{N}$, such that\n\\begin{align}\n H_{\\text{high-dim}} =&\\sum_{\\vec{k}\\cdot\\vec{c}_{1},\\cdots,\\vec{k}\\cdot\\vec{c}_{N}} \\sum_{\\vec{n},\\vec{m}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}} \\left[H_{\\vec{m}}\\right] \\psi_{\\vec{n},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}} \\nonumber \\\\\n & + \\sum_{\\vec{k}\\cdot\\vec{c}_{1},\\cdots,\\vec{k}\\cdot\\vec{c}_{N}} {\\sum_{\\vec{n},\\vec{l}}} \\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}} \\left[H_{\\vec{l},p_{i}}^{(i)}\\right] e^{ip_{i}\\vec{k}\\cdot\\vec{c}_{i}} e^{i p_{i} 2\\pi \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) } {\\psi}_{\\vec{n},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}} \\nonumber \\\\\n & + \\sum_{\\vec{k}\\cdot\\vec{c}_{1},\\cdots,\\vec{k}\\cdot\\vec{c}_{N}}\\sum_{\\vec{n}} \\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}} \\left[V^{(i)}_{p_{i}}\\right] e^{ip_{i}\\vec{k}\\cdot\\vec{c}_{i}} e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right)}{\\psi}_{\\vec{n},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}}, \\label{eq:non_ortho_3}\n\\end{align}\nwhere the summation of each $\\vec{k} \\cdot \\vec{c}_{i}$ is from $0$ to $2\\pi$. \nNotice that the summation over $\\{\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k} \\cdot \\vec{c}_{N} \\}$ is outside of $\\sum_{i=1}^{N} \\sum_{p_{i}} $ over all integer $p_{i}$ for each modulation $i = 1 ,\\ldots, N$. \nThis means that we are summing over $\\{\\vec{k}\\cdot \\vec{c}_{1},\\cdots, \\vec{k}\\cdot\\vec{c}_{N} \\}$ for each term in Eq.~(\\ref{eq:H_low_general_2}). \nAs mentioned in the main text, the physical motivation to sum over $\\{\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot\\vec{c}_{N} \\}$ is because a single set of $\\{\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot\\vec{c}_{N} \\}$ does not contain the full information of the lattice model in the promoted $(d+N)$D space. \nOnly when we consider all values of $\\{\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot\\vec{c}_{N} \\}$, can we obtain the exact form of the promoted lattice model, shown in Eq.~(\\ref{eq:non_ortho_4}) below. \nWe remind the readers that repeated indices are not implicitly summed.\n\nNote that ${\\psi}^{\\dagger}_{\\vec{n},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}}$ is the Fourier transform of $\\psi^{\\dagger}_{\\vec{n},\\vec{\\nu}}$ through\n\\begin{align}\n {\\psi}^{\\dagger}_{\\vec{n},\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N}} = \\frac{1}{\\sqrt{L}} \\sum_{\\vec{\\nu}} e^{i \\vec{k}\\cdot\\left( \\nu_{1}\\vec{c}_{1}+ \\cdots + \\nu_{N}\\vec{c}_{N} \\right)} \\psi^{\\dagger}_{\\vec{n},\\vec{\\nu}}, \\label{eq:FT_general}\n\\end{align}\nwhere $\\vec{\\nu} = (\\nu_{1},\\cdots,\\nu_{N}) \\in \\mathbb{Z}^{N}$ and $L$ is the size of the system in the additional dimensions (taken to infinity at the end of the calculation). We can thus take an inverse Fourier transform to find\n\\begin{align}\n H_{\\text{high-dim}} =& \\sum_{\\vec{n},\\vec{m},\\vec{\\nu}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{\\nu}} \\left[ H_{\\vec{m}}\\right] \\psi_{\\vec{n},\\vec{\\nu}} \\nonumber \\\\\n & + {\\sum_{\\vec{n},\\vec{l},\\vec{\\nu}}} \\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}} \\left[H_{\\vec{l},p_{i}}^{(i)}\\right] e^{i 2\\pi p_{i} \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) } {\\psi}_{\\vec{n},\\vec{\\nu}} \\nonumber \\\\\n & + \\sum_{\\vec{n},\\vec{\\nu}}\\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}} \\left[V^{(i)}_{p_{i}}\\right] e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right)}{\\psi}_{\\vec{n},\\vec{\\nu}}, \\label{eq:non_ortho_4}\n\\end{align}\nwhere ${\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}}$ denotes the electron creation operator for an electron at the lattice position $\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j}$, and ${\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{\\nu} - p_{i}\\hat{\\nu}_{i}}$ denotes the electron creation operator for an electron at the lattice position $\\sum_{j=1}^{d}(n_{j} + l_{j})\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} - p_{i} \\vec{c}_{i}$. \nNotice that a $d$D modulated system with phase offsets $\\{\\phi^{(1)},\\cdots,\\phi^{(N)}\\}$ is described by the Bloch Hamiltonian (see Eq.~(\\ref{eq:non_ortho_3})) of the promoted $(d+N)$D lattice model with fixed crystal momenta $\\{\\vec{k}\\cdot \\vec{c}_{1},\\cdots,\\vec{k}\\cdot \\vec{c}_{N} \\}$ via our identification of $\\vec{k} \\cdot \\vec{c}_{i}$ with $\\phi^{(i)}$. \nFor later convenience, we can also rewrite Eq.~(\\ref{eq:non_ortho_4}) as\n\\begin{align}\n H_{\\text{high-dim}} =& \\sum_{\\vec{n},\\vec{m},\\vec{\\nu}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{\\nu}} \\left[H_{\\vec{m}}\\right] \\psi_{\\vec{n},\\vec{\\nu}} \\label{eq:non_ortho_5_1} \\\\\n & + {\\sum_{\\vec{n},\\vec{l},\\vec{\\nu}}} \\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}} \\left[H_{\\vec{l},p_{i}}^{(i)}\\right]e^{-i \\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d} l_{j}\\vec{a}_{j} \\right) }e^{i 2\\pi p_{i} \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}\\left( n_{j} + \\frac{1}{2}l_{j} \\right)\\vec{a}_{j} \\right) } {\\psi}_{\\vec{n},\\vec{\\nu}} \\label{eq:non_ortho_5_2} \\\\\n & + \\sum_{\\vec{n},\\vec{\\nu}}\\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}} \\left[V^{(i)}_{p_{i}}\\right] e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right)}{\\psi}_{\\vec{n},\\vec{\\nu}}, \\label{eq:non_ortho_5_3}\n\\end{align}\nwhere we have used\n\\begin{align}\n \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} = \\sum_{j=1}^{d}\\left( n_{j} + \\frac{1}{2}l_{j} \\right)\\vec{a}_{j} - \\sum_{j=1}^{d} \\frac{1}{2}l_{j} \\vec{a}_{j}.\n\\end{align}\nThis will prove useful when we go to identify the gauge fields appearing in the hopping matrix elements. \nWe remind the readers again that the product $p_{i}\\vec{q}^{(i)}$ in Eqs.~(\\ref{eq:non_ortho_5_1}--\\ref{eq:non_ortho_5_3}) does not imply a summation over $i$, rather it denotes the product of integer $p_{i}$ and the $i^{th}$ modulation wave vector $\\vec{q}^{(i)}$. \nEqs.~(\\ref{eq:non_ortho_5_1}--\\ref{eq:non_ortho_5_3}) can then be interpreted as a $(d+N)$D lattice model with Hamiltonian\n\\begin{align}\n H_{\\text{high-dim}} =& \\sum_{\\vec{n},\\vec{m},\\vec{\\nu}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{\\nu}} \\left[H_{\\vec{m}}\\right] \\psi_{\\vec{n},\\vec{\\nu}} \\label{eq:non_ortho_6_1} \\\\\n & + {\\sum_{\\vec{n},\\vec{l},\\vec{\\nu}}} \\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}} \\left[H_{\\vec{l},p_{i}}^{(i)}\\right]e^{-i \\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d} l_{j}\\vec{a}_{j} \\right) }{\\psi}_{\\vec{n},\\vec{\\nu}} \\label{eq:non_ortho_6_2} \\\\\n & + \\sum_{\\vec{n},\\vec{\\nu}}\\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i}} \\left[V^{(i)}_{p_{i}}\\right] {\\psi}_{\\vec{n},\\vec{\\nu}}, \\label{eq:non_ortho_6_3}\n\\end{align}\ncoupled through a Peierls substitution\\cite{Peierls_substitution} to a $U(1)$ gauge field \n\\begin{align}\n \\vec{A} = \\frac{1}{2\\pi} \\sum_{i=1}^{N} \\sum_{j=1}^{d} \\vec{G}_{i} \\left( \\vec{q}^{(i)}\\cdot \\left[ \\left( \\vec{r} \\cdot \\vec{g}_{j} \\right) \\vec{a}_{j} \\right] \\right), \\label{eq:non_ortho_A}\n\\end{align}\nwhere $\\vec{r} \\in \\mathbb{R}^{d+N}$. \nNotice that Eqs.~(\\ref{eq:non_ortho_6_1}--\\ref{eq:non_ortho_6_3}) represent the $(d+N)$D model {\\it without} $U(1)$ gauge fields.\nAlthough there are complex phases $e^{-i \\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d} l_{j}\\vec{a}_{j} \\right) }$ in Eq.~(\\ref{eq:non_ortho_6_2}), they do not depend on the reduced coordinates $(\\vec{n},\\vec{\\nu}) \\in \\mathbb{Z}^{d+N}$ in the dimensionally-promoted lattice and so may be regarded as inherent phase factors in the $(d+N)$D model without $U(1)$ gauge field. \nTo validate the identification Eq.~(\\ref{eq:non_ortho_A}), we compute the various Peierls phase factors in the next section. \n\n\\subsubsection{\\label{sec:computing_Peierls_phase_1}Computation of Peierls phases}\n\nLet us consider the terms in Eq.~(\\ref{eq:non_ortho_6_3}) where the electrons hop from \n\\begin{equation}\n\\vec{r}_{i} = \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j}\n\\end{equation}\nto \n\\begin{equation}\n \\vec{r}_{f} =\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} - p_{i} \\vec{c}_{i}. \n\\end{equation}\nThe Peierls phase can be computed from a straight line integral from $\\vec{r}_{i}$ to $\\vec{r}_{f} \\in$ $\\mathbb{R}^{d+N}$ through\n\\begin{align}\n \\exp{-i \\int_{\\vec{r}_{i}}^{\\vec{r}_{f}} d\\vec{r} \\cdot \\vec{A}},\n\\end{align}\nwhere we have worked in unit $\\hbar = c= |e| = 1$ and the electron has charge $-|e| = -1$. \nThe straight line connecting $\\vec{r}_{i}$ to $\\vec{r}_{f}$ is\n\\begin{align}\n \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} - p_{i} \\vec{c}_{i}t,\n\\end{align}\nwhere $ t\\in [0,1]$ and the corresponding infinitesimal displacement vector is $-p_{i}\\vec{c}_{i}dt$. \nThe line integral can then be computed as follows:\n\\begin{align}\n \\exp{-i \\int_{\\vec{r}_{i}}^{\\vec{r}_{f}} d\\vec{r} \\cdot \\vec{A}} & = \\exp{i \\int_{0}^{1}dt p_{i} \\vec{c}_{i} \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\left( \\sum_{l=1}^{d}n_{l}\\vec{a}_{l} + \\sum_{l=1}^{N}\\nu_{l}\\vec{c}_{l} - p_{i} \\vec{c}_{i}t \\right) \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_1_1} \\\\\n & = \\exp{i \\int_{0}^{1}dt p_{i} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\delta_{ij} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\left( \\sum_{l=1}^{d}n_{l}\\vec{a}_{l} \\right) \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_1_2} \\\\\n & = \\exp{i 2\\pi \\int_{0}^{1}dt p_{i} \\sum_{k=1}^{d} \\left( \\vec{q}^{(i)}\\cdot \\left[ \\sum_{l=1}^{d}n_{l}\\delta_{lk} \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_1_3} \\\\\n & = \\exp{i 2\\pi \\int_{0}^{1}dt p_{i} \\left( \\vec{q}^{(i)}\\cdot \\sum_{k=1}^{d} n_{k} \\vec{a}_{k} \\right) } \\label{eq:Peierls_1_4} \\\\\n & = \\exp{i 2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\sum_{k=1}^{d} n_{k} \\vec{a}_{k} }. \\label{eq:Peierls_1_5}\n\\end{align}\nWe used Eq.~(\\ref{eq:reciprocal_2}) and Eq.~(\\ref{eq:reciprocal_4}) in going from Eq.~(\\ref{eq:Peierls_1_1}) to Eq.~(\\ref{eq:Peierls_1_2}), Eq.~(\\ref{eq:reciprocal_1}) in going from Eq.~(\\ref{eq:Peierls_1_2}) to Eq.~(\\ref{eq:Peierls_1_3}) and then finally do an integral $\\int_{0}^{1}dt=1$ to obtain Eq.~(\\ref{eq:Peierls_1_5}). \nEq.~(\\ref{eq:Peierls_1_5}) is exactly the additional phase factor in Eq.~(\\ref{eq:non_ortho_5_3}) compared to Eq.~(\\ref{eq:non_ortho_6_3}). \n\n\nNext, consider Eq.~(\\ref{eq:non_ortho_6_2}) where the electrons hop from \n\\begin{equation}\n\\vec{r}_{i} = \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j}\n\\end{equation}\nto \n\\begin{equation}\n\\vec{r}_{f} =\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\sum_{j=1}^{d}l_{j}\\vec{a}_{j} - p_{i} \\vec{c}_{i}.\n\\end{equation} \nThe straight line connecting $\\vec{r}_{i}$ to $\\vec{r}_{f}$ is\n\\begin{align}\n \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\left( \\sum_{j=1}^{d}l_{j}\\vec{a}_{j} - p_{i} \\vec{c}_{i} \\right) t,\n\\end{align}\nwhere $ t= [0,1]$ and the corresponding infinitesimal displacement vector is $\\left( \\sum_{j=1}^{d}l_{j}\\vec{a}_{j} - p_{i} \\vec{c}_{i} \\right) dt$. \nThe line integral can then be computed as follows:\n\\begin{align}\n & \\exp{-i \\int_{\\vec{r}_{i}}^{\\vec{r}_{f}} d\\vec{r} \\cdot \\vec{A}} \\\\\n & = \\exp{-i \\int_{0}^{1}dt \\left( \\sum_{r=1}^{d}l_{r}\\vec{a}_{r} - p_{i} \\vec{c}_{i} \\right) \\cdot \\frac{1}{2\\pi} \\sum_{j,k} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\left( \\sum_{r=1}^{d}n_{r}\\vec{a}_{r} + \\sum_{r=1}^{N}\\nu_{r}\\vec{c}_{r} + \\left( \\sum_{r=1}^{d}l_{r}\\vec{a}_{r} - p_{i} \\vec{c}_{i} \\right) t \\right) \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_1} \\\\\n & = \\exp{i \\int_{0}^{1}dt p_{i} \\vec{c}_{i} \\cdot \\frac{1}{2\\pi} \\sum_{j,k} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\left( \\sum_{r=1}^{d}\\left( n_{r} + l_{r}t \\right)\\vec{a}_{r} \\right) \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_2} \\\\\n & = \\exp{i 2\\pi \\int_{0}^{1}dt p_{i} \\sum_{j,k} \\delta_{ij} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\sum_{r=1}^{d}\\left( n_{r} + l_{r}t \\right)\\delta_{rk} \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_3} \\\\\n & = \\exp{i 2\\pi \\int_{0}^{1}dt p_{i} \\sum_{k=1}^{d} \\left( \\vec{q}^{(i)}\\cdot \\left[ \\left( n_{k} + l_{k}t \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_4} \\\\\n & = \\exp{i 2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \\sum_{k=1}^{d} \\left( n_{k} + \\frac{1}{2}l_{k} \\right) \\vec{a}_{k} \\right) }. \\label{eq:Peierls_2_5}\n\\end{align}\nWe used Eq.~(\\ref{eq:reciprocal_2}) and Eq.~(\\ref{eq:reciprocal_3}) in going from Eq.~(\\ref{eq:Peierls_2_1}) to Eq.~(\\ref{eq:Peierls_2_2}), Eq.~(\\ref{eq:reciprocal_1}) and Eq.~(\\ref{eq:reciprocal_4}) in going from Eq.~(\\ref{eq:Peierls_2_2}) to Eq.~(\\ref{eq:Peierls_2_3}) and then finally do integrals $\\int_{0}^{1}dt=1$ and $\\int_{0}^{1}tdt=\\frac{1}{2}$ to obtain Eq.~(\\ref{eq:Peierls_2_5}). \nEq.~(\\ref{eq:Peierls_2_5}) is exactly the additional phase factor in Eq.~(\\ref{eq:non_ortho_5_2}) compared to Eq.~(\\ref{eq:non_ortho_6_2}). \nCrucially, we see that our redefinition Eq.~(\\ref{eq:non_ortho_A}) accounts for the factor of $1\/2$ arising in the line integral of the vector potential.\n\n\nFinally, consider Eq.~(\\ref{eq:non_ortho_6_1}) where the electrons hop from \n\\begin{equation}\n\\vec{r}_{i} = \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j}\n\\end{equation}\nto\n\\begin{equation} \n \\vec{r}_{f} =\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\sum_{j=1}^{d}m_{j}\\vec{a}_{j}.\n\\end{equation}\nThe straight line connecting $\\vec{r}_{i}$ to $\\vec{r}_{f}$ is\n\\begin{align}\n \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\left( \\sum_{j=1}^{d}m_{j}\\vec{a}_{j} \\right) t,\n\\end{align}\nwhere $ t= [0, 1]$ and the corresponding infinitesimal displacement vector is $\\left( \\sum_{j=1}^{d}m_{j}\\vec{a}_{j} \\right) dt$. \nThe line integral can then be computed as follows:\n\\begin{align}\n & \\exp{-i \\int_{\\vec{r}_{i}}^{\\vec{r}_{f}} d\\vec{r} \\cdot \\vec{A}} \\\\\n & = \\exp{-i \\int_{0}^{1}dt \\left( \\sum_{r=1}^{d}m_{r}\\vec{a}_{r} \\right) \\cdot \\frac{1}{2\\pi} \\sum_{j,k} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\left( \\sum_{r=1}^{d}n_{r}\\vec{a}_{r} + \\sum_{r=1}^{N}\\nu_{r}\\vec{c}_{r} + \\left( \\sum_{r=1}^{d}m_{r}\\vec{a}_{r} \\right) t \\right) \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_3_1} \\\\\n & = 1. \\label{eq:Peierls_3_2}\n\\end{align}\nWe used Eq.~(\\ref{eq:reciprocal_3}) in going from Eq.~(\\ref{eq:Peierls_3_1}) to Eq.~(\\ref{eq:Peierls_3_2}). \nThis means that no additional phase factors arise if we compare Eq.~(\\ref{eq:non_ortho_5_1}) and Eq.~(\\ref{eq:non_ortho_6_1}). \nThus, we see that our dimensionally-promoted Hamiltonian is consistent with Eqs.~(\\ref{eq:non_ortho_6_1}--\\ref{eq:non_ortho_6_3}) coupled via a Peierls substitution to the vector potential Eq.~(\\ref{eq:non_ortho_A}).\n\n\\subsubsection{\\label{sec:remarks_consistency_check_and_summary}Remarks, consistency checks and summary}\n\nLet us briefly recap what we have developed. \nWe began with a $d$D system with modulated hoppings and on-site energies, together with lattice vectors $\\{\\vec{a}_{1},\\cdots,\\vec{a}_{d}\\}$. \nUpon dimensional promotion, we can choose {\\it any} linearly independent additional lattice vectors $\\{\\vec{c}_{1},\\cdots,\\vec{c}_{N}\\}$ which together with $\\{\\vec{a}_{1},\\cdots,\\vec{a}_{d}\\}$ span the promoted $(d+N)$D space. \nThe corresponding $U(1)$ gauge field is given in Eq.~(\\ref{eq:non_ortho_A}), which has linear dependence on $\\vec{r}$, the position vector in the promoted $(d+N)$D space. \nThis implies the field strength $F_{\\mu \\nu} = \\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu}$ is constant in space. \nIf we choose $\\{\\vec{a}_{1},\\cdots,\\vec{a}_{d},\\vec{c}_{1},\\cdots,\\vec{c}_{N} \\}$ to be orthogonal unit vectors in the $(d+N)$D Cartesian coordinates, and if the model only has modulated on-site energies, the general dimensional promotion procedure present here will reduce to the one present in Sec.~III of the main text.\n\nIn this more general construction in terms of non-orthogonal lattice vectors with modulated hopping terms, the mapping from the $d$D modulated system to the promoted $(d+N)$D lattice coupled to a $U(1)$ gauge field also requires no additional parameters. \nThe Hamiltonians before and after the dimensional promotion, together with the convention for the Fourier series expansions are summarized in Table~\\ref{tab:model_summary}. \nThe hopping matrices in $(d+N)$D space, and the corresponding Peierls phases are summarized in Table~\\ref{tab:hopping_1}. \nTo use Table~\\ref{tab:hopping_1} we multiply the hopping matrix entry and the Peierls phases to obtain the Hamiltonian matrix elements (with background $U(1)$ gauge fields) in the promoted $(d+N)$D space in Eqs.~(\\ref{eq:non_ortho_5_1}--\\ref{eq:non_ortho_5_3}). \nWe remind the readers that there are phase factors $e^{-i p_{i} \\pi \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d}l_{j}\\vec{a}_{j} \\right)}$ in the hopping matrix from $(\\vec{n},\\vec{\\nu})$ to $(\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i})$, as shown in Table~\\ref{tab:hopping_1}. \nThis means that these phase factors are included in the definition of the hopping matrices in the $(d+N)$D model, in addition to the Peierls phase factor. \nIn addition, we reemphasize that the phases $\\phi^{(i)}$ correspond to $\\vec{k}\\cdot \\vec{c}_{i}$ where $\\vec{k}$ is the crystal momentum $\\in \\mathbb{T}^{N}$ in the dimensionally-promoted Brillouin zone.\nThe modulation wave vectors $\\vec{q}^{(i)}$ enter the definition of the $(d+N)$D $U(1)$ gauge fields in Eq.~(\\ref{eq:non_ortho_A}).\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\nOriginal $d$D modulated system & Eq.~(\\ref{eq:H_low_general}) \\\\\n\\hline\nFourier expansion convention of on-site and hopping modulations & Eqs.~(\\ref{eq:general_FT_1}--\\ref{eq:general_FT_2})\\\\\n\\hline\nPromoted $(d+N)$D system with $U(1)$ gauge fields & Eqs.~(\\ref{eq:non_ortho_5_1}--\\ref{eq:non_ortho_5_3}) \\\\\n\\hline\nPromoted $(d+N)$D system without $U(1)$ gauge fields & Eqs.~(\\ref{eq:non_ortho_6_1}--\\ref{eq:non_ortho_6_3}) \\\\\n\\hline\n\\end{tabular}\n\\caption{Relevant equations in the general dimensional promotion formalism.}\n\\label{tab:model_summary}\n\\end{table}\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nHopping from $(\\vec{n},\\vec{\\nu})$ to & Hopping matrices & Peierls phases \\\\\n\\hline\n\\hline\n$(\\vec{n}+\\vec{m},\\vec{\\nu})$ & $\\left[H_{\\vec{m}}\\right]$ & $1$ \\\\\n\\hline\n$(\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i})$ & $ \\left[V^{(i)}_{p_{i}}\\right] $ & $e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right)}$ \\\\\n\\hline\n$(\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i})$ & $\\left[H_{\\vec{l},p_{i}}^{(i)}\\right]e^{-i \\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d} l_{j}\\vec{a}_{j} \\right) }$ & $e^{i 2\\pi p_{i} \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}\\left( n_{j} + \\frac{1}{2}l_{j} \\right)\\vec{a}_{j} \\right) }$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Hopping terms in the promoted $(d+N)$D model and the corresponding Peierls phases, expressed in terms of parameters from the $d$D modulated system in Eq.~(\\ref{eq:H_low_general}). \nNotice that $p_{i}\\hat{\\nu}_{i}$ does not imply a summation over $i$.}\n\\label{tab:hopping_1}\n\\end{table}\n\nBefore moving on, some additional remarks are in order. \nFirst, the vector potential in Eq.~(\\ref{eq:non_ortho_A}) satisfies\n\\begin{align}\n \\vec{A}\\left( \\vec{r} \\right) = \\vec{A}\\left( \\vec{r} + \\sum_{j=1}^{N} \\nu_{j} \\vec{c}_{j} \\right)\n\\end{align}\nfor any $\\{\\nu_{1},\\cdots,\\nu_{N} \\} \\in \\mathbb{Z}^{N}$, which is a direct consequence of Eq.~(\\ref{eq:reciprocal_2}), and corresponds to a generalized Landau gauge condition. \nThis allows us to recover our low dimensional modulated system, by Fourier transforming the higher-dimensional model Eq.~(\\ref{eq:non_ortho_4}) along $\\vec{k}$ in the subspace spanned by $\\{\\vec{G}_{1},\\cdots,\\vec{G}_{N} \\} \\in \\mathbb{T}^{N}$ and regarding $\\vec{k} \\cdot \\vec{c}_{i}$ as $\\phi^{(i)}$. \nAny fixed value of $\\{ \\phi^{(1)},\\cdots,\\phi^{(N)} \\}$ then describes a lower-dimensional modulated system with fixed phase offsets. \nIn other words, we can use a $d$D modulated system with controllable phase offsets $\\{ \\phi^{(1)},\\cdots,\\phi^{(N)} \\}$ to map out, by varying the values of $\\phi^{(i)}$, the whole spectrum of the promoted $(d+N)$D model coupled to a $U(1)$ gauge field. \nSecond, the constraints Eq.~(\\ref{eq:Hermitian_1}), Eq.~(\\ref{eq:Hermitian_4}) and Eq.~(\\ref{eq:Hermitian_5}) ensures that $H_{\\text{high-dim}}$ in Eq.~(\\ref{eq:non_ortho_4}) with Peierls phases, as well as Eqs.~(\\ref{eq:non_ortho_6_1}--\\ref{eq:non_ortho_6_3}) without Peierls phases are Hermitian. \nWe will make use of this in the subsequent examples and only list non-redundant matrix elements of the Hamiltonian. \nThird, we have assumed so far that for a given modulation $\\vec{q}^{(i)}$, the phase offsets of the on-site and hopping modulation are the same (see Eqs.~(\\ref{eq:H_form_1}) and (\\ref{eq:H_form_2})). \nHowever, it is possible for a system to develop incoherence between the modulation of the on-site energy and hopping terms. \nIn these cases, we can relax our requirement that all $\\vec{q}^{(i)}$ are mutually incommensurate, and regard the modulated hopping terms and on-site energy as being described by two distinct but numerically equal modulation wave vectors. \nThis will increase the number of additional dimensions necessary to represent the system.\nWe demonstrate how this situation can be handled using the 1D Rice-Mele chain later in Sec.~\\ref{sec:1D_RM_chain_incoherence}. \nFourth, in Refs.~\\onlinecite{LL_fragile_Lian_Biao,Herzog_Hof_topo}, it is shown that in general the line integral of the Peierls phase should be taken on a piecewise linear path along which the Wannier functions have greatest overlap, as opposed to the linear path we used here. \nIn our present situation, this means we are implicitly assuming that the total Hilbert space (occupied plus unoccupied states) for our dimensionally promoted model can be described by topologically trivial Wannier functions. \nThe topological properties of the Wannier functions in the promoted $(d+N)$D space are an interesting direction for future investigation, including investigating whether topologically non-trivial $(d+N)$D Wannier functions can imply any physical properties in the low dimensional modulated system. \nLastly, we emphasize that our interpretation of the promoted models in $(d+N)$D, given in Eq.~(\\ref{eq:non_ortho_4}) is not unique. \nOur interpretation is fixed by the requirement that the $U(1)$ gauge fields (Eq.~(\\ref{eq:non_ortho_A})) have a spatially constant field strength $F_{\\mu\\nu}$ in $(d+N)$D. \nTherefore, this allows us to easily construct the continuum theory describing low energy dynamics in $(d+N)$D, as we know the underlying $U(1)$ gauge field, written as a function of $\\vec{r} \\in \\mathbb{R}^{d+N}$. \nThis completes our description on how to compute the $U(1)$ gauge field in a $(d+N)$D lattice with non-orthogonal lattice vectors, and the corresponding lattice model in $(d+N)$D space.\n\n{\n\\subsubsection{Generalization to systems with arbitrary orbital positions in the original $d$D space}\n\nIn Sec.~\\ref{subsubsec:__dimensional_promotion}--\\ref{sec:remarks_consistency_check_and_summary}, we have implicitly chosen to promote the dimension of a $d$D modulated lattice model whose orbitals are located exactly at the $d$D lattice points $\\sum_{j=1}^{d}n_{j}\\vec{a}_{j}$ constructed from $\\{ \\vec{a}_{1},\\ldots,\\vec{a}_{d}\\}$. \nThis choice makes the emergence of the $U(1)$ gauge field and the identification of Peierls phases transparent. \nHowever, in many models of practical interest, not all of the orbitals are located at the $d$D lattice points. \nAs the computation of Peierls phases depends on the actual orbital positions, we now examine the effect of orbital positions in our dimensional promotion method.\n\nWe consider a $d$D modulated system whose $\\alpha^{\\text{th}}$ orbital is located at the position $\\vec{r}_{\\alpha}$ measured from the origin of the unit cell. \nWe will assume that $\\vec{r}_{\\alpha}$ can be written as a linear combination of the $d$D lattice vectors $\\{\\vec{a}_{1},\\ldots,\\vec{a}_{d}\\}$, namely\n\\begin{align}\n \\vec{r}_{\\alpha} = \\sum_{j=1}^{d} x^{j}_{\\alpha} \\vec{a}_{j}, \\label{eq:r_alpha_expansion}\n\\end{align}\nwhere $x^{j}_{\\alpha} \\in [0,1)$ denotes the fractional component of $\\vec{r}_{\\alpha}$ along $\\vec{a}_{j}$ (this excludes certain quasi-$d$-dimensional models). \nTherefore, the actual position of the $\\alpha^{\\text{th}}$ orbital in the unit cell labelled by $\\vec{n} \\in \\mathbb{Z}^{d}$ is $\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\vec{r}_{\\alpha}$. \nWhen all the orbitals are located at the lattice points $\\sum_{j=1}^{d}n_{j}\\vec{a}_{j}$, we will have $\\vec{r}_{\\alpha} = 0$ for all $\\alpha$. \nThe generic Hamiltonian for our $d$D system is still given by Eq.~(\\ref{eq:H_low_general}). \nFor later convenience, here we rewrite Eq.~(\\ref{eq:H_low_general}) with an explicit summation over the orbital components as\n\\begin{equation}\n H_{\\text{low-dim}} = \\sum_{\\alpha,\\beta}\\sum_{\\vec{n},\\vec{m}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\alpha} \\left[H_{\\vec{m}}\\right]_{\\alpha,\\beta} {\\psi}_{\\vec{n},\\beta} + \\sum_{\\alpha,\\beta}{\\sum_{\\vec{n},\\vec{l}}} \\sum_{i=1}^{N} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\alpha} \\left[H_{\\vec{l},\\vec{n}}^{(i)} \\right]_{\\alpha,\\beta} {\\psi}_{\\vec{n},\\beta} + \\sum_{\\alpha,\\beta}\\sum_{\\vec{n}} \\sum_{i=1}^{N}{\\psi}^{\\dagger}_{\\vec{n},\\alpha} \\left[ V^{(i)}_{\\vec{n}} \\right]_{\\alpha,\\beta} {\\psi}_{\\vec{n},\\beta}, \\label{eq:H_low_general_alpha}\n\\end{equation}\nwhere ${\\psi}^{\\dagger}_{\\vec{n},\\alpha}$ and ${\\psi}_{\\vec{n},\\alpha}$ are the creation and annihilation operator of the $\\alpha^{\\text{th}}$ orbital at the unit cell labelled by $\\vec{n} \\in \\mathbb{Z}^{d}$. \nAs before, $\\left[H_{\\vec{m}}\\right]_{\\alpha,\\beta}$, $\\left[H_{\\vec{l},\\vec{n}}^{(i)} \\right]_{\\alpha,\\beta}$ and $\\left[ V^{(i)}_{\\vec{n}} \\right]_{\\alpha,\\beta}$ are the $(\\alpha,\\beta)$ entries for the unmodulated Hamiltonian $\\left[H_{\\vec{m}}\\right]$, the hopping modulations $\\left[H_{\\vec{l},\\vec{n}}^{(i)} \\right]$ and the on-site modulations $\\left[ V^{(i)}_{\\vec{n}} \\right]$, respectively. \nThe dimensional promotion procedure is identical to that in Sec.~\\ref{subsubsec:__dimensional_promotion} and thus we obtain the same $(d+N)$D model given by Eq.~(\\ref{eq:non_ortho_4}). \nAgain, for later convenience we rewrite Eq.~(\\ref{eq:non_ortho_4}) with an explicit summation over the orbital components as\n\\begin{align}\n H_{\\text{high-dim}} =& \\sum_{\\alpha,\\beta}\\sum_{\\vec{n},\\vec{m},\\vec{\\nu}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{\\nu},\\alpha} \\left[ H_{\\vec{m}}\\right]_{\\alpha,\\beta} \\psi_{\\vec{n},\\vec{\\nu},\\beta} \\label{eq:H_high_dim_normal_hopping_alpha} \\\\\n & + \\sum_{\\alpha,\\beta}{\\sum_{\\vec{n},\\vec{l},\\vec{\\nu}}} \\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i},\\alpha} \\left[H_{\\vec{l},p_{i}}^{(i)}\\right]_{\\alpha,\\beta} e^{i 2\\pi p_{i} \\vec{q}^{(i)} \\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right) } {\\psi}_{\\vec{n},\\vec{\\nu},\\beta} \\label{eq:H_high_dim_hopping_modulation_alpha} \\\\\n & + \\sum_{\\alpha,\\beta}\\sum_{\\vec{n},\\vec{\\nu}}\\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i},\\alpha} \\left[V^{(i)}_{p_{i}}\\right]_{\\alpha,\\beta} e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left(\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} \\right)}{\\psi}_{\\vec{n},\\vec{\\nu},\\beta}. \\label{eq:H_high_dim_onsite_modulation_alpha}\n\\end{align}\nWe then take the actual position of the $\\alpha^{\\text{th}}$ orbital in the unit cell labelled by $(\\vec{n},\\vec{\\nu}) \\in (\\mathbb{Z}^{d},\\mathbb{Z}^{N})$ to be $\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\vec{r}_{\\alpha}$ where we have assumed that there are no components of $\\vec{r}_\\alpha$ along $\\vec{c}_{j}$ and Eq.~(\\ref{eq:r_alpha_expansion}) still holds. \nHowever, due to the generic position $\\vec{r}_{\\alpha}$ of the orbitals, the separation of phase factors in Eqs.~(\\ref{eq:H_high_dim_normal_hopping_alpha}--\\ref{eq:H_high_dim_onsite_modulation_alpha}) between periodic hopping and the Peierls phase due to the gauge field in Eq.~(\\ref{eq:non_ortho_A}) needs to be modified compared with those in Sec.~\\ref{sec:computing_Peierls_phase_1}.\n\nWe now compute the various Peierls phases associated with terms in Eqs.~(\\ref{eq:H_high_dim_normal_hopping_alpha}--\\ref{eq:H_high_dim_onsite_modulation_alpha}) due to the gauge field in Eq.~(\\ref{eq:non_ortho_A}). First, we consider the terms in Eq.~(\\ref{eq:H_high_dim_onsite_modulation_alpha}) where the electrons hop from \n\\begin{equation}\n\\vec{r}_{i} = \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\vec{r}_{\\beta}\n\\end{equation}\nto \n\\begin{equation}\n \\vec{r}_{f} =\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} - p_{i} \\vec{c}_{i} + \\vec{r}_{\\alpha}. \n\\end{equation}\nThe straight line connecting $\\vec{r}_{i}$ to $\\vec{r}_{f}$ is\n\\begin{align}\n \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\vec{r}_{\\beta} + \\left(- p_{i} \\vec{c}_{i} + \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t,\n\\end{align}\nwhere $ t\\in [0,1]$ and the corresponding infinitesimal displacement vector is $\\left(- p_{i} \\vec{c}_{i} + \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)dt$. The corresponding Peierls phase with a line integral can then be computed as follows:\n\\begin{align}\n & \\exp{-i \\int_{\\vec{r}_{i}}^{\\vec{r}_{f}} d\\vec{r} \\cdot \\vec{A}} \\\\\n & = \\exp{-i \\int_{0}^{1}dt \\left(- p_{i} \\vec{c}_{i} + \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right) \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\begin{bmatrix}\n \\sum_{l=1}^{d}n_{l}\\vec{a}_{l} + \\sum_{l=1}^{N}\\nu_{l}\\vec{c}_{l} + \\vec{r}_{\\beta} \\\\\n +\\left(- p_{i} \\vec{c}_{i} + \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t\n \\end{bmatrix}\n \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_1_1_alpha}\\\\\n & = \\exp{+i \\int_{0}^{1}dt p_{i} \\vec{c}_{i} \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left(\n \\left(\n \\sum_{l=1}^{d}n_{l}\\vec{a}_{l} + \\vec{r}_{\\beta} + \\left( \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t\n \\right)\n \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_1_2_alpha} \\\\\n & = \\exp{+i \\int_{0}^{1}dt p_{i} \\vec{c}_{i} \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left(\n \\left(\n \\sum_{l=1}^{d}\\left( n_{l} + x_{\\beta}^{l} + t x_{\\alpha}^{l} - t x_{\\beta}^{l} \\right)\\vec{a}_{l} \n \\right)\n \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_1_3_alpha}\\\\\n & = \\exp{+i2\\pi \\int_{0}^{1}dt p_{i} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\delta_{ij} \\left( \\vec{q}^{(j)}\\cdot \\left[ \n \\sum_{l=1}^{d}\\left(n_{l} + x_{\\beta}^{l}+tx^{l}_{\\alpha} - t x^{l}_{\\beta}\\right)\\delta_{lk} \n \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_1_4_alpha} \\\\\n & = \\exp{+i2\\pi \\int_{0}^{1}dt p_{i} \\left( \\vec{q}^{(i)}\\cdot \\left[ \n \\sum_{l=1}^{d}\\left(n_{l} + x_{\\beta}^{l}+tx^{l}_{\\alpha} - t x^{l}_{\\beta}\\right)\n \\vec{a}_{l} \\right] \\right) } \\label{eq:Peierls_1_5_alpha} \\\\\n & = \\exp{+i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left[ \n \\sum_{l=1}^{d}\\left(n_{l} + x_{\\beta}^{l}+\\frac{1}{2}x^{l}_{\\alpha} - \\frac{1}{2} x^{l}_{\\beta}\\right)\n \\vec{a}_{l} \\right] }\\label{eq:Peierls_1_6_alpha} \\\\\n & = \\exp{+i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left[ \n \\sum_{l=1}^{d}\\left(n_{l} + \\frac{x^{l}_{\\alpha} + x^{l}_{\\beta}}{2} \\right)\n \\vec{a}_{l} \\right] } \\label{eq:Peierls_1_7_alpha} .\n\\end{align}\nTo go from Eq.~(\\ref{eq:Peierls_1_1_alpha}) to Eq.~(\\ref{eq:Peierls_1_2_alpha}) we have used Eq.~(\\ref{eq:reciprocal_2}) and the fact that $(\\vec{r}_{\\alpha} - \\vec{r}_{\\beta}) \\cdot \\vec{G}_{j} = 0$ because of Eqs.~(\\ref{eq:reciprocal_3}) and (\\ref{eq:r_alpha_expansion}). \nWe then use Eq.~(\\ref{eq:r_alpha_expansion}) to go from Eq.~(\\ref{eq:Peierls_1_2_alpha}) to Eq.~(\\ref{eq:Peierls_1_3_alpha}). \nTo go from Eq.~(\\ref{eq:Peierls_1_3_alpha}) to Eq.~(\\ref{eq:Peierls_1_4_alpha}), we have used Eqs.~(\\ref{eq:reciprocal_1}) and (\\ref{eq:reciprocal_4}). \nPerforming the summation over $j$ and $k$ we obtain Eq.~(\\ref{eq:Peierls_1_5_alpha}). \nUsing $\\int_{0}^{1}dt = 1$ and $\\int_{0}^{1}tdt = \\frac{1}{2}$ we arrive at Eqs.~(\\ref{eq:Peierls_1_6_alpha}--\\ref{eq:Peierls_1_7_alpha}).\n\nNext, let us consider the terms in Eq.~(\\ref{eq:H_high_dim_hopping_modulation_alpha}) where the electrons hop from \n\\begin{equation}\n\\vec{r}_{i} = \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\vec{r}_{\\beta}\n\\end{equation}\nto \n\\begin{equation}\n\\vec{r}_{f} =\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\sum_{j=1}^{d}l_{j}\\vec{a}_{j} - p_{i} \\vec{c}_{i} + \\vec{r}_{\\alpha}.\n\\end{equation}\nThe straight line connecting $\\vec{r}_{i}$ to $\\vec{r}_{f}$ is\n\\begin{align}\n \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\vec{r}_{\\beta} + \\left( \\sum_{j=1}^{d}l_{j}\\vec{a}_{j}- p_{i} \\vec{c}_{i} + \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t,\n\\end{align}\nwhere $ t\\in [0,1]$ and the corresponding infinitesimal displacement vector is $\\left( \\sum_{j=1}^{d}l_{j}\\vec{a}_{j}- p_{i} \\vec{c}_{i} + \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)dt$. \nThe corresponding Peierls phase with a line integral can then be computed as follows:\n\\begin{align}\n & \\exp{-i \\int_{\\vec{r}_{i}}^{\\vec{r}_{f}} d\\vec{r} \\cdot \\vec{A}} \\\\\n & = \\exp{-i \\int_{0}^{1}dt \\left( \\sum_{r=1}^{d}l_{r}\\vec{a}_{r}- p_{i} \\vec{c}_{i} + \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right) \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\begin{bmatrix}\n \\sum_{r=1}^{d}n_{r}\\vec{a}_{r} + \\sum_{r=1}^{N}\\nu_{r}\\vec{c}_{r} + \\vec{r}_{\\beta} \\\\\n + \\left( \\sum_{r=1}^{d}l_{r}\\vec{a}_{r}- p_{i} \\vec{c}_{i} + \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t\n \\end{bmatrix}\n \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_1_alpha}\\\\\n & = \\exp{+i \\int_{0}^{1}dt p_{i} \\vec{c}_{i} \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\begin{bmatrix}\n \\sum_{r=1}^{d}n_{r}\\vec{a}_{r} + \\vec{r}_{\\beta} \\\\\n + \\left( \\sum_{r=1}^{d}l_{r}\\vec{a}_{r} + \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t\n \\end{bmatrix}\n \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_2_alpha}\\\\\n & = \\exp{+i \\int_{0}^{1}dt p_{i} \\vec{c}_{i} \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\left( \\sum_{r=1}^{d}\\left( n_{r} + tl_{r} + x^{r}_{\\beta} + t x^{r}_{\\alpha} - t x^{r}_{\\beta} \\right)\\vec{a}_{r} \\right)\n \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_3_alpha}\\\\\n & = \\exp{+i2\\pi \\int_{0}^{1}dt p_{i} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\delta_{ij} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\sum_{r=1}^{d}\\left( n_{r} + tl_{r} + x^{r}_{\\beta} + t x^{r}_{\\alpha} - t x^{r}_{\\beta} \\right)\\delta_{rk} \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_2_4_alpha}\\\\\n & = \\exp{+i2\\pi \\int_{0}^{1}dt p_{i} \\vec{q}^{(i)}\\cdot \\left[ \\sum_{r=1}^{d}\\left( n_{r} + tl_{r} + x^{r}_{\\beta} + t x^{r}_{\\alpha} - t x^{r}_{\\beta} \\right) \\vec{a}_{r} \\right] } \\label{eq:Peierls_2_5_alpha}\\\\\n & = \\exp{+i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left[ \\sum_{r=1}^{d}\\left( n_{r} + \\frac{1}{2}l_{r} + x^{r}_{\\beta} + \\frac{1}{2} x^{r}_{\\alpha} - \\frac{1}{2} x^{r}_{\\beta} \\right) \\vec{a}_{r} \\right] } \\label{eq:Peierls_2_6_alpha}\\\\\n & = \\exp{+i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left[ \\sum_{r=1}^{d}\\left( n_{r} + \\frac{l_{r} +x^{r}_{\\alpha} + x^{r}_{\\beta} }{2} \\right) \\vec{a}_{r} \\right] }. \\label{eq:Peierls_2_7_alpha}\n\\end{align}\nTo go from Eq.~(\\ref{eq:Peierls_2_1_alpha}) to Eq.~(\\ref{eq:Peierls_2_2_alpha}) we have used Eqs.~(\\ref{eq:reciprocal_2}--\\ref{eq:reciprocal_3}), and again the fact that $(\\vec{r}_{\\alpha} - \\vec{r}_{\\beta}) \\cdot \\vec{G}_{j} = 0$ because of Eqs.~(\\ref{eq:reciprocal_3}) and (\\ref{eq:r_alpha_expansion}).\n We then use Eq.~(\\ref{eq:r_alpha_expansion}) to go from Eq.~(\\ref{eq:Peierls_2_2_alpha}) to Eq.~(\\ref{eq:Peierls_2_3_alpha}). \n To go from Eq.~(\\ref{eq:Peierls_2_3_alpha}) to Eq.~(\\ref{eq:Peierls_2_4_alpha}), we have used Eqs.~(\\ref{eq:reciprocal_1}) and (\\ref{eq:reciprocal_4}). \n Performing the summation over $j$ and $k$ we obtain Eq.~(\\ref{eq:Peierls_2_5_alpha}). \n Using $\\int_{0}^{1}dt = 1$ and $\\int_{0}^{1}tdt = \\frac{1}{2}$ we arrive at Eqs.~(\\ref{eq:Peierls_2_6_alpha}--\\ref{eq:Peierls_2_7_alpha}).\n\nFinally, let us consider the terms in Eq.~(\\ref{eq:H_high_dim_normal_hopping_alpha}) where the electrons hop from \n\\begin{equation}\n\\vec{r}_{i} = \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\vec{r}_{\\beta}\n\\end{equation}\nto \n\\begin{equation}\n\\vec{r}_{f} =\\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\sum_{j=1}^{d}m_{j}\\vec{a}_{j} + \\vec{r}_{\\alpha}.\n\\end{equation}\nThe straight line connecting $\\vec{r}_{i}$ to $\\vec{r}_{f}$ is\n\\begin{align}\n \\sum_{j=1}^{d}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{N}\\nu_{j}\\vec{c}_{j} + \\vec{r}_{\\beta} + \\left( \\sum_{j=1}^{d}m_{j}\\vec{a}_{j}+ \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t,\n\\end{align}\nwhere $ t\\in [0,1]$ and the corresponding infinitesimal displacement vector is $\\left( \\sum_{j=1}^{d}m_{j}\\vec{a}_{j}+ \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)dt$. \nThe corresponding Peierls phase with a line integral can then be computed as follows:\n\\begin{align}\n & \\exp{-i \\int_{\\vec{r}_{i}}^{\\vec{r}_{f}} d\\vec{r} \\cdot \\vec{A}} \\\\\n & = \\exp{-i \\int_{0}^{1}dt \\left( \\sum_{r=1}^{d}m_{r}\\vec{a}_{r}+ \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right) \\cdot \\frac{1}{2\\pi} \\sum_{j=1}^{N} \\sum_{k=1}^{d} \\vec{G}_{j} \\left( \\vec{q}^{(j)}\\cdot \\left[ \\left( \\begin{bmatrix}\n \\sum_{r=1}^{d}n_{r}\\vec{a}_{r} + \\sum_{r=1}^{N}\\nu_{r}\\vec{c}_{r} + \\vec{r}_{\\beta} \\\\\n + \\left( \\sum_{r=1}^{d}m_{r}\\vec{a}_{r}+ \\vec{r}_{\\alpha} - \\vec{r}_{\\beta} \\right)t\n \\end{bmatrix}\n \\cdot \\vec{g}_{k} \\right) \\vec{a}_{k} \\right] \\right) } \\label{eq:Peierls_3_1_alpha}\\\\ \n & = 1. \\label{eq:Peierls_3_2_alpha}\n\\end{align}\nTo go from Eq.~(\\ref{eq:Peierls_3_1_alpha}) to Eq.~(\\ref{eq:Peierls_3_2_alpha}) we have used Eq.~(\\ref{eq:reciprocal_3}) and again the fact that $(\\vec{r}_{\\alpha} - \\vec{r}_{\\beta}) \\cdot \\vec{G}_{j} = 0$ because of Eqs.~(\\ref{eq:reciprocal_3}) and (\\ref{eq:r_alpha_expansion}).\n\n\nWith this knowledge of the Peierls phases in Eqs.~(\\ref{eq:Peierls_1_7_alpha}), (\\ref{eq:Peierls_2_7_alpha}), and (\\ref{eq:Peierls_3_2_alpha}), we can rewrite $H_{\\text{high-dim}}$ in Eqs.~(\\ref{eq:H_high_dim_normal_hopping_alpha}--\\ref{eq:H_high_dim_onsite_modulation_alpha}) as \n\\begin{align}\n H_{\\text{high-dim}} =& \\sum_{\\alpha,\\beta}\\sum_{\\vec{n},\\vec{m},\\vec{\\nu}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{\\nu},\\alpha} \\left[ H_{\\vec{m}}\\right]_{\\alpha,\\beta} \\psi_{\\vec{n},\\vec{\\nu},\\beta} \\label{eq:H_high_dim_normal_hopping_alpha_split} \\\\\n & + \\sum_{\\alpha,\\beta}{\\sum_{\\vec{n},\\vec{l},\\vec{\\nu}}} \\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i},\\alpha} \\left[H_{\\vec{l},p_{i}}^{(i)}\\right]_{\\alpha,\\beta} e^{-i\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left(\\sum_{j=1}^{d}\\left( l_{j} +x^{j}_{\\alpha} + x^{j}_{\\beta} \\right) \\vec{a}_{j} \\right) } e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d}\\left( n_{j} + \\frac{l_{j} +x^{j}_{\\alpha} + x^{j}_{\\beta} }{2} \\right) \\vec{a}_{j} \\right) } {\\psi}_{\\vec{n},\\vec{\\nu},\\beta} \\label{eq:H_high_dim_hopping_modulation_alpha_split} \\\\\n & + \\sum_{\\alpha,\\beta}\\sum_{\\vec{n},\\vec{\\nu}}\\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i},\\alpha} \\left[V^{(i)}_{p_{i}}\\right]_{\\alpha,\\beta} e^{-i\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left(\n \\sum_{j=1}^{d}\\left( x^{j}_{\\alpha} + x^{j}_{\\beta} \\right)\n \\vec{a}_{j} \\right) } \n e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \n \\sum_{j=1}^{d}\\left(n_{j} + \\frac{x^{j}_{\\alpha} + x^{j}_{\\beta}}{2} \\right)\n \\vec{a}_{j} \\right) } \n {\\psi}_{\\vec{n},\\vec{\\nu},\\beta}. \\label{eq:H_high_dim_onsite_modulation_alpha_split}\n\\end{align}\nEqs.~(\\ref{eq:H_high_dim_normal_hopping_alpha_split}--\\ref{eq:H_high_dim_onsite_modulation_alpha_split}) can then be interpreted as a $(d+N)$D lattice model with Hamiltonian\n\\begin{align}\n H_{\\text{high-dim}} =& \\sum_{\\alpha,\\beta}\\sum_{\\vec{n},\\vec{m},\\vec{\\nu}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{m},\\vec{\\nu},\\alpha} \\left[ H_{\\vec{m}}\\right]_{\\alpha,\\beta} \\psi_{\\vec{n},\\vec{\\nu},\\beta} \\label{eq:H_high_dim_normal_hopping_alpha_no_gauge} \\\\\n & + \\sum_{\\alpha,\\beta}{\\sum_{\\vec{n},\\vec{l},\\vec{\\nu}}} \\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i},\\alpha} \\left[H_{\\vec{l},p_{i}}^{(i)}\\right]_{\\alpha,\\beta} e^{-i\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left(\\sum_{j=1}^{d}\\left( l_{j} +x^{j}_{\\alpha} + x^{j}_{\\beta} \\right) \\vec{a}_{j} \\right) } {\\psi}_{\\vec{n},\\vec{\\nu},\\beta} \\label{eq:H_high_dim_hopping_modulation_alpha_no_gauge} \\\\\n & + \\sum_{\\alpha,\\beta}\\sum_{\\vec{n},\\vec{\\nu}}\\sum_{i=1}^{N} \\sum_{p_{i}} {\\psi}^{\\dagger}_{\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i},\\alpha} \\left[V^{(i)}_{p_{i}}\\right]_{\\alpha,\\beta} e^{-i\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left(\n \\sum_{j=1}^{d}\\left( x^{j}_{\\alpha} + x^{j}_{\\beta} \\right)\n \\vec{a}_{j} \\right) } \n {\\psi}_{\\vec{n},\\vec{\\nu},\\beta}, \\label{eq:H_high_dim_onsite_modulation_alpha_no_gauge}\n\\end{align}\nwhich is periodic with lattice vectors $\\{\\vec{a}_{1},\\ldots,\\vec{a}_{d},\\vec{c}_{1},\\ldots,\\vec{c}_{N} \\}$, coupled through a Peierls substitution\\cite{Peierls_substitution} to a $U(1)$ gauge field given in Eq.~(\\ref{eq:non_ortho_A}). \nNotice that we have regarded\n\\begin{align}\n e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d}\\left( n_{j} + \\frac{l_{j} +x^{j}_{\\alpha} + x^{j}_{\\beta} }{2} \\right) \\vec{a}_{j} \\right) } \n\\end{align}\nand\n\\begin{align}\n e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \n \\sum_{j=1}^{d}\\left(n_{j} + \\frac{x^{j}_{\\alpha} + x^{j}_{\\beta}}{2} \\right)\n \\vec{a}_{j} \\right) } \n\\end{align}\nin Eqs.~(\\ref{eq:H_high_dim_hopping_modulation_alpha_split}) and (\\ref{eq:H_high_dim_onsite_modulation_alpha_split}) as the Peierls phases (see Eqs.~(\\ref{eq:Peierls_2_7_alpha}) and (\\ref{eq:Peierls_1_7_alpha})) due to the $U(1)$ gauge field in Eq.~(\\ref{eq:non_ortho_A}). \nNotice that there are no Peierls phases induced from the gauge field in Eq.~(\\ref{eq:H_high_dim_normal_hopping_alpha_split}). \n\nWe have thus generalized our dimensional promotion method to $d$D modulated lattice models whose orbitals are not located at the $d$D lattice points $\\sum_{j=1}^{d}n_{j}\\vec{a}_{j}$. \nSimilar to Tables~\\ref{tab:model_summary} and \\ref{tab:hopping_1}, we have summarized the Hamiltonian before and after the dimensional promotion in Table~\\ref{tab:model_summary_general_orbital_positions}, and the hopping matrix elements together with the Peierls phases in Table~\\ref{tab:hopping_1_general_orbital_positions}. \nA notable feature is that now both the hopping matrix elements of the $(d+N)$D lattice model {\\it without} the gauge field and the Peierls phases due to the gauge field in Eq.~(\\ref{eq:non_ortho_A}) encode information of the orbital positions. \nWhen all the orbitals are located right at the lattice points we have $\\vec{r}_{\\alpha} = 0$ for all $\\alpha$, namely $x^{j}_{\\alpha}=0$ for all $j$ and $\\alpha$ in Eq.~(\\ref{eq:r_alpha_expansion}). \nIn such cases, Table~\\ref{tab:hopping_1_general_orbital_positions} effectively reduces to Table~\\ref{tab:hopping_1}.\n\n\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\nOriginal $d$D modulated system & Eq.~(\\ref{eq:H_low_general_alpha}) \\\\\n\\hline\nPromoted $(d+N)$D system with $U(1)$ gauge fields & Eqs.~(\\ref{eq:H_high_dim_normal_hopping_alpha_split}--\\ref{eq:H_high_dim_onsite_modulation_alpha_split}) \\\\\n\\hline\nPromoted $(d+N)$D system without $U(1)$ gauge fields & Eqs.~(\\ref{eq:H_high_dim_normal_hopping_alpha_no_gauge}--\\ref{eq:H_high_dim_onsite_modulation_alpha_no_gauge}) \\\\\n\\hline\n\\end{tabular}\n\\caption{Relevant equations in the general dimensional promotion formalism with arbitrary orbital positions.}\n\\label{tab:model_summary_general_orbital_positions}\n\\end{table}\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nHopping from $\\beta^{\\text{th}}$ orbital in unit cell $(\\vec{n},\\vec{\\nu})$ to & Hopping matrix elements & Peierls phases \\\\\n\\hline\n\\hline\n$\\alpha^{\\text{th}}$ orbital in unit cell $(\\vec{n}+\\vec{m},\\vec{\\nu})$ & $\\left[H_{\\vec{m}}\\right]_{\\alpha,\\beta}$ & $1$ \\\\\n\\hline\n$\\alpha^{\\text{th}}$ orbital in unit cell $(\\vec{n},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i})$ & $ \\left[V^{(i)}_{p_{i}}\\right]_{\\alpha,\\beta} e^{-i\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d}\\left( x^{j}_{\\alpha} + x^{j}_{\\beta} \\right) \\vec{a}_{j} \\right) } $ & $e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d}\\left(n_{j} + \\frac{x^{j}_{\\alpha} + x^{j}_{\\beta}}{2} \\right) \\vec{a}_{j} \\right) } $ \\\\\n\\hline\n$\\alpha^{\\text{th}}$ orbital in unit cell $(\\vec{n}+\\vec{l},\\vec{\\nu}-p_{i}\\hat{\\nu}_{i})$ & $\\left[H_{\\vec{l},p_{i}}^{(i)}\\right]_{\\alpha,\\beta} e^{-i\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left(\\sum_{j=1}^{d}\\left( l_{j} +x^{j}_{\\alpha} + x^{j}_{\\beta} \\right) \\vec{a}_{j} \\right) } $ & $e^{i2\\pi p_{i} \\vec{q}^{(i)}\\cdot \\left( \\sum_{j=1}^{d}\\left( n_{j} + \\frac{l_{j} +x^{j}_{\\alpha} + x^{j}_{\\beta} }{2} \\right) \\vec{a}_{j} \\right) } $ \\\\\n\\hline\n\\end{tabular}\n\\caption{Hopping terms in the promoted $(d+N)$D model with arbitrary orbital positions and the corresponding Peierls phases, expressed in terms of parameters from the $d$D modulated system in Eq.~(\\ref{eq:H_low_general}). \nNotice that $p_{i}\\hat{\\nu}_{i}$ does not imply a summation over $i$.}\n\\label{tab:hopping_1_general_orbital_positions}\n\\end{table}\n\n}\n\n\\subsection{\\label{subsec:examples_general_dim_promotion}Examples}\n\nTo demonstrate our general construction, we will use Table~\\ref{tab:model_summary} and Table~\\ref{tab:hopping_1} to consider four examples: \n(1) promoting the 1D Rice-Mele chain to a 2D square lattice with $\\pi$-flux, \n(2) promoting the 1D Rice-Mele chain with incoherent phase offsets in on-site and hopping modulations to a 3D cubic lattice coupled to a $U(1)$ gauge field, \n(3) promoting a 1D modulated system to a 2D hexagonal lattice with a perpendicular magnetic field, and \n(4) promoting a 2D modulated system with hexagonal lattice to a 3D hexagonal lattice coupled to a $U(1)$ gauge field.\n\n\\subsubsection{\\label{sec:ex_1D_RM_chain} 1D Rice-Mele chain $\\to$ 2D square lattice with $\\pi$-flux}\n\nThe Hamiltonian of the 1D Rice-Mele\\cite{RiceMele} chain oriented along the $x$-axis is given by\n\\begin{align}\n H_{\\text{Rice-Mele}} = \\sum_{n} \\left( t + \\delta t (-1)^{n} \\cos{\\phi} \\right)\\psi^{\\dagger}_{n+1}\\psi_{n} + \\text{h.c.} + \\sum_{n} (-1)^{n+1} \\Delta \\sin{\\phi} \\psi^{\\dagger}_{n}\\psi_{n},\n \\label{eq:supp_Rice_Mele_1}\n\\end{align}\nwhere $\\psi^{\\dagger}_{n}$ is the creation operator for an electron at position $n$, and $\\text{h.c.}$ denotes the Hermitian conjugate of all terms before it. \nWe can rewrite Eq.~(\\ref{eq:supp_Rice_Mele_1}) as\n\\begin{align}\n H_{\\text{Rice-Mele}} & = \\sum_{n} \\left( t + \\delta t \\cos{(\\pi n + \\phi)} \\right)\\psi^{\\dagger}_{n+1}\\psi_{n} + \\text{h.c.} - \\sum_{n} \\Delta \\sin{(\\pi n + \\phi)} \\psi^{\\dagger}_{n}\\psi_{n} \n \\label{eq:supp_Rice_Mele_2} \\\\\n & = \\sum_{n} \\left( t + \\delta t \\cdot \\frac{e^{i\\left( \\pi n + \\phi\\right)} + e^{-i\\left( \\pi n + \\phi\\right)}}{2} \\right)\\psi^{\\dagger}_{n+1}\\psi_{n} + \\text{h.c.} - \\sum_{n} \\Delta \\cdot \\frac{e^{i\\left( \\pi n + \\phi\\right)} - e^{-i\\left( \\pi n + \\phi\\right)}}{2i} \\psi^{\\dagger}_{n}\\psi_{n} .\\label{eq:supp_Rice_Mele_3}\n\\end{align}\nThus we can see that 1D Rice-Mele chain is a 1D modulated system with modulation wave vector $q = 1\/2$ ($2\\pi q = \\pi$). \nWe now choose the second, synthetic lattice vector to be $\\hat{y}$. \nThe promote 2D system will then have\n\\begin{align}\n \\vec{a}_{1} = (1,0),\\ \\vec{c}_{1} = (0,1),\\ \\vec{g}_{1} = 2\\pi(1,0) \\text{ and } \\vec{G}_{1} = 2\\pi (0,1),\n\\end{align}\nin terms of the notation from Eqs.~(\\ref{eq:reciprocal_1}--\\ref{eq:reciprocal_4}). \nNotice that $\\vec{a}_{1}$ and $\\vec{c}_{1}$ describe a square lattice. \nUsing the Fourier expansion convention in Table~\\ref{tab:model_summary} and Table~\\ref{tab:hopping_1} with $\\vec{q} = (1\/2,0)$ and Eq.~(\\ref{eq:supp_Rice_Mele_3}), we obtain the hopping terms summarized in Table~\\ref{tab:hopping_2}. \nMultiplying the entries for hopping matrices and Peierls phases, the Hamiltonian for the promoted 2D model is\n\\begin{align}\n H_{\\text{2D}} = \\sum_{n,m} \\left( t\\psi^{\\dagger}_{n+1,m}\\psi_{n,m}- \\frac{i\\delta t}{2}e^{i\\pi \\left(n+\\frac{1}{2} \\right)} \\psi^{\\dagger}_{n+1,m-1}\\psi_{n,m}+ \\frac{i\\delta t}{2}e^{-i\\pi \\left(n+\\frac{1}{2} \\right)} \\psi^{\\dagger}_{n+1,m+1}\\psi_{n,m}- \\frac{i\\Delta}{2}e^{-i\\pi n} \\psi^{\\dagger}_{n,m+1}\\psi_{n,m} + \\text{h.c.} \\right), \\label{eq:1D_Rice_Mele_to_2D_2}\n\\end{align}\nwhere $\\psi^{\\dagger}_{n,m}$ is the creation operator for an electron at position $(n,m)$, and the vector potential to which this 2D model is coupled is \n\\begin{align}\n \\vec{A} = \\frac{1}{2\\pi} \\vec{G}_{1} \\left( \\vec{q}\\cdot \\left[ \\left( \\vec{r} \\cdot \\vec{g}_{1}\\right) \\vec{a}_{1} \\right] \\right) = (0,\\pi x).\n\\end{align}\nThis $\\vec{A}$ produces a perpendicular magnetic field $\\vec{B} = \\pi \\hat{z}$ in the promoted 2D system such that there is a $\\pi$-flux per plaquette. \n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nHopping from $(n,m)$ to & Hopping matrices & Peierls phases \\\\\n\\hline\n\\hline\n$(n+1,m)$ & $t$ & $1$ \\\\\n\\hline\n$(n,m+1)$ & $ \\Delta \/ (2i) = -i\\Delta \/ 2$ & $e^{-i \\pi n}$ \\\\\n\\hline \n$(n+1,m+1)$ & $(\\delta t\/ 2) \\cdot e^{i \\pi \/2} = i\\delta t\/ 2 $ & $e^{-i \\pi \\left( n + \\frac{1}{2} \\right)}$ \\\\\n\\hline\n$(n+1,m-1)$ & $(\\delta t\/ 2) \\cdot e^{-i \\pi \/2} = -i\\delta t\/ 2 $ & $e^{i \\pi \\left( n + \\frac{1}{2} \\right)}$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Hopping matrices and the corresponding Peierls phases for the promoted $2$D model from a 1D Rice-Mele chain.\nThe hopping matrices along the opposite directions of those listed here are omitted and can be obtained through Hermitian conjugation.}\n\\label{tab:hopping_2}\n\\end{table}\n\n\\subsubsection{\\label{sec:1D_RM_chain_incoherence}1D Rice-Mele chain with phase offset incoherence $\\to$ 3D cubic lattice coupled to a $U(1)$ gauge field}\n\nConsider again a 1D Rice-Mele chain oriented along the $x$-axis as in Sec.~\\ref{sec:ex_1D_RM_chain}. \nNow, however, we assume that the phase offsets in the hopping and on-site modulation can be different.\nThe $1$D Hamiltonian then reads\n\\begin{align}\n H_{\\text{Rice-Mele}} = \\sum_{n} \\left( t + \\delta t (-1)^{n} \\cos{\\phi^{(1)}} \\right)\\psi^{\\dagger}_{n+1}\\psi_{n} + \\text{h.c.} + \\sum_{n} (-1)^{n+1} \\Delta \\sin{\\phi^{(2)}} \\psi^{\\dagger}_{n}\\psi_{n},\n \\label{eq:Rice_Mele_incoherence}\n\\end{align}\nwhere $\\psi^{\\dagger}_{n}$ is the creation operator for an electron at position $n$, and $\\text{h.c.}$ means the Hermitian conjugate of all terms before it. \nThis situation arises when we are able to tune the phase offsets of the on-site and hopping modulations independently. \nWe can write Eq.~(\\ref{eq:Rice_Mele_incoherence}) as\n\\begin{align}\n H_{\\text{Rice-Mele}} & = \\sum_{n} \\left( t + \\delta t \\cos{(\\pi n + \\phi^{(1)})} \\right)\\psi^{\\dagger}_{n+1}\\psi_{n} + \\text{h.c.} - \\sum_{n} \\Delta \\sin{(\\pi n + \\phi^{(2)})} \\psi^{\\dagger}_{n}\\psi_{n} \n \\label{eq:Rice_Mele_incoherence_2} \\\\\n & = \\sum_{n} \\left( t + \\delta t \\cdot \\frac{e^{i\\left( \\pi n + \\phi^{(1)}\\right)} + e^{-i\\left( \\pi n + \\phi^{(1)}\\right)}}{2} \\right)\\psi^{\\dagger}_{n+1}\\psi_{n} + \\text{h.c.} - \\sum_{n} \\Delta \\cdot \\frac{e^{i\\left( \\pi n + \\phi^{(2)}\\right)} - e^{-i\\left( \\pi n + \\phi^{(2)}\\right)}}{2i} \\psi^{\\dagger}_{n}\\psi_{n}. \\label{eq:Rice_Mele_incoherence_3}\n\\end{align}\nWe now choose the second and third synthetic lattice vectors to be $\\hat{y}$ and $\\hat{z}$. \nThe promote 3D system will then have\n\\begin{align}\n & \\vec{a}_{1} = (1,0,0),\\ \\vec{c}_{1} = (0,1,0),\\ \\vec{c}_{2} = (0,0,1) \\\\\n & \\vec{g}_{1} = 2\\pi(1,0,0),\\ \\vec{G}_{1} = 2\\pi(0,1,0),\\ \\vec{G}_{2} = 2\\pi(0,0,1),\n\\end{align}\nin terms of the notation from Eqs.~(\\ref{eq:reciprocal_1}--\\ref{eq:reciprocal_4}). \nNotice that $\\{\\vec{a}_{1},\\vec{c}_{1},\\vec{c}_{2} \\}$ describe a 3D cubic lattice. \nUsing the Fourier expansion convention in Table~\\ref{tab:model_summary} and Table~\\ref{tab:hopping_1} with $\\vec{q}^{(1)}= \\vec{q}^{(2)} = (1\/2,0,0)$ and Eq.~(\\ref{eq:Rice_Mele_incoherence_3}), we obtain the hopping terms in this case summarized in Table~\\ref{tab:hopping_2_incoherence}. \nMultiplying the entries for hopping matrices and Peierls phases, the Hamiltonian for the promoted 3D model is\n\\begin{equation}\n\\hspace*{-0cm}\n H_{\\text{3D}} = \\sum_{n,m,l} \\left( t\\psi^{\\dagger}_{n+1,m,l}\\psi_{n,m,l}- \\frac{i\\delta t}{2}e^{i\\pi \\left(n+\\frac{1}{2} \\right)} \\psi^{\\dagger}_{n+1,m-1,l}\\psi_{n,m,l}+ \\frac{i\\delta t}{2}e^{-i\\pi \\left(n+\\frac{1}{2} \\right)} \\psi^{\\dagger}_{n+1,m+1,l}\\psi_{n,m,l}- \\frac{i\\Delta}{2}e^{-i\\pi n} \\psi^{\\dagger}_{n,m,l+1}\\psi_{n,m,l} + \\text{h.c.} \\right),\n\\end{equation}\nwhere $\\psi^{\\dagger}_{n,m,l}$ is the creation operator for an electron at position $(n,m,l)$, and the vector potential to which this 3D model is coupled is \n\\begin{align}\n \\vec{A} = \\frac{1}{2\\pi} \\left( \\vec{G}_{1} \\left(\\vec{q}^{(1)}\\cdot\\left[ \\left( \\vec{r}\\cdot \\vec{g}_{1} \\right)\\vec{a}_{1} \\right] \\right) + \\vec{G}_{2} \\left(\\vec{q}^{(2)}\\cdot\\left[ \\left( \\vec{r}\\cdot \\vec{g}_{1} \\right)\\vec{a}_{1} \\right] \\right) \\right) = (0,\\pi x, \\pi x).\n\\end{align}\nThis $\\vec{A}$ produces a magnetic field $\\vec{B} = (0,-\\pi,\\pi)$ in the promoted 3D system such that there is a $\\pi$-flux threading through the plaquettes in $zx$- and $xy$-planes. \nIn contrast to Sec.~\\ref{sec:ex_1D_RM_chain} where the promoted 2D model is a Chern insulator (see Sec.~\\ref{sec:Thouless_pump_1D_Rice_Mele}), the 1D Rice-Mele chain with phase offset incoherence promotes to a 3D gapless model. \nWe can see this by noting that with $\\phi^{(1)} = \\pi \/2$ and $\\phi^{(2)} = 0$, Eq.~(\\ref{eq:Rice_Mele_incoherence}) becomes\n\\begin{align}\n \\sum_{n} \\left( t \\psi^{\\dagger}_{n+1}\\psi_{n} + \\text{h.c.} \\right),\n\\end{align}\nwhich is the Hamiltonian for the 1D Rice-Mele chain at the gapless critical point.\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nHopping from $(n,m,l)$ to & Hopping matrices & Peierls phases \\\\\n\\hline\n\\hline\n$(n+1,m,l)$ & $t$ & $1$ \\\\\n\\hline\n$(n,m,l+1)$ & $ \\Delta \/ (2i) = -i\\Delta \/ 2$ & $e^{-i \\pi n}$ \\\\\n\\hline \n$(n+1,m+1,l)$ & $(\\delta t\/ 2) \\cdot e^{i \\pi \/2} = i\\delta t\/ 2 $ & $e^{-i \\pi \\left( n + \\frac{1}{2} \\right)}$ \\\\\n\\hline\n$(n+1,m-1,l)$ & $(\\delta t\/ 2) \\cdot e^{-i \\pi \/2} = -i\\delta t\/ 2 $ & $e^{i \\pi \\left( n + \\frac{1}{2} \\right)}$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Hopping matrices and the corresponding Peierls phases for the promoted $3$D model for a $1$D Rice-Mele chain with an incoherence between the phase offsets of on-site and hopping modulations. \nThe hopping matrices along the opposite directions of those listed here are omitted and can be obtained through Hermitian conjugation.}\n\\label{tab:hopping_2_incoherence}\n\\end{table}\n\n\\subsubsection{\\label{sec:1D_to_2D_hexa}1D modulated system $\\to$ 2D hexagonal lattice under a perpendicular magnetic field}\n\nLet us now explore what happens when our dimensionally-promoted lattice vectors are non-orthogonal. \nConsider a $1$D modulated system with both on-site and nearest-neighbor hopping modulations, with Hamiltonian\n\\begin{align}\n H_{1D} = & \\sum_{n_{1}} \\left( \\psi^{\\dagger}_{n_{1}+1} [H_{\\vec{a}_{1}}] \\psi_{n_{1}} + \\psi^{\\dagger}_{n_{1}+1} e^{i \\left( 2\\pi \\vec{q} \\cdot \\left( n_{1} + \\frac{1}{2} \\right)\\vec{a}_{1} + \\phi \\right)} [H_{\\vec{a}_{1}-\\vec{a}_{2}}] \\psi_{n_{1}} + \\text{h.c.} \\right) \\label{eq:1D_hexagonal_1} \\\\\n & + \\sum_{n_{1}} \\left( \\psi^{\\dagger}_{n_{1}} [H_{0}] \\psi_{n_{1}} + \\psi^{\\dagger}_{n_{1}} [V_{n_{1}}] \\psi_{n_{1}} \\right), \\label{eq:1D_hexagonal_2}\n\\end{align}\nwhere $\\text{h.c.}$ denotes hermitian conjugation, and $\\psi^{\\dagger}_{n_{1}}$ is the creation operator for an electron at position $n_{1}\\vec{a}_{1}$. \nAdditionally, $[V_{n_{1}}]$ is a modulated on-site interaction which can be decomposed into\n\\begin{align}\n [V_{n_{1}}] = e^{-i \\left( 2\\pi \\vec{q} \\cdot n_{1}\\vec{a}_{1} + \\phi\\right)} [H_{\\vec{a}_{2}}] + e^{i \\left( 2\\pi \\vec{q} \\cdot n_{1}\\vec{a}_{1} + \\phi \\right)} [H_{\\vec{a}_{2}}]^{\\dagger}, \\label{eq:1D_hexagonal_3}\n\\end{align}\nand $\\vec{a}_{1} = (1,0)$ and $\\vec{q} = (q,0)$. \nWe also have a modulated nearest-neighbor hopping term from site $n_{1}$ to $n_{1} + 1$ given by the second term\n\\begin{equation}\ne^{i \\left( 2\\pi \\vec{q} \\cdot \\left( n_{1} + \\frac{1}{2} \\right)\\vec{a}_{1} + \\phi \\right)} [H_{\\vec{a}_{1}-\\vec{a}_{2}}]\n\\end{equation}\nin Eq.~(\\ref{eq:1D_hexagonal_1}). \nThe $\\vec{a}_{2}$ in Eq.~(\\ref{eq:1D_hexagonal_1}--\\ref{eq:1D_hexagonal_3}) will become useful when we promote the dimension: \nAt this stage, $\\vec{a}_{2}$ is an unspecified label, though we have anticipated that it will be identified with the synthetic direction in the promoted lattice. \n\nWe promote the dimension of this 1D modulated system to 2D and choose the second, synthetic lattice vector as $\\vec{a}_{2} = (1\/2, \\sqrt{3}\/2)$. \nThus, $\\{\\vec{a}_{1},\\vec{a}_{2} \\}$ forms a 2D hexagonal lattice. \nThe reciprocal lattice vectors in this promoted 2D space are then\n\\begin{align}\n & \\vec{g}_{1} = 2\\pi \\left( 1 , -\\frac{1}{\\sqrt{3}} \\right), \\\\\n & \\vec{g}_{2} = 2\\pi \\left( 0,\\frac{2}{\\sqrt{3}}\\right),\n\\end{align}\nwhere we bear in mind that, if compared with our previous notation from Eqs.~(\\ref{eq:reciprocal_1}--\\ref{eq:reciprocal_4}), we should identify $\\vec{a}_{2} \\leftrightarrow \\vec{c}_{1}$ and $\\vec{g}_{2} \\leftrightarrow \\vec{G}_{1}$. \nUsing the Fourier expansion convention in Table~\\ref{tab:model_summary} and Table~\\ref{tab:hopping_1} with $\\vec{q} = (q,0)$, and Eqs.~(\\ref{eq:1D_hexagonal_1}--\\ref{eq:1D_hexagonal_3}), we obtain the hopping terms in this case in Table~\\ref{tab:hopping_3}. \nMultiplying the entries for hopping matrices and Peierls phases, the Hamiltonian for the promoted 2D model is\n\\begin{align}\n H_{2D}= & \\sum_{n_{1},n_{2}} \\left( \\psi^{\\dagger}_{n_{1}+1,n_{2}} [H_{\\vec{a}_{1}}] \\psi_{n_{1},n_{2}} + \\psi^{\\dagger}_{n_{1},n_{2}+1} [H_{\\vec{a}_{2}}]e^{-i 2\\pi \\vec{q} \\cdot n_{1}\\vec{a}_{1}} \\psi_{n_{1},n_{2}} + \\psi^{\\dagger}_{n_{1}+1,n_{2}-1} [H_{\\vec{a}_{1}-\\vec{a}_{2}}] e^{i 2\\pi \\vec{q} \\cdot \\left( n_{1} + \\frac{1}{2} \\right)\\vec{a}_{1}} \\psi_{n_{1},n_{2}} + \\text{h.c.} \\right) \\\\\n & + \\sum_{n_{1},n_{2}} \\psi^{\\dagger}_{n_{1},n_{2}} [H_{0}] \\psi_{n_{1},n_{2}},\n\\end{align}\nwhere $\\psi^{\\dagger}_{n_{1},n_{2}}$ is the creation operator for an electron at position $n_{1}\\vec{a}_{1}+ n_{2}\\vec{a}_{2}$, and the vector potential to which this 2D model is coupled is \n\\begin{align}\n \\vec{A} = \\frac{1}{2\\pi} \\vec{g}_{2} \\left( \\vec{q}\\cdot \\left[ \\left( \\vec{r} \\cdot \\vec{g}_{1}\\right) \\vec{a}_{1} \\right] \\right) = \\left( 0 , \\frac{4\\pi q}{\\sqrt{3}} \\left( x - \\frac{y}{\\sqrt{3}} \\right) \\right).\n\\end{align}\nThis $\\vec{A}$ reproduces a perpendicular magnetic field $\\vec{B} = \\frac{4\\pi q}{\\sqrt{3}} \\hat{z}$ in the promoted 2D system. \nWe thus see that this 1D modulated system may be used to map out the Hofstadter spectrum of a hexagonal lattice with an irrational magnetic flux.\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nHopping from $n_{1}\\vec{a}_{1} + n_{2}\\vec{a}_{2}$ to & Hopping matrices & Peierls phase \\\\\n\\hline\n\\hline\n$n_{1}\\vec{a}_{1} + n_{2}\\vec{a}_{2}$ & $[H_{0}]$ & $1$ \\\\\n\\hline\n$(n_{1}+1)\\vec{a}_{1} + n_{2}\\vec{a}_{2}$ & $[H_{\\vec{a}_{1}}]$ & $1$ \\\\\n\\hline\n$n_{1}\\vec{a}_{1} + (n_{2}+1)\\vec{a}_{2}$ & $[H_{\\vec{a}_{2}}]$ & $e^{-i 2\\pi \\vec{q} \\cdot n_{1}\\vec{a}_{1}}$ \\\\\n\\hline\n$(n_{1}+1)\\vec{a}_{1} + (n_{2}-1)\\vec{a}_{2}$ & $e^{i 2\\pi \\vec{q} \\cdot \\frac{1}{2}\\vec{a}_{1}} [H_{\\vec{a}_{1}-\\vec{a}_{2}}]e^{-i \\pi \\vec{q}\\cdot \\vec{a}_{1}} = [H_{\\vec{a}_{1}-\\vec{a}_{2}}]$ & $e^{i 2\\pi \\vec{q}\\cdot \\left( n_{1} + \\frac{1}{2} \\right)\\vec{a}_{1}}$\\\\\n\\hline\n\\end{tabular}\n\\caption{Hopping matrices and the corresponding Peierls phases for the promoted $2$D model for a 1D modulated chain described by Eqs.~(\\ref{eq:1D_hexagonal_1}--\\ref{eq:1D_hexagonal_3}). \nThe hopping matrices along the opposite directions for those listed here are omitted and can be obtained through Hermitian conjugate.}\n\\label{tab:hopping_3}\n\\end{table}\n\n\\subsubsection{2D hexagonal lattice $\\to$ 3D hexagonal lattice coupled to a $U(1)$ gauge field}\n\nConsider a 2D modulated system with one on-site modulation, hexagonal lattice and Hamiltonian\n\\begin{align}\n H_{2D}= \\sum_{\\vec{n},\\vec{m}} \\psi^{\\dagger}_{\\vec{n}+\\vec{m}} [H_{\\vec{m}}] \\psi_{\\vec{n}} + \\sum_{\\vec{n}} \\psi^{\\dagger}_{\\vec{n}} [V_{\\vec{n}}] \\psi_{\\vec{n}}, \\label{eq:2D_hexa_to_3D_hexa_1}\n\\end{align}\nwhere $\\vec{n} = (n_{1},n_{2})$, $\\vec{m} = (m_{1},m_{2})$, $\\vec{a}_{1} = \\left(1\/2,\\sqrt{3}\/2\\right)$, $\\vec{a}_{2} = \\left(-1\/2,\\sqrt{3}\/2\\right)$, and $\\psi^{\\dagger}_{\\vec{n}}$ is the creation operator for an electron at lattice position $n_{1} \\vec{a}_{1} + n_{2} \\vec{a}_{2}$.\nThe matrix $[H_{\\vec{m}}]$ describes hopping terms from position $n_{1} \\vec{a}_{1} + n_{2} \\vec{a}_{2}$ to $(n_{1}+m_{1}) \\vec{a}_{1} + (n_{2}+m_{2})\\vec{a}_{2}$ which can include long range hopping terms. \nThe on-site modulation can be expanded as \n\\begin{align}\n [V_{\\vec{n}}] = \\sum_{p} [V_{p}] e^{i p \\left( 2\\pi \\vec{q} \\cdot \\left(\\sum_{j=1}^{2}n_{j}\\vec{a}_{j} \\right) + \\phi \\right) }, \\label{eq:2D_hexa_to_3D_hexa_2}\n\\end{align}\nwhere $p \\in \\mathbb{Z}$, $[V_{p}]$ is the $p^{\\text{th}}$ Fourier coefficient of $[V_{\\vec{n}}]$ and $\\vec{q} = (q_{x},q_{y})$ is the modulation wave vector parallel to the 2D system. \nWe next promote the dimension of this 2D system to 3D and choose the third, synthetic lattice vector $\\vec{a}_{3}$ as $(0,0,1)$. \nThe lattice vectors in the promoted 3D space are then\n\\begin{align}\n \\vec{a}_{1} = \\left( \\frac{1}{2},\\frac{\\sqrt{3}}{2},0 \\right),\\ \\vec{a}_{2} = \\left( -\\frac{1}{2},\\frac{\\sqrt{3}}{2},0 \\right),\\ \\vec{a}_{3} = \\left( 0,0,1 \\right),\n\\end{align}\nwhich describes a 3D hexagonal lattice. \nThe corresponding reciprocal lattice vectors are\n\\begin{align}\n \\vec{g}_{1} = 2\\pi \\left( 1, \\frac{1}{\\sqrt{3}},0 \\right),\\ \\vec{g}_{2} = 2\\pi \\left( -1, \\frac{1}{\\sqrt{3}},0 \\right),\\ \\vec{g}_{3} = 2\\pi (0,0,1).\n\\end{align}\nAs in Sec.~\\ref{sec:1D_to_2D_hexa}, we bear in mind that in comparing with Eqs.~(\\ref{eq:reciprocal_1}--\\ref{eq:reciprocal_4}), we should identify $\\vec{a}_{3} \\leftrightarrow \\vec{c}_{1} $ and $\\vec{g}_{3} \\leftrightarrow \\vec{G}_{1}$. \nUsing the Fourier expansion convention in Table~\\ref{tab:model_summary} and Table~\\ref{tab:hopping_1} with $\\vec{q} = (q_{x},q_{y})$, and Eqs.~(\\ref{eq:2D_hexa_to_3D_hexa_1}--\\ref{eq:2D_hexa_to_3D_hexa_2}), we obtain the hopping terms given in Table~\\ref{tab:hopping_4}. \nMultiplying the entries for hopping matrices and Peierls phases, the Hamiltonian for the promoted 3D model is\n\\begin{align}\n H_{3D} = \\sum_{\\vec{n},\\vec{m}} \\psi^{\\dagger}_{\\vec{n} + \\vec{m}} [H_{\\vec{m}}] \\psi_{\\vec{n}} + \\sum_{\\vec{n},p} \\psi^{\\dagger}_{\\vec{n} - (0,0,p)} [V_{p}] e^{i 2\\pi p \\vec{q} \\cdot \\left( \\sum_{j=1}^{2}n_{j}\\vec{a}_{j} \\right)} \\psi_{\\vec{n}},\n\\end{align}\nwhere $\\vec{n} = (n_{1},n_{2},n_{3})$, $\\vec{m }= (m_{1},m_{2},0)$, $\\psi^{\\dagger}_{\\vec{n}}$ is the creation operator for an electron at lattice position $\\sum_{j=1}^{3}n_{j}\\vec{a}_{j}$, and the vector potential to which this 3D model is coupled is \n\\begin{align}\n \\vec{A} = \\frac{1}{2\\pi} \\vec{g}_{3} \\left( \\vec{q}\\cdot \\left[ \\left( \\vec{r} \\cdot \\vec{g}_{1}\\right) \\vec{a}_{1} + \\left( \\vec{r} \\cdot \\vec{g}_{2}\\right) \\vec{a}_{2} \\right] \\right) = \\left( 0,0,2\\pi\\left(q_{x}x + q_{y}y \\right) \\right).\n\\end{align}\nThis $\\vec{A}$ produces a magnetic field $\\vec{B} = \\left(2\\pi q_{y},-2\\pi q_{x},0 \\right)$ in the promoted 3D space.\n\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nHopping from $\\sum_{j=1}^{3}n_{j}\\vec{a}_{j}$ to & Hopping matrices & Peierls phase \\\\\n\\hline\n\\hline\n$\\sum_{j=1}^{3}n_{j}\\vec{a}_{j} + \\sum_{j=1}^{2} m_{j}\\vec{a}_{j}$ & $[H_{\\vec{m}}]$ & $1$ \\\\\n\\hline\n$\\sum_{j=1}^{3}n_{j}\\vec{a}_{j} - p \\vec{a}_{3}$ & $[V_{p}]$ & $e^{i2\\pi p \\vec{q} \\cdot \\left(\\sum_{j=1}^{2}n_{j}\\vec{a}_{j} \\right)}$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Hopping matrices and the corresponding Peierls phases for the promoted $3$D model from a 2D hexagonal lattice. \nThe hopping matrices along the opposite directions of those listed here are omitted and can be obtained through Hermitian conjugation.}\n\\label{tab:hopping_4}\n\\end{table}\n\n\n\n\\section{\\label{sec:Thouless_pump_1D_Rice_Mele}Thouless pump and dimensional promotion for 1D Rice-Mele model}\n\nHere we review how topological properties such as the Thouless pump\\cite{Thouless_pump_original_paper} in the 1D Rice-Mele model\\cite{RiceMele} (SSH chain\\cite{su1979solitons}) can be attributed to a 2D lattice model coupled to a $U(1)$ gauge field, via our dimensional promotion procedure. \nConsider the 1D Rice-Mele model\n\\begin{align}\n H_{\\text{Rice-Mele}} = \\sum_{n} \\left( t + \\delta t (-1)^{n} \\cos{\\phi} \\right)c^{\\dagger}_{n+1}c_{n} + \\text{h.c.} + \\sum_{n} (-1)^{n+1} \\Delta \\sin{\\phi} c^{\\dagger}_{n}c_{n},\n \\label{eq:supp_Rice_Mele}\n\\end{align}\nwhere $c^{\\dagger}_{n}$ is the creation operator for an electron at 1D sites $n \\in \\mathbb{Z}$, and $\\text{h.c.}$ means the Hermitian conjugate of all terms before it. \nThis Hamiltonian describes a $1$D chain with a twofold (Peierls) CDW distortion. \nSuppose we prepare the ground state (with Fermi level $E_{F}=0$) for $H_{\\text{Rice-Mele}}$ with $t = 1$, $\\delta t=-0.1$, $\\Delta = 0.5$ at $\\phi = 0$. \nBefore dimensional promotion, let us first review the properties of this 1D Hamiltonian. \nAs the phase $\\phi$ adiabatically changes from $0$ to $2\\pi$, the single valence band Wannier center\\cite{Kohn59,Brouder2007,Marzari2012,ksv,fu2006time} in the bulk is pumped by one unit cell (two lattice sites), as shown in Fig.~\\ref{Fig_Rice_Mele} (c). \nThe bulk polarization then changes by $(-1)\\times d = -d$. \nThis bulk polarization change is quantized\\cite{ksv,resta1994macroscopic} in units of the unit cell length $d=2$. \nThis is the classic realization of the Thouless pump\\cite{Thouless_pump_original_paper} in 1D. \nIn Fig.~\\ref{Fig_Rice_Mele} (a) we show the $\\phi$-sliding spectrum (defined in Sec.~IV of the main text) for a finite chain with size $100$ and positions $n = 1 ,\\ldots, 100$. \nWe see that in addition to the gapped bulk states, there are two linearly dispersing modes that cross the bulk gap. \nThis indicates that during the pumping process, localized charges at the right and left end (see Figs.~\\ref{Fig_Rice_Mele} (b) and (d)) of the chain flow out of and into the occupied state subspace, respectively. \nThis is the celebrated topological origin of the quantized polarization change in the bulk, and the existence of boundary states crossing the energy gap.\n\\begin{figure}[h]\n \\includegraphics[scale=0.4]{Fig_1D_Rice_Mele.pdf}\n \\caption{(a) $\\phi$-sliding spectrum of the Rice-Mele model with $t = 1$, $\\delta t=-0.1$ and $\\Delta = 0.5$. \n (c) The shifting of the valence band Wannier center (blue dots) as a function of $\\phi$. \n The vertical black lines denote the boundary of the unit cell, which has length $2$. \n The orange dots denote the positions of the tight-binding basis orbitals. \n The origin of the $x$-axis is placed at the middle of the bond with hopping $t - \\delta t \\cos{\\phi}$ where $t = 1$ and $\\delta t = -0.1$. \n In other words, the unit cell is formed by the sites $n=1$ and $n=2$ in Eq.~(\\ref{eq:supp_Rice_Mele}) and the origin is placed at the midpoint between these two sites. \n The inversion centers at $\\phi=0$ and $\\pi$ lie at integer values of $x$, and as such the Wannier centers at these two $\\phi$ are located either at the center of the unit cell ($\\phi=0$) or the boundaries between unit cells ($\\phi=\\pi)$\\cite{Aris2014}. \n Also, this unit cell choice is commensurate with the finite size system in (a), (b) and (d), where we choose $n = 1 ,\\ldots, 100$. \n (b) $\\&$ (d) Probability distribution of localized modes around the two ends at $\\phi = 0.9 \\pi$ and energy $E = -0.1545$ $\\&$ $+0.1545$. \n Notice that during the adiabatic pumping from $\\phi =0$ to $2\\pi$, it is the boundary state at the left [right] end, see (d) [(b)], that flows into [out of] the subspace of occupied states. \n This is consistent with (c) where the Wannier center flows toward the right and reappear at the left boundary of the unit cell at $\\phi = \\pi$.}\n \\label{Fig_Rice_Mele}\n\\end{figure}\n\nLet us now see how these properties emerge in our 2D dimensionally-promoted picture. \nIdentifying $\\phi$ as the crystal momentum $k_{y}$ along the second, synthetic dimension $y$, $H_{\\text{Rice-Mele}}$ is equivalent to the Bloch Hamiltonian of a 2D model (see Sec.~\\ref{sec:ex_1D_RM_chain} for the detailed derivation) \n\\begin{align}\n H_{\\text{2D}} = \\sum_{n,m} \\left( tc^{\\dagger}_{n+1,m}c_{n,m}- \\frac{i\\delta t}{2}e^{i\\pi \\left(n+\\frac{1}{2} \\right)} c^{\\dagger}_{n+1,m-1}c_{n,m}+ \\frac{i\\delta t}{2}e^{-i\\pi \\left(n+\\frac{1}{2} \\right)} c^{\\dagger}_{n+1,m+1}c_{n,m}- \\frac{i\\Delta}{2}e^{-in\\pi} c^{\\dagger}_{n,m+1}c_{n,m} \\right)+ \\text{h.c.}, \\label{eq:1D_Rice_Mele_to_2D_1}\n\\end{align}\nwith a fixed $k_{y}$. Here $c^{\\dagger}_{n,m}$ is the creation operator for an electron at 2D site $(n,m)$ on a square lattice. \nIf, during the adiabatic process, $H_{\\text{Rice-Mele}}$ is always gapped, the quantized polarization change is determined by the Chern number\\cite{tknn,niu1984quantised,niu1985quantized,Aris2014,fu2006time,bernevigbook} of the occupied bands of $H_{\\text{2D}}$. \nWe can thus characterize the topological properties shown in Fig.~\\ref{Fig_Rice_Mele} by a Chern number $C=1$. \nWe notice that in addition to hopping terms from site $(n,m)$ to $(n\\pm1,m)$ and $(n,m\\pm1)$, $H_{\\text{2D}}$ also has hopping terms going from site $(n,m)$ to $(n\\pm1,m+1)$ and $(n\\pm1,m- 1)$. \nThis is in contrast to the standard textbook correspondence between the SSH chain and a 2D Chern insulator, where a re-embedding of orbitals within the enlarged unit cell is typically used to obtain a 2D model with only perpendicular hoppings along the $\\hat{x}$ and $\\hat{y}$ directions.\nNote also that $H_{\\text{2D}}$ is a 2D lattice model coupled to a $U(1)$ gauge field $\\vec{A} = (0,\\pi x)$ through the Peierls substitution\\cite{Peierls_substitution} assuming the electron has charge $-1$. \nThis $\\vec{A}$ produces a uniform $U(1)$ magnetic field threading $\\pi$-flux per plaquette, since the magnetic field $\\vec{B} = \\vec{\\nabla} \\cross \\vec{A} = \\pi \\hat{z}$. \nReinserting factors of $\\hbar$ and $|e|$, this corresponds to half a flux quantum $\\Phi_{0} = 2\\pi \\hbar \/ |e|$ per plaquette. \nThe localized and mid-gap states in $H_{\\text{Rice-Mele}}$ are then identified as the chiral edge modes due to the quantum Hall effect in $H_{\\text{2D}}$.\n\n\\section{\\label{sec:promoted_lattice_model} Promoted lattice models}\n\nFor completeness, in this section we give the promoted lattice models in position space corresponding to the 2D modulated system with helical sliding modes and the 3D Weyl semimetal with mean-field charge-density waves (CDWs) in Secs.~V and VI of the main text, respectively.\n\n\\subsection{\\label{sec:promoted_helical_lattice_model}2D modulated system with helical sliding modes}\n\nThe promoted 3D lattice model\\cite{Wieder_spin_decoupled_helical_HOTI} for the 2D modulated system with helical sliding modes in the main text is\n\\begin{align}\n H = \\sum_{\\vec{n}} & \\left( \\psi^{\\dagger}_{\\vec{n}+\\hat{x}} [H_{+\\hat{x}}]\\psi_{\\vec{n}} + \\psi^{\\dagger}_{\\vec{n}-\\hat{x}} [H_{+\\hat{x}}]^{\\dagger}\\psi_{\\vec{n}} +\\psi^{\\dagger}_{\\vec{n}+\\hat{z}} [H_{+\\hat{z}}]\\psi_{\\vec{n}} + \\psi^{\\dagger}_{\\vec{n}-\\hat{z}} [H_{+\\hat{z}}]^{\\dagger}\\psi_{\\vec{n}}+ \\psi^{\\dagger}_{\\vec{n}} [H_{\\text{on-site}}] \\psi_{\\vec{n}} \\nonumber \\right. \\\\\n & \\left. + \\psi^{\\dagger}_{\\vec{n}+\\hat{y}} [H_{+\\hat{y}}] e^{-i\\text{A}_{\\vec{n}+\\hat{y},\\vec{n}}}\\psi_{\\vec{n}} + \\psi^{\\dagger}_{\\vec{n}-\\hat{y}} e^{i\\text{A}_{\\vec{n},\\vec{n}-\\hat{y}}}[H_{+\\hat{y}}]^{\\dagger} \\psi_{\\vec{n}} \\right) \\label{eq:lattice_model_helical_sliding} \n\\end{align}\nwith\n\\begin{align}\n & [H_{+\\hat{x}}] = \\frac{v_{x}}{2}\\tau_{z}\\mu_{0}\\sigma_{0} - \\frac{u_{x}}{2i}\\tau_{y}\\mu_{y}\\sigma_{0},\\\\\n & [H_{+\\hat{y}}] = \\frac{v_{y}}{2}\\tau_{z}\\mu_{0}\\sigma_{0} - \\frac{v_{H}}{2 i} \\tau_{y}\\mu_{z}\\sigma_{z},\\\\\n & [H_{+\\hat{z}}] = \\frac{v_{z}}{2}\\tau_{z}\\mu_{0}\\sigma_{0} - \\frac{u_{z}}{2i}\\tau_{x}\\mu_{0}\\sigma_{0}, \\\\\n & [H_{\\text{on-site}}] = m_{1}\\tau_{z}\\mu_{0}\\sigma_{0}+m_{2}\\tau_{z}\\mu_{x}\\sigma_{0} + m_{3}\\tau_{z}\\mu_{z}\\sigma_{0}+m_{v_{1}}\\tau_{0}\\mu_{z}\\sigma_{0} + m_{v_{2}}\\tau_{0}\\mu_{x}\\sigma_{0}.\n\\end{align}\nThe $SU(2)$ lattice gauge field is given by\n\\begin{align}\n \\text{A}_{\\vec{n}+\\hat{y},\\vec{n}} =2\\pi \\left( q_{x}n_{x}+q_{z}n_{z} \\right) \\tau_{0}\\mu_{0}\\sigma_{z}. \\label{eq:lattice_SU2_A}\n\\end{align}\nThe $SU(2)$ gauge field in the continuous coordinate representation is then (dropping the $\\tau_{0}\\mu_{0}$)\n\\begin{align}\n \\vec{A} = (0,2\\pi(q_{x}x+q_{z}z)\\sigma_{z},0). \\label{eq:A_SU2}\n\\end{align}\nThe corresponding $SU(2)$ magnetic field $\\vec{B} = \\vec{\\nabla} \\cross \\vec{A} - i \\vec{A} \\cross \\vec{A}$ is\\cite{Estienne_2011}\n\\begin{align}\n \\vec{B} = (-2\\pi q_{z}\\sigma_{z},0,2\\pi q_{x}\\sigma_{z}). \\label{eq:B_SU2_1}\n\\end{align}\nEq.~(\\ref{eq:lattice_model_helical_sliding}) describes a helical higher-order topological insulator (HOTI) coupled to a $SU(2)$ gauge field. \nIf we Fourier transform Eq.~(\\ref{eq:lattice_model_helical_sliding}) along $y$ and regard $k_{y}$ (wavenumber along $y$) as the sliding phase $\\phi$, we can obtain the 2D modulated system in the main text.\n\n\\subsection{\\label{sec:promoted_4D_lattice_model}3D Weyl semimetal with mean-field charge density waves}\n\nThe promoted 4D lattice model for the 3D Weyl semimetal with mean-field CDW order\\cite{dynamical_axion_insulator_BB} in the main text is\n\\begin{align}\n H&=\\left(\\sum_{\\vec{n}}\\left[-it_x\\psi^\\dag_{\\vec{n}+\\hat{x}}\\sigma_x \\psi_{\\vec{n}}-it_y \\psi^\\dag_{\\vec{n}+\\hat{y}}\\sigma_y \\psi_{\\vec{n}}+t_z\\psi^\\dag_{\\vec{n}+\\hat{z}}\\sigma_z \\psi_{\\vec{n}} + |\\Delta| e^{-i2 \\pi q n_{z}} \\psi^\\dag_{\\vec{n}+\\hat{w}} \\sigma_{z} \\psi_{\\vec{n}} \\right]\\right. \\nonumber \\\\\n & \\left. +\\sum_{\\vec{n}}\\frac{m}{2}\\left(\\psi^\\dag_{\\vec{n}+\\hat{x}}\\sigma_z \\psi_{\\vec{n}} + \\psi^\\dag_{\\vec{n}+\\hat{y}}\\sigma_z \\psi_{\\vec{n}} -2 \\psi^\\dag_{\\vec{n}}\\sigma_z \\psi_{\\vec{n}} \\right) -\\sum_{\\vec{n}} t_z \\left(\\cos (\\pi q)\\right) \\psi^\\dag_{\\vec{n}}\\sigma_z \\psi_{\\vec{n}}\\right) +\\mathrm{h.c.}, \\label{eq:lattice_model_4D_nodal_line}\n\\end{align}\nwhere the phase factors correspond to Peierls substitution of a 4D $U(1)$ gauge field\n\\begin{align}\n \\vec{A} = (0,0,0,2\\pi q z).\n\\end{align}\nThe only non-zero components of the $U(1)$ field strength $F_{\\mu \\nu} = \\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu}$ is\n\\begin{align}\n F_{zw} = -F_{wz} = \\partial_{z}A_{w} - \\partial_{w}A_{z}=2 \\pi q, \\label{eq:4D_U1_Fzw}\n\\end{align}\nwhich threads through the $zw$ plane. \nWithout coupling to a 4D $U(1)$ gauge field, the Bloch Hamiltonian of Eq.~(\\ref{eq:lattice_model_4D_nodal_line}) with $q=0$ is\n\\begin{align}\n H(\\vec{k})=& -2[t_x \\sin (k_x) \\sigma_x +t_y \\sin (k_y) \\sigma_y] +2t_z[\\cos (k_z) -\\cos (\\pi q)] \\sigma_{z} -m[2-\\cos (k_x) - \\cos (k_y)] \\sigma_{z} + 2|\\Delta|\\cos{(k_{w})}\\sigma_{z},\n\\label{eq:lattice_model_4D_nodal_line_bloch}\n\\end{align}\nwhich has a nodal line in the $k_{z}$-$k_{w}$ plane (with $k_{x}=k_{y}=0$) defined by\n\\begin{align}\n t_{z}\\cos(k_{z}) + |\\Delta| \\cos(k_{w}) = t_{z}\\cos{(\\pi q)}. \\label{eq:lattice_nodal_line}\n\\end{align}\nTherefore, Eq.~(\\ref{eq:lattice_model_4D_nodal_line}) describes a 4D nodal line system coupled to a 4D $U(1)$ gauge field where the corresponding field strength Eq.~(\\ref{eq:4D_U1_Fzw}) threads through the area enclosed by the nodal line defined in Eq.~(\\ref{eq:lattice_nodal_line}). \nIf we Fourier transform Eq.~(\\ref{eq:lattice_model_4D_nodal_line}) along $w$ and regard $k_{w}$ as the sliding phase $\\phi$, we obtain the model for a 3D Weyl semimetal with mean-field CDW order given in the main text.\n\n\\section{\\label{sec:inv_sym_gauge_tr}Inversion symmetry up to a gauge transformation}\n \nIn this section we will show that if a lattice has inversion symmetry, such as the 3D models promoted from our examples of 2D modulated systems with chiral and helical sliding modes, then upon coupling to a $U(1)$ or $SU(2)$ gauge field producing a constant magnetic field, the inversion symmetry is still preserved {\\it up to a gauge transformation}. \nWe will do this in details in the simplest case, which is a 2D square lattice coupled to a perpendicular magnetic field. \nWe will briefly mention the generalization to the 3D cases, which corresponds to the dimensional promotion from 2D modulated systems.\n\n\\subsection{2D system with inversion symmetry}\n\nLet us consider a 2D square lattice with only one degree of freedom within each unit cell labelled by $(n,m) \\in \\mathbb{Z}^{2}$\n\\begin{align}\n H = -t\\sum_{n,m} \\left( \\psi^{\\dagger}_{n+1,m}\\psi_{n,m} +\\psi^{\\dagger}_{n-1,m}\\psi_{n,m}+ e^{-i2\\pi bn}\\psi^{\\dagger}_{n,m+1}\\psi_{n,m}+ e^{+i2\\pi bn}\\psi^{\\dagger}_{n,m-1}\\psi_{n,m} \\right). \\label{eq:inv_sym_gauge_tr_1}\n\\end{align}\nEq.~(\\ref{eq:inv_sym_gauge_tr_1}) is coupled to a $U(1)$ gauge field $\\vec{A} = 2\\pi bx \\hat{y}$ through Peierls substitution where we have assumed the particle carries charge $-1$. \nThis $U(1)$ gauge field produces a constant perpendicular magnetic field $\\vec{B} = 2\\pi b \\hat{z}$. \nIf we turn off the $U(1)$ gauge field by setting $b = 0$, Eq.~(\\ref{eq:inv_sym_gauge_tr_1}) will have inversion symmetry, where the inversion operators are defined by \n\\begin{align}\n & \\mathcal{I}_{n_{c},m_{c}} \\psi^{\\dagger}_{n,m} \\mathcal{I}_{n_{c},m_{c}}^{-1} = \\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m}, \\label{eq:inv_sym_gauge_tr_2} \\\\\n & \\mathcal{I}_{n_{c},m_{c}} \\psi_{n,m} \\mathcal{I}_{n_{c},m_{c}}^{-1} = \\psi_{2n_{c}-n,2m_{c}-m}. \\label{eq:inv_sym_gauge_tr_3}\n\\end{align}\nHere $\\mathcal{I}_{n_{c},m_{c}}$ is a unitary operator and $(n_{c},m_{c})$ is the inversion center (which can be any lattice point). We then have\n\\begin{align}\n \\mathcal{I}_{n_{c},m_{c}} H \\mathcal{I}_{n_{c},m_{c}}^{-1} = H\\ \\forall (n_{c},m_{c})\n\\end{align}\nprovided that the summation over $n$ and $m$ in Eq.~(\\ref{eq:inv_sym_gauge_tr_1}) goes from $- \\infty$ to $+\\infty$. \nWhen $b \\ne 0$, we must modify our unitary inversion operations in Eq.~(\\ref{eq:inv_sym_gauge_tr_2}) and Eq.~(\\ref{eq:inv_sym_gauge_tr_3}) to be\n\\begin{align}\n & \\mathcal{I}_{n_{c},m_{c}} \\psi^{\\dagger}_{n,m} \\mathcal{I}_{n_{c},m_{c}}^{-1} = \\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m}e^{i4\\pi b n_{c}(m-m_{c})}, \\label{eq:inv_sym_gauge_tr_4} \\\\\n & \\mathcal{I}_{n_{c},m_{c}} \\psi_{n,m} \\mathcal{I}_{n_{c},m_{c}}^{-1} = \\psi_{2n_{c}-n,2m_{c}-m}e^{-i4\\pi b n_{c}(m-m_{c})}, \\label{eq:inv_sym_gauge_tr_5}\n\\end{align}\nwhich acts as inversion through the center $(n_{c},m_{c})$ together with a gauge transformation.\nWith these modified inversion operations, the following three identities can be proved: \n\\begin{align}\n & \\mathcal{I}_{n_{c},m_{c}} H \\mathcal{I}_{n_{c},m_{c}}^{-1} = H \\ \\forall (n_{c},m_{c}), \\label{eq:inv_sym_gauge_tr_6} \\\\\n & \\left( \\mathcal{I}_{n_{c},m_{c}} \\right)^{2} \\psi^{\\dagger}_{n,m} \\left( \\mathcal{I}_{n_{c},m_{c}}^{-1} \\right)^{2} = \\psi^{\\dagger}_{n,m}, \\label{eq:inv_sym_gauge_tr_7}\\\\\n & \\left( \\mathcal{I}_{n_{c},m_{c}} \\right)^{2} \\psi_{n,m} \\left( \\mathcal{I}_{n_{c},m_{c}}^{-1} \\right)^{2} = \\psi_{n,m}. \\label{eq:inv_sym_gauge_tr_8}\n\\end{align}\nWe prove Eq.~(\\ref{eq:inv_sym_gauge_tr_6}) as follows:\n\\begin{align}\n \\mathcal{I}_{n_{c},m_{c}} H \\mathcal{I}_{n_{c},m_{c}}^{-1} & = -t \\sum_{n,m} \\begin{bmatrix}\n \\psi^{\\dagger}_{2n_{c}-n-1,2m_{c}-m} e^{i 4\\pi b n_{c}(m-m_{c})} \\psi_{2n_{c}-n,2m_{c}-m} e^{-i 4\\pi b n_{c}(m-m_{c})} \\\\\n + \\psi^{\\dagger}_{2n_{c}-n+1,2m_{c}-m} e^{i 4\\pi b n_{c}(m-m_{c})} \\psi_{2n_{c}-n,2m_{c}-m} e^{-i 4\\pi b n_{c}(m-m_{c})} \\\\\n + e^{-i2\\pi b n } \\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m-1}e^{i 4\\pi b n_{c}(m+1-m_{c})} \\psi_{2n_{c}-n,2m_{c}-m} e^{-i 4\\pi b n_{c}(m-m_{c})} \\\\\n + e^{+i2\\pi b n } \\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m+1}e^{i 4\\pi b n_{c}(m-1-m_{c})} \\psi_{2n_{c}-n,2m_{c}-m} e^{-i 4\\pi b n_{c}(m-m_{c})}\n \\end{bmatrix} \\label{eq:pf_inv_1}\\\\\n & = -t \\sum_{n,m} \\begin{bmatrix}\n \\psi^{\\dagger}_{2n_{c}-n-1,2m_{c}-m} \\psi_{2n_{c}-n,2m_{c}-m} \\\\\n + \\psi^{\\dagger}_{2n_{c}-n+1,2m_{c}-m} \\psi_{2n_{c}-n,2m_{c}-m} \\\\\n + e^{-i2\\pi b n } \\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m-1}e^{i 4\\pi b n_{c}} \\psi_{2n_{c}-n,2m_{c}-m} \\\\\n + e^{+i2\\pi b n } \\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m+1}e^{-i 4\\pi b n_{c}} \\psi_{2n_{c}-n,2m_{c}-m} \n \\end{bmatrix} \\label{eq:pf_inv_2} \\\\\n & = -t \\sum_{n,m} \\begin{bmatrix}\n \\psi^{\\dagger}_{n-1,m} \\psi_{n,m} \\\\\n + \\psi^{\\dagger}_{n+1,m} \\psi_{n,m} \\\\\n + e^{-i2\\pi b (-n+2n_{c}) } \\psi^{\\dagger}_{n,m-1}e^{i 4\\pi b n_{c}} \\psi_{n,m} \\\\\n + e^{+i2\\pi b (-n+2n_{c}) } \\psi^{\\dagger}_{n,m+1}e^{-i 4\\pi b n_{c}} \\psi_{n,m} \n \\end{bmatrix} \\label{eq:pf_inv_3} \\\\\n & = -t \\sum_{n,m} \\begin{bmatrix}\n \\psi^{\\dagger}_{n-1,m} \\psi_{n,m} \\\\\n + \\psi^{\\dagger}_{n+1,m} \\psi_{n,m} \\\\\n + e^{+i2\\pi bn } \\psi^{\\dagger}_{n,m-1}\\psi_{n,m} \\\\\n + e^{-i2\\pi bn } \\psi^{\\dagger}_{n,m+1} \\psi_{n,m} \n \\end{bmatrix} \\label{eq:pf_inv_4} \\\\\n & = H. \\label{eq:pf_inv_5}\n\\end{align}\nIn Eq.~(\\ref{eq:pf_inv_1}) we apply the inversion operation to each of the creation and annihilation operators according to Eq.~(\\ref{eq:inv_sym_gauge_tr_4}) and Eq.~(\\ref{eq:inv_sym_gauge_tr_5}). \nThe first to fourth terms in Eq.~(\\ref{eq:pf_inv_1}) correspond to the (transformed) first to fourth terms in Eq.~(\\ref{eq:inv_sym_gauge_tr_1}). \nIn Eq.~(\\ref{eq:pf_inv_2}) we cancel out redundant exponential phase factors and reindex $n \\to -n + 2n_{c}$ and $m\\to -m+2m_{c}$ in Eq.~(\\ref{eq:pf_inv_3}). \nEq.~(\\ref{eq:pf_inv_4}) and Eq.~(\\ref{eq:pf_inv_5}) then show that the transformed Hamiltonian is the same. \nWe can also prove Eq.~(\\ref{eq:inv_sym_gauge_tr_7}) as follows:\n\\begin{align}\n \\left( \\mathcal{I}_{n_{c},m_{c}} \\right)^{2} \\psi^{\\dagger}_{n,m} \\left( \\mathcal{I}_{n_{c},m_{c}}^{-1} \\right)^{2} & = \\mathcal{I}_{n_{c},m_{c}} \\mathcal{I}_{n_{c},m_{c}} \\psi^{\\dagger}_{n,m} \\mathcal{I}_{n_{c},m_{c}}^{-1} \\mathcal{I}_{n_{c},m_{c}}^{-1} \\\\\n & = \\mathcal{I}_{n_{c},m_{c}} \\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m}e^{i4\\pi b n_{c}(m-m_{c})} \\mathcal{I}_{n_{c},m_{c}}^{-1}\\\\\n & = \\psi^{\\dagger}_{2n_{c}-(2n_{c}-n),2m_{c}-(2m_{c}-m)} e^{i4\\pi b n_{c}((2m_{c}-m)-m_{c})} e^{i4\\pi b n_{c}(m-m_{c})}\\\\\n & = \\psi^{\\dagger}_{n,m} e^{i4\\pi b n_{c}(m_{c}-m)} e^{i4\\pi b n_{c}(m-m_{c})} \\\\\n & = \\psi^{\\dagger}_{n,m} .\n\\end{align}\nWe have used Eq.~(\\ref{eq:inv_sym_gauge_tr_4}) twice above, first acting on $\\psi^{\\dagger}_{n,m}$ and then on $\\psi^{\\dagger}_{2n_{c}-n,2m_{c}-m}$. \nEq.~(\\ref{eq:inv_sym_gauge_tr_8}) can also be proved in a similar way as follows:\n\\begin{align}\n \\left( \\mathcal{I}_{n_{c},m_{c}} \\right)^{2} \\psi_{n,m} \\left( \\mathcal{I}_{n_{c},m_{c}}^{-1} \\right)^{2} & = \\mathcal{I}_{n_{c},m_{c}} \\mathcal{I}_{n_{c},m_{c}} \\psi_{n,m} \\mathcal{I}_{n_{c},m_{c}}^{-1} \\mathcal{I}_{n_{c},m_{c}}^{-1} \\\\\n & = \\mathcal{I}_{n_{c},m_{c}} \\psi_{2n_{c}-n,2m_{c}-m}e^{-i4\\pi b n_{c}(m-m_{c})} \\mathcal{I}_{n_{c},m_{c}}^{-1}\\\\\n & = \\psi_{2n_{c}-(2n_{c}-n),2m_{c}-(2m_{c}-m)} e^{-i4\\pi b n_{c}((2m_{c}-m)-m_{c})} e^{-i4\\pi b n_{c}(m-m_{c})}\\\\\n & = \\psi_{n,m} e^{-i4\\pi b n_{c}(m_{c}-m)} e^{-i4\\pi b n_{c}(m-m_{c})} \\\\\n & = \\psi_{n,m} .\n\\end{align}\nEq.~(\\ref{eq:inv_sym_gauge_tr_6}) implies that if the $H$ in Eq.~(\\ref{eq:inv_sym_gauge_tr_1}) has non-zero $b$ with the summation over $n$ and $m$ going from $- \\infty$ to $+\\infty$, then $H$ is invariant under the inversion operation $\\mathcal{I}_{n_{c},m_{c}}$ defined by Eqs.~(\\ref{eq:inv_sym_gauge_tr_4}) and (\\ref{eq:inv_sym_gauge_tr_5}). \nAlso, since $\\psi^{\\dagger}_{n,m}$ and $\\psi_{n,m}$ denote the creation and annihilation operators for electronic states spanning the whole Hilbert space, Eq.~(\\ref{eq:inv_sym_gauge_tr_7}) and Eq.~(\\ref{eq:inv_sym_gauge_tr_8}) imply that $\\left( \\mathcal{I}_{n_{c},m_{c}} \\right)^{2}$ is the identity operation. \n\nTo complete the proof, we give a construction for the unitary operator $\\mathcal{I}_{n_c,m_c}$.\nThe matrix representation of $\\mathcal{I}_{n_{c},m_{c}}$ in Eq.~(\\ref{eq:inv_sym_gauge_tr_5}) for sites at $(n,m)$ and $(2n_{c}-n,2m_{c}-m)$ is given by\n\\begin{align}\n \\begin{bmatrix}\n 0 & e^{-i4\\pi b n_{c}(m-m_{c})} \\\\\n e^{-i4\\pi b n_{c}(m_{c}-m)} & 0\n \\end{bmatrix}, \\label{eq:mat_I_nc_mc}\n\\end{align}\nsince\n\\begin{align}\n & \\mathcal{I}_{n_{c},m_{c}} \\psi_{n,m} \\mathcal{I}_{n_{c},m_{c}}^{-1} = \\psi_{2n_{c}-n,2m_{c}-m}e^{-i4\\pi b n_{c}(m-m_{c})}, \\\\\n & \\mathcal{I}_{n_{c},m_{c}} \\psi_{2n_{c}-n,2m_{c}-m} \\mathcal{I}_{n_{c},m_{c}}^{-1} = \\psi_{n,m}e^{-i4\\pi b n_{c}(m_{c}-m)}.\n\\end{align}\nAs we can see Eq.~(\\ref{eq:mat_I_nc_mc}) is in fact a unitary matrix. \nTherefore, even though in this 2D lattice we have a constant perpendicular magnetic field which couples to the lattice through a Peierls substitution, there still exist unitary inversion operators with inversion centers at every lattice site which square to the identity and commute with the Hamiltonian.\n\n\\subsection{3D system with inversion symmetry}\n\nWe have shown that by defining unitary inversion operations up to a gauge transformation in Eq.~(\\ref{eq:inv_sym_gauge_tr_4}) and Eq.~(\\ref{eq:inv_sym_gauge_tr_5}), the inversion symmetry of Eq.~(\\ref{eq:inv_sym_gauge_tr_1}) is preserved. \nLet us now discuss how this extends to our $3$D promoted systems. \nSince our $3$D model of a chiral HOTI from the main text is also coupled to a constant $U(1)$ gauge field given by $\\vec{A} = (0,0,2\\pi(q_{x}x+q_{y}y))$, we can construct the proper unitary inversion operators squaring to the identity, given by\n\\begin{align}\n & \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}} \\psi^{\\dagger}_{n_{x},n_{y},n_{z}} \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}}^{-1} = \\psi^{\\dagger}_{2n_{x}^{c}-n_{x},2n_{y}^{c}-n_{y},2n_{z}^{c}-n_{z}} [\\mathcal{I}]^{-1} e^{i 4\\pi \\left( q_{x}n_{x}^{c} + q_{y}n_{y}^{c} \\right) \\left( n_{z} - n_{z}^{c} \\right)} , \\\\\n & \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}} \\psi_{n_{x},n_{y},n_{z}} \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}}^{-1} = [\\mathcal{I}] \\psi_{2n_{x}^{c}-n_{x},2n_{y}^{c}-n_{y},2n_{z}^{c}-n_{z}} e^{-i 4\\pi \\left( q_{x}n_{x}^{c} + q_{y}n_{y}^{c} \\right) \\left( n_{z} - n_{z}^{c} \\right)}.\n\\end{align}\nIn these expressions $\\psi^{\\dagger}_{n_{x},n_{y},n_{z}}$ is the 4-component creation operator for an electron at site $(n_{x},n_{y},n_{z})$, $(n_{x}^{c},n_{y}^{c},n_{z}^{c})$ is the inversion center at any lattice point, and $[\\mathcal{I}] = \\tau_{z}\\sigma_{0}$ is the unitary inversion matrix which also squares to the identity and acts on the degrees of freedom within a unit cell. \nFor the case of our helical model, recall that our example of Eq.~(\\ref{eq:lattice_model_helical_sliding}) in Sec.~\\ref{sec:promoted_helical_lattice_model} is spin-decoupled. \nThus the $SU(2)$ gauge field given by $\\vec{A} = (0,2\\pi (q_{x}x+q_{z}z)\\sigma_{z},0)$ to which the model is coupled acts effectively as oppositely oriented $U(1)$ magnetic fields for spin up and down electrons. \nWe can thus define the unitary inversion operations squaring to identity up to a spin-dependent ($SU(2)$) gauge transformation\n\\begin{align}\n & \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}} \\psi^{\\dagger}_{n_{x},n_{y},n_{z},\\sigma} \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}}^{-1} = \\psi^{\\dagger}_{2n_{x}^{c}-n_{x},2n_{y}^{c}-n_{y},2n_{z}^{c}-n_{z},\\sigma} [\\mathcal{I}]^{-1} e^{i 4\\sigma\\pi \\left( q_{x}n_{x}^{c} + q_{z}n_{z}^{c} \\right) \\left( n_{y} - n_{y}^{c} \\right)} , \\\\\n & \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}} \\psi_{n_{x},n_{y},n_{z},\\sigma} \\mathcal{I}_{n_{x}^{c},n_{y}^{c},n_{z}^{c}}^{-1} = [\\mathcal{I}] \\psi_{2n_{x}^{c}-n_{x},2n_{y}^{c}-n_{y},2n_{z}^{c}-n_{z},\\sigma} e^{-i 4\\sigma\\pi \\left( q_{x}n_{x}^{c} + q_{z}n_{z}^{c} \\right) \\left( n_{y} - n_{y}^{c} \\right)},\n\\end{align}\nwhere $\\psi^{\\dagger}_{n_{x},n_{y},n_{z},\\sigma}$ is the 4-component creation operator for an electron at site $(n_{x},n_{y},n_{z})$ for a fixed spin $\\sigma = \\pm$ and $[\\mathcal{I}] = \\tau_{z}\\mu_{0}$. \nTherefore, the 3D chiral (helical) HOTI coupled to a $U(1)$ ($SU(2)$) gauge field in our examples preserves inversion symmetry.\n\n\n\\section{\\label{bigsec:chiral_sliding}Low energy bulk theory of chiral higher-order topological sliding modes}\n\nIn this section, we consider the low energy bulk theory of electrons in a 3D chiral HOTI minimally coupled to a $U(1)$ gauge field, shown in the main text and verify that it can capture several qualitative properties in numerical results of the lattice model calculation with $q_{y} = 0.02$.\n\nAs mentioned in the main text, the relevant low energy theory is \n\\begin{align}\n H_{\\text{bulk}} = m_{\\text{bulk}} \\tau_{z}\\sigma_{0} + \\tau_{x}\\vec{p} \\cdot \\vec{\\sigma} + \\tau_{0} \\vec{M} \\cdot \\vec{\\sigma}. \\label{eq:3D_chiral_HOTO_bulk}\n\\end{align}\nCoupling this $H_{\\text{bulk}}$ to a vector potential $\\vec{A} = By \\hat{z}$ such that $p_{z} \\to p_{z} + By$, and defining \n\\begin{align}\n a^{\\dagger}_{k_{z}} = \\frac{1}{\\sqrt{2B}}\\left( k_{z} + By - ip_{y} \\right), \\label{eq:3D_chiral_bulk_U1_ladder}\n\\end{align}\nwe can rewrite the Hamiltonian as \n\\begin{align}\n H_{\\text{bulk}}(k_{x},k_{z}) = m_{\\text{bulk}} \\tau_{z}\\sigma_{0} + \\tau_{x} \\begin{bmatrix}\n \\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}+a^{\\dagger}_{k_{z}} \\right) & k_{x} -\\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}-a^{\\dagger}_{k_{z}} \\right) \\\\\n k_{x} +\\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}-a^{\\dagger}_{k_{z}} \\right) & -\\sqrt{\\frac{B}{2}} \\left(a_{k_{z}}+a^{\\dagger}_{k_{z}} \\right)\n \\end{bmatrix} + \\tau_{0} \\vec{M} \\cdot \\vec{\\sigma}. \\label{eq:3D_bulk_LL_Hamiltonian}\n\\end{align}\n\nFig.~\\ref{SM_chiral_bulk_low_energy} (b) shows the numerically computed energy spectrum $E_{k_{x},k_{z}}$ for Eq.~(\\ref{eq:3D_bulk_LL_Hamiltonian}) as a function of $k_{x}$. \nNote that $E_{k_{x},k_{z}}$ does not depend on $k_{z}$, giving rise to flat Landau levels (LLs) along $k_{z}$. \nWe now relate this low energy bulk theory to the $\\phi$-sliding spectrum of the 2D modulated system by identifying $k_{z}$ in Eq.~(\\ref{eq:3D_bulk_LL_Hamiltonian}) with $\\Delta \\phi = \\phi - \\pi$, as we have done in the main text. \nTo assist the following discussion, we also show again the $\\phi$-sliding spectrum of the 2D modulated system which can be promoted to a 3D chiral HOTI coupled to a $U(1)$ gauge field with $\\vec{q} = (0,q_{y})=(0,0.02)$ in Fig.~\\ref{SM_chiral_bulk_low_energy} (a). \nThe flat bands corresponding to bulk-confined modes in Figs.~\\ref{SM_chiral_bulk_low_energy} (c) and (d) are marked in green ($E = -0.5144$) and orange ($E = +0.4656$).\n\n\nFirst, the band edges for the valence and conduction in Fig.~\\ref{SM_chiral_bulk_low_energy} (b) are at energy $E = -0.5248$ and $+0.4661$.\nThis indicates that the particle-hole symmetry in the bulk of a 3D chiral HOTI is generally broken upon $U(1)$ Landau quantization.\nThis is also reflected in the energy eigenvalues of the bulk flat bands in the lattice model Fig.~\\ref{SM_chiral_bulk_low_energy} (a), with probability density shown in Figs.~\\ref{SM_chiral_bulk_low_energy} (c) and (d). \nWe see that the energies ($E = -0.5144$ for Fig.~\\ref{SM_chiral_bulk_low_energy} (c) and $E = +0.4656$ in Fig.~\\ref{SM_chiral_bulk_low_energy} (d)) of these states are not symmetric about 0. \n\n\nSecond, the Hamiltonian in Eq.~(\\ref{eq:3D_bulk_LL_Hamiltonian}) has eigenstates that are extended along $x$ and $z$ while confined along $y$. \nThe confinement along $y$ can be understood through the definition of the ladder operator in Eq.~(\\ref{eq:3D_chiral_bulk_U1_ladder}), which creates simple harmonic oscillator (SHO) states along $y$. \nProjecting the 3D eigenfunction to the 2D modulated system, only the probability distributions along $x$ and $y$ are physically meaningful. \nTherefore in 2D we expect to see flat bands along $k_{z}$, which is identified as $\\Delta \\phi = \\phi - \\pi$ in the low energy model. The wave functions with these energies are confined along $y$ while extended along $x$. \nThis is consistent with Figs.~\\ref{SM_chiral_bulk_low_energy} (a) and (c)--(d). \nThe above discussion shows that our low energy bulk theory in 3D does qualitatively explain the existence of bulk confined modes in 2D and their flat dispersion along the $\\phi$-axis.\n\n\n\\begin{figure}[h]\n\\includegraphics[scale=0.4]{SM_chiral_bulk_low_energy_add_horizontal_line_at_pm_0_5_new.pdf}\n\\caption{(a) $\\phi$-sliding spectrum of the 2D modulated system that can be promoted to a 3D chiral HOTI\\cite{pozo2019quantization} coupled to a $U(1)$ gauge field with $\\vec{q} = (0,q_{y})$ and $q_{y} = 0.02$. \n(b) Bulk Landau levels of Eq.~(\\ref{eq:3D_bulk_LL_Hamiltonian}) for the low energy model of a 3D chiral HOTI with $m_{\\text{bulk}} = 0.5$, $B = 0.2$ and $\\vec{M} = (0.03,0.03,0.03)$. \nThe band edges are given by $E = -0.5248$ and $0.4661$. \nThe orange, black and green horizontal lines correspond to energies $E = +0.5$, $0$ and $-0.5$, respectively. \n(c) $\\&$ (d) Probability distribution of bulk-confined modes in the flat bands of (a) at $\\phi = \\pi$ with energies $-0.5144$ and $0.4656$ marked by green and orange respectively.\nWe note that (c) and (d) share similar probability distributions, for example both are confined along $y$ and centered around the middle of the finite sample. This is consistent with the low energy theory prediction of energy-asymmetry in Eq.~(\\ref{eq:3D_bulk_LL_Hamiltonian}) and (b). The darker (black) color in (c)--(d) implies higher probability density. \nIn (c) and (d), the $x$- and $y$-coordinate both range from $-15, \\ldots, +15$.}\n\\label{SM_chiral_bulk_low_energy}\n\\end{figure}\n\n\n\\section{\\label{bigsec:helical_sliding}Low energy theory of helical higher-order topological sliding modes}\n\nIn this section, we construct various low energy theories for the dimensionally-promoted 3D helical HOTI coupled to an $SU(2)$ gauge field. \nWe use these low energy models to explain corner, edge and bulk states in the 2D modulated system with helical sliding modes.\n\n\\subsection{2D corner modes $\\leftrightarrow$ 3D hinge modes with $SU(2)$ gauge field}\n\nIn our promoted 3D model, different spin ($\\vec{\\sigma}$) subspaces are decoupled, see Sec.~\\ref{sec:promoted_helical_lattice_model}. \nTherefore, all eigenstates are spin-polarized.\nThe 2D spin-polarized corner modes are the projection of the helical hinge modes in the 3D lattice. \nIf we denote the third, synthetic dimension as $y$ and the corresponding crystal momentum as $k_{y}$, the low energy theory of the helical hinge mode will be\n\\begin{align}\n H_{\\text{hinge}}= v_{F} \\left( k_{y}\\sigma_{z}' + 2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{z}z_{\\text{hinge}} \\right) \\sigma_{0}' \\right), \\label{3D_helical_hinge_mode_Hamiltonian}\n\\end{align}\nwhere we have minimally coupled the Hamiltonian $v_{F}k_{y}\\sigma_{z}'$ to a $SU(2)$ gauge field \n\\begin{align}\n \\vec{A} = (0,2\\pi (q_{x}x+q_{z}z)\\sigma_{z}',0). \\label{SU2_A}\n\\end{align}\nNotice that we denote our basis as $\\vec{\\sigma}'$. \nAlthough the eigenstates of $\\sigma'_z$ have opposite spins, they might contain non-trivial orbital and sub-lattice textures. \nAlso notice that we have $2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{z}z_{\\text{hinge}} \\right) \\sigma_{0}'$ in Eq.~(\\ref{3D_helical_hinge_mode_Hamiltonian}) instead of $2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{z}z_{\\text{hinge}} \\right) \\sigma_{z}'$. \nSince as we replace $k_{y}$ by $k_{y} + 2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{z}z_{\\text{hinge}} \\right) \\sigma_{z}'$ through the minimal coupling, we will have \n\\begin{align}\n v_{F}k_{y}\\sigma_{z}' = v_{F}\\begin{bmatrix}\n k_{y} & 0 \\\\ \n 0 & -k_{y}\n \\end{bmatrix} & \\to v_{F}\\begin{bmatrix}\n \\left( k_{y} + 2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{z}z_{\\text{hinge}} \\right) \\right) & 0 \\\\ \n 0 & -\\left( k_{y} - 2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{z}z_{\\text{hinge}} \\right) \\right)\n \\end{bmatrix} \\\\\n & = v_{F}k_{y}\\sigma_{z}' + v_{F} \\cdot 2\\pi \\left( q_{x}x_{\\text{hinge}} + q_{z}z_{\\text{hinge}} \\right)\\sigma_{0}'.\n\\end{align}\nIn Eq.~(\\ref{3D_helical_hinge_mode_Hamiltonian}), we have assumed that: \n(1) there is only one pair of helical hinge modes along this hinge,\n(2) the magnitude of the group velocity is $v_{F}$, \n(3) the electron has charge $-1$, and \n(4) the fixed position of the hinge along $y$ is at $(x_{\\text{hinge}},z_{\\text{hinge}})$ which is set by the coordinate system. \nEq.~(\\ref{SU2_A}) again implies that the modulation wave vector $\\vec{q} = (q_{x},q_{z})$ enters the definition of the $SU(2)$ gauge field. \nAs we can see in Eq.~(\\ref{3D_helical_hinge_mode_Hamiltonian}), if we tune $\\vec{q}$, we are effectively shifting the hinge mode dispersion along the $k_{y}$-axis for spin up [down] electrons by an amount $-2\\pi \\left(q_{x}x_{\\text{hinge}}+q_{z}z_{\\text{hinge}}\\right)$ [$+2\\pi \\left(q_{x}x_{\\text{hinge}}+q_{z}z_{\\text{hinge}}\\right)$]. \nSince we use $q_{x}=0$ and $q_{z}\\ne0$ in the main text for helical sliding modes, the following discussion will focus on this case. \nThe generalization to other combinations of $q_{x}$ and $q_{z}$ follows the same procedure.\n\n\nTo connect the low energy theory of the $\\phi$-sliding spectrum to the shifting of spin-polarized corner mode dispersion, we identified $k_{y}$ in Eq.~(\\ref{3D_helical_hinge_mode_Hamiltonian}) as $\\phi$, since the center of the $\\phi$-sliding spectrum is at $\\phi = 0$, as shown in Fig.~\\ref{SM_Fig_5} (a) for $q_{z} = 0$. \nUpon projection to 2D, the position $(x_{\\text{hinge}},z_{\\text{hinge}})$ of the hinge along $y$ again becomes the position $(x_{\\text{corner}},z_{\\text{corner}})$ of the 2D corner. \nWe thus obtain\n\\begin{align}\n H_{\\text{corner}}= v_{F} \\left( \\phi \\sigma_{z}' + 2\\pi \\left( q_{x}x_{\\text{corner}} + q_{z}z_{\\text{corner}} \\right) \\sigma_{0}' \\right) \\label{eq:H_helical_hinge_1}\n\\end{align}\nfrom the main text. \nWe now examine this low energy theory through numerical simulations. \nWe will be focusing on how the doubly-degenerate gap-crossing bands corresponding to spin-polarized corner modes respond as we increase magnitude of $\\vec{q}$.\n\nWe show in Fig.~\\ref{SM_Fig_5} (b) the $\\phi$-sliding spectrum for $q_{z} = 0.02$. \nComparing Fig.~\\ref{SM_Fig_5} (b) with Fig.~\\ref{SM_Fig_5} (a), we see that as we turn on $q_{z}$, the two doubly-degenerate bands cross the gap with opposite slopes. \nTo explain this, we notice that each of the doubly-degenerate gap-crossing states corresponds to a localized pair of modes at inversion-related corners with opposite spins, as they are related by the $\\mathcal{I}\\mathcal{T}$-symmetry (see main text Sec.~V) and shown in Figs.~\\ref{SM_Fig_5} (c) and (d). \nIn our coordinate system, the corner modes are localized at positions $(x_{\\text{corner}},z_{\\text{corner}})$ = $\\pm(L\/2,L\/2)$, where $L=30$. \nThe corresponding 3D helical hinge mode dispersion relations are\n\\begin{align}\n & H_{\\text{hinge 1}}= v_{F} \\left( k_{y}\\sigma_{z}' - \\pi q_{z}L \\sigma_{0}' \\right), \\\\\n & H_{\\text{hinge 2}} = -v_{F} \\left( k_{y}\\sigma_{z}' + \\pi q_{z}L \\sigma_{0}' \\right),\n\\end{align}\nwhere inversion symmetry requires that the eigenstate of $H_{\\text{hinge 1}}$ and $H_{\\text{hinge 2}}$ with same spins have opposite group velocities. \nIdentifying $k_{y}$ as $\\phi$, we have that the Hamiltonians of the bands crossing the gap are given by\n\\begin{align}\n & H_{\\text{corner 1}}= v_{F} \\left( \\phi \\sigma_{z}' - \\pi q_{z}L \\sigma_{0}' \\right), \\label{eq:corner_1_SU2}\\\\\n & H_{\\text{corner 2}} = -v_{F} \\left( \\phi \\sigma_{z}' + \\pi q_{z}L \\sigma_{0}' \\right). \\label{eq:corner_2_SU2}\n\\end{align}\nWe then see that as we increase the $q_{z}$, the band in $H_{\\text{corner 1}}$ along $\\phi$ with slope $+v_{F}$ and spin $\\uparrow$ moves in the same direction as the band in $H_{\\text{corner 2}}$ with group velocity $+v_{F}$ and spin $\\downarrow$, which is consistent with Fig.~\\ref{SM_Fig_5} (b). \nIn fact, a detailed comparison between Figs.~\\ref{SM_Fig_5} (a) and (b) shows that the corner mode dispersion shifts along $\\phi$ by $\\pi q_{z}L \\approx 0.6\\pi $ for $q_{z} = 0.02$ and $L = 30$, implying that our low energy theory describes both the 2D corner modes and the corresponding 3D helical hinge modes. \nThis confirms that we can tune the range of $\\phi$ where the spin-polarized corner modes emerge from the bulk bands by modifying the periodicity of the modulation, as we have stated in the main text. \nSimilar to the chiral sliding modes in the main text, when $\\pi q_{z}L > 2\\pi$, the gap-crossing bands will be folded back within the range $\\phi = [0,2\\pi)$. \nThis happens in Fig. 3 (a) of the main text, where $q_{z} = 0.11957$.\n\n\\begin{figure}[h]\n \\includegraphics[scale=0.4]{SM_Fig_5_change_ticking_freq_add_color.pdf}\n \\caption{(a) $\\phi$-sliding spectrum of Eq.~(35) in the main text with parameters $m_{1} = -3$, $m_{2} = 0.3$, $m_{3} = 0.2$, $m_{v_{1}} = -0.4$, $m_{v_{2}} = 0.2$, $v_{x}=v_{z}=u_{x}=u_{z} = 1$, $v_{y} = 2$, $v_{H} = 1.2$ and $\\vec{q} = (0,q_{z})$ where $q_{z} = 0$. \n This reduces to the $y$-rod band structure of a 3D helical HOTI\\cite{Wieder_spin_decoupled_helical_HOTI} without $SU(2)$ gauge fields. \n Notice the Rashba-like shifting of the surface band away from $\\phi = 0$, near $E \\approx \\pm 0.2$. \n (b) $\\phi$-sliding spectrum with the same parameters as (a) but with $q_{z} = 0.02$. \n We identify the flat dispersion in (b) marked by orange, green and red as $E^{-}_{\\sigma',k_{y},n = 1}$, $E^{-}_{\\sigma',k_{y},n =0}$ and $E^{+}_{\\sigma',k_{y},n = 1}$ in Eq.~(\\ref{wvfn_top}), respectively. \n (c)$\\&$(d) Summation of probability distribution of the doubly-degenerate corners modes related to each other by the $\\mathcal{I}\\mathcal{T}$-symmetry at $\\phi = 0.6\\pi$ and $1.4\\pi$ with both energies equal to $-0.0124$. \n The two corner modes in each of (c) and (d) are localized at inversion-related corners and have opposite spins. \n The darker (black) color in (c)--(d) implies higher probability density. \n In (c) and (d), the $x$- and $z$-coordinate both range from $-15,\\ldots,+15$.}\n \\label{SM_Fig_5}\n\\end{figure}\n\n\\subsection{\\label{sec:surface_SU2}2D edge-confined modes $\\leftrightarrow$ 3D surface modes with $SU(2)$ gauge field}\n\nAs stated in the main text, we may understand the flat bands corresponding to edge-confined modes using a low energy theory in the promoted 3D lattice. \nIn this subsection we construct a surface theory minimally coupled to an $SU(2)$ gauge field to describe the projected 2D edge states in the modulated system.\n\nDue to the shifted band edge of the surface bands in the corresponding 3D helical HOTI (see Fig.~\\ref{SM_Fig_5} (a) and its caption), we may consider the surface theory by stacking a low energy Chern insulator with spin $\\uparrow$ and its time-reversal counterpart, with a relative shift in momentum space. \nThe low energy theory of a Chern insulator is given by $H_{\\text{Chern}}=\\vec{p}_{\\parallel}\\cdot \\vec{\\tau}_{\\parallel}' - m\\hat{n} \\cdot \\vec{\\tau}'$, where $\\vec{\\tau}'$ is the basis describing spin $\\uparrow$ electron with some orbital and sub-lattice textures, $\\parallel$ is the parallel component along the surface, $\\hat{n}$ is the surface normal vector and $m$ is the mass term. \nAs shown in Eq.~(\\ref{eq:B_SU2_1}), if $q_{x} = 0$, which is the case we consider in Fig.~\\ref{SM_Fig_6} below, we have $\\vec{B} = \\vec{\\nabla} \\cross \\vec{A} - i \\vec{A} \\cross \\vec{A}$ parallel to $\\hat{x}$. \nWe then consider a surface theory on the $yz$-plane whose surface normal is $\\hat{x}$. \nThe corresponding surface Hamiltonian without an $SU(2)$ gauge field reads $H_{\\text{surf}} = p_{y}\\tau_{y}'\\sigma_{0}'+p_{z}\\tau_{z}'\\sigma_{z}'+(\\Delta k_{y})\\tau_{y}'\\sigma_{z}'-m\\tau_{x}'\\sigma_{0}'$, where $\\vec{\\sigma}'$ again denotes the spin degrees of freedom, and $\\Delta k_{y}$ is a {\\it real constant} denoting the shift of low energy surface band minima from $\\vec{k}=\\Gamma$. \nTo facilitate the subsequent analysis, we perform a basis transformation through a $-2\\pi\/3$ radian rotation $U$ along the $[1,1,1]$ axis in the {\\it orbital} space $\\vec{\\tau}'$ such that $U^{\\dagger}(\\tau_{x}',\\tau_{y}',\\tau_{z}')U = (\\tau_{z}',\\tau_{x}',\\tau_{y}')$. \nThe transformed surface Hamiltonian then reads\n\\begin{align}\n H_{\\text{surf}} = p_{y}\\tau_{x}'\\sigma_{0}'+p_{z}\\tau_{y}'\\sigma_{z}'+(\\Delta k_{y})\\tau_{x}'\\sigma_{z}'-m\\tau_{z}'\\sigma_{0}'.\n \\label{eq:top_surf_SU2_helical_transformed}\n\\end{align}\nNotice that this surface theory is spin-decoupled. \nIn a general helical HOTI the spins are coupled. \nHowever, the general procedure will be the same: we first construct a low energy surface theory, and then couple it to the desired $SU(2)$ gauge field. \n\nWe now couple Eq.~(\\ref{eq:top_surf_SU2_helical_transformed}) to an $SU(2)$ gauge field $\\vec{A} = (0,Bz \\sigma_{z}',0)$ which produces a $SU(2)$ magnetic field $\\vec{B} = (-B\\sigma_{z}',0,0)$. \nTherefore opposite spins experience opposite magnetic fields. \nThe minimally-coupled surface theory is\n\\begin{align}\n H_{\\text{surf}} = \\begin{bmatrix}\n \\left(p_{y}+\\Delta k_{y} + Bz \\right)\\tau_{x}' + p_{z}\\tau_{y}' - m \\tau_{z}' & 0 \\\\ 0 & \\left(p_{y}-\\Delta k_{y} - Bz \\right)\\tau_{x}' - p_{z}\\tau_{y}' - m \\tau_{z}'\n \\end{bmatrix}, \\label{eq:helical_surface_before_rewrite}\n\\end{align}\nwhere we have assumed that both $B$ and $m$ are positive. \nFourier transforming to replace $p_{y}$ by the wavenumber $k_{y}$, and defining spin- and $k_{y}$-dependent ladder operators\n\\begin{align}\n a^{\\dagger}_{k_{y},\\uparrow} = \\frac{1}{\\sqrt{2B}}\\left(k_{y} + \\Delta k_{y} + Bz - ip_{z} \\right) \\text{ and } a^{\\dagger}_{k_{y},\\downarrow} = \\frac{1}{\\sqrt{2B}}\\left(k_{y} - \\Delta k_{y} - Bz + ip_{z} \\right), \\label{eq:top_ladders_SU2}\n\\end{align}\nwe can rewrite Eq.~(\\ref{eq:helical_surface_before_rewrite}) in each spin subspace ($\\sigma'=\\pm$) as\n\\begin{align}\n H_{\\text{surf}}(k_{y},\\sigma') = \\begin{bmatrix}\n -m & \\sqrt{2B} a_{k_{y},\\sigma'}^{\\dagger} \\\\\n \\sqrt{2B} a_{k_{y},\\sigma'} & m\n \\end{bmatrix}.\n\\end{align}\nWe can solve for the eigenstates and energies to find\n\\begin{align}\n & \\psi^{-}_{\\sigma',k_{y},n=0} = e^{ik_{y}y}\\begin{bmatrix}\n \\ket{0,k_{y},\\sigma'} \\\\ 0\n \\end{bmatrix},\\ E^{-}_{\\sigma',k_{y},n=0} = -m, \\nonumber \\\\\n & \\psi^{-}_{\\sigma',k_{y},n>0} = e^{ik_{y}y}\\begin{bmatrix}\n \\ket{n,k_{y},\\sigma'} \\\\ \\alpha_{-}(n)\\ket{n-1,k_{y},\\sigma'}\n \\end{bmatrix},\\ E^{-}_{\\sigma',k_{y},n>0} = -\\sqrt{m^{2} + 2Bn}, \\nonumber \\\\\n & \\psi^{+}_{\\sigma',k_{y},n>0} = e^{ik_{y}y}\\begin{bmatrix}\n \\ket{n,k_{y},\\sigma'} \\\\ \\alpha_{+}(n)\\ket{n-1,k_{y},\\sigma'}\n \\end{bmatrix},\\ E^{+}_{\\sigma',k_{y},n>0} = +\\sqrt{m^{2} + 2Bn}, \\nonumber \\\\\n & {\\color{black}{\\text{where }}} \\alpha_{\\pm}(n) = \\frac{1}{\\sqrt{2Bn}}\\left(\\pm \\sqrt{m^{2} + 2Bn} + m \\right). \\label{wvfn_top}\n\\end{align}\nHere $\\sigma' = \\pm$ is the spin quantum number, $k_{y}$ is the wavenumber along $y$, $n$ is an non-negative integer {\\color{black}{labelling}} the $SU(2)$ LLs, and $\\ket{n,k_{y},\\sigma'}$ is the $n^{\\text{th}}$ simple harmonic oscillator (SHO) eigenstate along $z$ defined using $a^{\\dagger}_{k_{y},\\sigma'}$. \nNotice that the coefficient $\\alpha_{\\pm}(n)$ does not depend on which spin sector we are considering. \nWe now connect the above surface theory to the flat bands and corresponding wavefunctions in the $\\phi$-sliding spectrum of the 2D modulated system by identifying $k_{y}$ as $\\phi$ and $B$ as $2\\pi q_{z}$. \nThis is because in our 3D promoted lattice Eq.~(\\ref{eq:lattice_model_helical_sliding}) and the numerical examples, we have $q_{x} = 0$ such that $\\vec{A} = (0,2\\pi q_{z}z\\sigma_{z}',0)$ produces an $SU(2)$ magnetic field with strength $2\\pi q_{z}$. \nWe will mainly focus on the two [one] flat bands in Fig.~\\ref{SM_Fig_5} (b) with negative [positive] energy closest to the zero. \nThese correspond to $E^{-}_{\\sigma',k_{y},n \\le 1}$ [$E^{+}_{\\sigma',k_{y},n=1}$].\n\nFirst, notice that the spectrum in Eq.~(\\ref{wvfn_top}) breaks particle-hole symmetry in both spin sectors, as there are no $+m$ energy eigenvalues. \nThis can be observed in Fig.~\\ref{SM_Fig_5} (b) where there are no flat bands of edge-confined modes around $E\\approx +0.2$ corresponding to $E = +m$. \nWe thus identify the flat bands in Fig.~\\ref{SM_Fig_5} (b) marked by orange, green and red as $E^{-}_{\\sigma',k_{y},n = 1}$, $E^{-}_{\\sigma',k_{y},n =0}$ and $E^{+}_{\\sigma',k_{y},n = 1}$ in Eq.~(\\ref{wvfn_top}), respectively.\n\nNext, the probability densities $|\\psi^{-}_{\\sigma',k_{y},n=0}|^2$ and $|\\psi^{\\pm}_{\\sigma',k_{y},n=1}|^2$ are respectively proportional to $\\left|\\varphi_{0,B}(z+(\\Delta k_{y} + \\sigma' k_{y})\/B) \\right|^{2}$ and $\\left| \\alpha_{\\pm}(1) \\right|^{2}\\left|\\varphi_{0,B}(z+(\\Delta k_{y} + \\sigma' k_{y})\/B) \\right|^{2} + \\left|\\varphi_{1,B}(z+(\\Delta k_{y} + \\sigma' k_{y})\/B) \\right|^{2}$, where $\\varphi_{n,B}(z)$ is the $n^{\\text{th}}$ SHO eigenstate along $z$. \nNotice that we have put explicit $B$-dependence on $\\varphi_{n,B}(z)$ since the cyclotron frequency and the localization of wave functions depend on the field strength.\nThis implies that: \n(1) the probability density computed from $\\psi^{-}_{\\sigma',k_{y},n=0}$ has a pure Gaussian distribution along $z$, and \n(2) $\\psi^{-}_{\\sigma',k_{y},n=1}$ has a larger contribution from the SHO first excited state than $\\psi^{+}_{\\sigma',k_{y},n=1}$, since $\\left| \\alpha_{-}(1) \\right|^{2} < \\left| \\alpha_{+}(1) \\right|^{2}$ and we have assumed both $B$ and $m$ are positive. \nFigs.~\\ref{SM_Fig_6} (a)--(c) show the 2D probability distribution at $\\phi = 0$ for edge-confined modes in different LLs together with the insets showing the probability integrated over non-negative $x$-coordinates. \nWhile both Figs.~\\ref{SM_Fig_6} (a) and (c) corresponds to $n = 1$ LL, Fig.~\\ref{SM_Fig_6} (a) is from the negative energy branch and Fig.~\\ref{SM_Fig_6} (c) is from the positive energy branch. \nTherefore Fig.~\\ref{SM_Fig_6} (a) is more characteristic of the SHO first excited state than Fig.~\\ref{SM_Fig_6} (c). \nIn contrast, Fig.~\\ref{SM_Fig_6} (b), being the $n=0$ LL wave function, shows a Gaussian probability distribution characteristic of the SHO ground state. \nWe notice that, if we restrict ourselves to a single edge, the wave function in Fig.~\\ref{SM_Fig_6} (a) contains a slightly asymmetric SHO first excited state. \nThis is due to the complicated on-site and hopping energies in our model, which break extraneous symmetries, such that the wave function in Fig.~\\ref{SM_Fig_6} (a) has nonzero penetration to bulk. \nThis can be compared with Fig.~\\ref{SM_Fig_6} (b) where the edge-confined modes are much more localized on the edges. \nThus, the surface theory present above should be recognized as an effective theory we use to extract out qualitative properties of wave functions, and it suffices to identify the SHO first excited state character of Fig.~\\ref{SM_Fig_6} (a) corresponding to $\\psi^{-}_{\\sigma',k_{y},n=1}$ with energy level $E^{-}_{\\sigma',k_{y},n = 1}$ in Eq.~(\\ref{wvfn_top}).\n\nThird, the definition of the ladder operator in Eq.~(\\ref{eq:top_ladders_SU2}) predicts that even when we have $\\phi = 0$ (which corresponds to $k_{y} = 0$) the center of the SHO eigenstate will be shifted from the center of the coordinate system by a distance $|\\Delta k_{y}\/ B|$ along $z$. \nRecall that $\\Delta k_{y}$ is the slight Rashba-like shift of the surface bands away from $\\Gamma$ (see Fig.~\\ref{SM_Fig_5} (a) and its caption). \nIn the 2D modulated system, this is equivalent to saying that the edge-confined modes at $\\phi = 0$ will not have wavefunctions centered at the middle of the edge along $z$. \nThe probability distribution at $\\phi = 0$ shown in Fig.~\\ref{SM_Fig_6} (a) to (c) confirms this, as no states are centered around the middle of the edges.\nGoing further, we also expect from Eq.~(\\ref{eq:top_ladders_SU2}) that the center of wave functions will be shifted along $z$ by $(-\\Delta k_{y} - \\sigma' k_{y})\/B$ for a given $k_{y}$. \nIdentifying $k_{y}$ as $\\phi$ and $B$ as $2\\pi q_{z}$, we deduce that the shifting of the edge-confined modes from the center of the edges is spin-dependent. \nThe center of each probability distribution is given by $l_{\\sigma'} = (-\\Delta k_{y} - \\sigma' \\phi)\/(2\\pi q_{z}) $. \nNotice that the edge-confined modes in Figs.~\\ref{SM_Fig_6} (d) and (e) are shifted by $\\approx \\pm 2.5$ lattice constants comparing with Fig.~\\ref{SM_Fig_6} (b). \nThis is because Figs.~\\ref{SM_Fig_6} (d) and (e) correspond to $\\Delta \\phi = \\phi - 0 = 0.1\\pi$, $q_{z} = 0.02$, and wave functions with opposite spins will be shifted in the opposite directions, according to the expression of $l_{\\sigma'}$ given above and $a^{\\dagger}_{k_{y},\\sigma'}$ in Eq.~(\\ref{eq:top_ladders_SU2}). \nTo be more precise, we estimate the center of the wave functions in Figs.~\\ref{SM_Fig_6} (b), (d) and (e) to be around $z \\approx 1.5$, $4$ and $-1$, respectively.\n\nFinally, similar to the chiral sliding modes in the 2D modulated system, we also have an additional degeneracies in the flat band states due to zone-folding (see Sec.~IV of the main text). \nUp to the degeneracy due to zone folding, the universal property we expect in these 2D helical sliding systems is that as we vary $\\phi$, which corresponds to sliding of density waves (DWs) with spin-orbit coupled interactions, there will be flat bands together with edge-confined modes that are projected from the surface of a helical HOTI with $SU(2)$ Landau quantization. \nWe have seen that the qualitative properties of the wave functions are all consistent with the low energy surface theory.\n\n\\begin{figure}[h]\n\\includegraphics[scale=0.4]{SM_helical_sliding_edge_modes.pdf}\n\\caption{(a)--(c) Average of the probability distribution for the four-fold degenerate edge-confined modes in the flat bands shown in Fig.~\\ref{SM_Fig_5} (b). \nThe four-fold degeneracy comes considering the two edges and two spins. \n(a)--(c) are edge-confined modes at $\\phi = 0$ with $E = -0.4227$, $-0.1711$ and $0.3675$ which are marked orange, green and red in Fig.~\\ref{SM_Fig_5} (b). \nThe corresponding energy levels are $E^{-}_{\\sigma^\\prime,k_{y}=0,n=1}$, $E^{-}_{\\sigma^\\prime,k_{y}=0,n=0}$ and $E^{+}_{\\sigma^\\prime,k_{y}=0,n=1}$in Eq.~(\\ref{wvfn_top}). \n(d)$\\&$(e) Edge modes at $\\phi = 0.1 \\pi$ with $E = -0.1711$ confined at the right edge.\nThey correspond to $E^{-}_{\\sigma^\\prime,k_{y}=0.1\\pi,n=0}$ with $\\sigma^\\prime = \\pm$. \nWe notice that in addition to (d) and (e) there are two other edge-confined modes with nearly identical energies localized on the left edge.\nThe darker (black) color in (a)--(e) implies higher probability density. \nThe inset in (a)--(e) is the probability distribution integrated over non-negative $x$-coordinates, from $x= 0,\\ldots, L\/2$ where $L=30$. \nIn (a)--(e), the $x$- and $z$-coordinate both range from $-15,\\ldots, +15$.}\n\\label{SM_Fig_6}\n\\end{figure}\n\n\n\\subsection{2D bulk-confined modes $\\leftrightarrow$ 3D bulk modes with $SU(2)$ gauge field}\n\nHaving accounted for the edge-confined modes in the 2D modulated system with helical sliding, we move on to consider the flat bands corresponding to bulk-confined modes. \nSimilar to Sec.~\\ref{sec:surface_SU2}, we will construct a low energy theory and couple it to an $SU(2)$ gauge field.\n\nWe consider the Bloch Hamiltonian of the promoted 3D helical HOTI around the $\\Gamma$ point (see Sec.~\\ref{sec:promoted_helical_lattice_model}), which is\n\\begin{align}\n H_{\\text{bulk}} = & m_{\\text{bulk}} \\tau_{z}\\mu_{0}\\sigma_{0} + m_{2} \\tau_{z}\\mu_{x}\\sigma_{0} + m_{3} \\tau_{z}\\mu_{z}\\sigma_{0} + m_{v_{1}}\\tau_{0}\\mu_{z}\\sigma_{0} + m_{v_{2}} \\tau_{0}\\mu_{x}\\sigma_{0} \\nonumber \\\\\n & + u_{x} p_{x} \\tau_{y}\\mu_{y}\\sigma_{0} + u_{z}p_{z}\\tau_{x}\\mu_{0}\\sigma_{0} + v_{H}p_{y} \\tau_{y}\\mu_{z}\\sigma_{z}. \\label{eq:3D_helical_HOTI_bulk}\n\\end{align}\nWe have redefined several parameters in the original model in Sec.~\\ref{sec:promoted_helical_lattice_model} for convenience. \nFor example, $m_{\\text{bulk}}$ is effectively $m_{1} +v_{x} + v_{y} + v_{z}$ from Eq.~(A1) \nin Ref.~\\onlinecite{Wieder_spin_decoupled_helical_HOTI}. \nWe now couple this $H_{\\text{bulk}}$ to the $SU(2)$ vector potential $\\vec{A} = Bz \\tau_{0}\\mu_{0}\\sigma_{z} \\hat{y}$, which is equivalent to Eq.~(\\ref{SU2_A}) with $q_{x} = 0$. \nTherefore, the term $v_{H}p_{y} \\tau_{y}\\mu_{z}\\sigma_{z}$ becomes\n\\begin{align}\n v_{H}p_{y} \\tau_{y}\\mu_{z}\\sigma_{z} \\to v_{H}\\tau_{y}\\mu_{z} \\begin{bmatrix}\n p_{y} + Bz & 0 \\\\ 0 & -(p_{y}-Bz)\n \\end{bmatrix},\n\\end{align}\nwhere the matrix acts in spin space. \nFourier transforming along $x$ and $y$ and defining the $k_{y}$- and spin($\\sigma = \\pm$)-dependent ladder operators\n\\begin{align}\n a^{\\dagger}_{\\sigma,k_{y}} = \\frac{1}{\\sqrt{2B}}\\left( \\sigma k_{y} + Bz - ip_{z} \\right), \\label{eq:ladder_bulk_helical_SU2}\n\\end{align}\nwe can rewrite Eq.~(\\ref{eq:3D_helical_HOTI_bulk}) coupled to $\\vec{A} = Bz \\tau_{0}\\mu_{0}\\sigma_{z} \\hat{y}$ in different spin sectors as\n\\begin{align}\n H_{\\sigma,k_{x},k_{y}} &= m_{\\text{bulk}} \\tau_{z}\\mu_{0} + m_{2} \\tau_{z}\\mu_{x} + m_{3}\\tau_{z}\\mu_{z} + + m_{v_{1}} \\tau_{0}\\mu_{z} + m_{v_{2}}\\tau_{0}\\mu_{x} \\nonumber \\\\\n & + u_{x}k_{x}\\tau_{y}\\mu_{y} + v_{H}\\sqrt{\\frac{B}{2}} \\left( a_{\\sigma,k_{y}} + a^{\\dagger}_{\\sigma,k_{y}} \\right)\\tau_{y}\\mu_{z} -i u_{z} \\sqrt{\\frac{B}{2}}\\left( a_{\\sigma,k_{y}} - a^{\\dagger}_{\\sigma,k_{y}} \\right)\\tau_{x}\\mu_{0}. \\label{eq:3D_bulk_SU2_LL_Hamiltonian}\n\\end{align}\nIn Fig.~\\ref{SM_Fig_7} (a) we show the numerically computed spectrum $E_{\\sigma,k_{x},k_{y}}$ as a function of $k_{x}$. \nNote that $E_{\\sigma,k_{x},k_{y}}$ does not depend on $\\sigma$ and $k_{y}$, giving rise to flat Landau levels as a function of $k_{y}$. \nWe relate this low energy bulk theory to the $\\phi$-sliding spectrum of the 2D modulated system by identifying $k_{y}$ in Eq.~(\\ref{eq:3D_bulk_SU2_LL_Hamiltonian}) with $\\phi$. \nWe now examine the consequences of this correspondence.\n\n\nFirst, examining the band edges for the valence and conduction bands in Fig.~\\ref{SM_Fig_7} (a) we notice that there is no particle-hole symmetry in the low energy spectrum of helical HOTI with $SU(2)$ Landau quantization.\nThis is also reflected in the energy eigenvalues and eigenstate probability distribution for the flat bands in Fig.~\\ref{SM_Fig_5} (b) corresponding to bulk-confined modes shown in Figs.~\\ref{SM_Fig_7} (b) and (c), which are in the bulk flat continuum directly below and above the edge flat dispersion marked in orange and red in Fig.~\\ref{SM_Fig_5} (b). \nThe figure caption of Fig.~\\ref{SM_Fig_7} gives the asymmetric energies of the two bulk states in Figs.~\\ref{SM_Fig_7} (b) and (c).\n\nSecond, we note that the Hamiltonian in Eq.~(\\ref{eq:3D_bulk_SU2_LL_Hamiltonian}) has eigenstates that are extended along $x$ and $y$ while confined along $z$. \nThe confinement along $z$ can also be understood through the definition of the spin-dependent ladder operators in Eq.~(\\ref{eq:ladder_bulk_helical_SU2}), which create SHO states along $z$. \nProjecting this 3D wave function to the 2D modulated system, only the probability distribution along $x$ and $z$ is preserved. \nTherefore in 2D we expect to see flat bands along $k_{y}$--which is identified as $\\phi$--and with corresponding wave functions confined along $z$ while extended along $x$. \nThis is consistent with Figs.~\\ref{SM_Fig_7} (b) and (c). \nThe above discussion shows that our low energy bulk theory in 3D qualitatively explains the existence of bulk confined modes in 2D. \nThe universal property of these types of 2D helical sliding systems, even with larger $|\\vec{q}|$ presented in the main text, is that there will be flat bands as we vary $\\phi$ with bulk-confined modes that are projected from the bulk of helical HOTI with $SU(2)$ Landau quantization.\n\n\n\\begin{figure}[h]\n \\includegraphics[scale=0.4]{SM_Fig_7_add_horizontal_pm_0_6_new.pdf}\n \\caption{(a) Bulk $SU(2)$ Landau levels of Eq.~(\\ref{eq:3D_bulk_SU2_LL_Hamiltonian}) for the low energy model of a 3D helical HOTI\\cite{Wieder_spin_decoupled_helical_HOTI} with $m_{\\text{bulk}} = 1$, $m_{2} = 0.3$, $m_{3} = 0.2$, $m_{v_{1}} = -0.4$, $m_{v_{2}} = 0.2$, $u_{x} = 1$, $v_{H}=1.2$, $u_{z} = 1$ and $B = 0.2$. \n The band edges occur at $E = -0.6618$ and $0.601$. \n The green, black and orange horizontal lines correspond to energy $+0.6$, $0$ and $-0.6$, respectively. \n Notice that there is in general no particle-hole symmetry in the bulk $SU(2)$ LL spectrum of a 3D helical HOTI. \n (b)$\\&$(c) Probability distribution of bulk-confined modes in the flat band of Fig.~\\ref{SM_Fig_5} (b) at $\\phi = 0$ with energy $-0.5393$ and $0.4835$ respectively with similar wave function confinement. \n These two modes are at the flat bulk continuum right below and above those flat bands for {\\it edge-confined} modes corresponding to $E^{-}_{\\sigma',k_{y},n=1}$ and $E^{+}_{\\sigma',k_{y},n=1}$ in Eq.~(\\ref{wvfn_top}), which we have marked in orange and red in Fig.~\\ref{SM_Fig_5} (b).\n We note that both (b) and (c) show modes confined along $z$, which is consistent with the ladder operators defined in Eq.~(\\ref{eq:ladder_bulk_helical_SU2}). \n However, the absolute value of energies for (b) and (c) are not same, indicating the spectrum is not particle-hole symmetric, and the minimal continuum model calculation in (a) captures this trend. \n The darker (black) color in (b)--(c) implies higher probability density. \n In (b) and (c), the $x$- and $z$-coordinate both range from $-15,\\ldots, +15$.}\n \\label{SM_Fig_7}\n\\end{figure}\n\n\n\n\\section{\\label{sec:numerical_method}Numerical methods to identify non-trivial Chern insulating layers in 3D Weyl-CDW systems}\n\n\nIn this section, we describe the numerical methods we used to identify layers (hybrid Wannier functions) and corresponding Bloch states in our 3D Weyl-CDW model. \nIn the following, we use the term $xy$-slab to denote a sample of 3D Weyl-CDW system infinite along the $x$ and $y$ directions and finite along $z$ with size $L_{z}$. \nIn addition, we use $y$-rod to denote a sample of 3D Weyl-CDW system infinite along $y$ and finite along $x$ and $z$ with size $L_{x} \\times L_{z}$. \nIn this section, we will present the numerical methods based on Berry phase, Berry curvature, and hybrid Wannier functions to identify the non-trivial bands in the $xy$-slab for $q = 1\/5$, $L_{z} = 25 $, $t_{x} = -t_{y} = t_{z} = 1$, $m=2$, $2|\\Delta|=0.75$ with $\\phi = 0$ and $\\phi = \\pi$ in the main text. \nThe numerical methods can also be applied to systems such as incommensurate CDWs with $q = \\tau \/4$ where $\\tau = (1+\\sqrt{5})\/2$ is the golden ratio (see main text).\n\nAs mentioned in the main text, we can understand the existence of quantum anomalous Hall (QAH) surface states by viewing the $xy$-slab as composed of a layered stack of Chern insulators (CIs). \nWhen we cut the $xy$-slab into the corresponding $y$-rod, those states in the $xy$-slab with non-zero Chern number will each contribute a number of chiral edge modes, depending on the Chern number of the bands. \nThese chiral edge modes collectively form the QAH surface states. The number of chiral edge modes in the $y$-rod on each edge is equal to the Chern number of occupied bands in the corresponding $xy$-slab. Specifically, the low energy theory in 4D predicts that topologically non-trivial bands in the $xy$-slab have a layered structure in position space, in which non-trivial bands will be localized along $z$ and separated by some amount of lattice constants\\cite{NicoDavidAXI2,dynamical_axion_insulator_BB}. \nIn the following, we will use a combination of Berry phase, Berry curvature and hybrid Wannier function calculations to identify those non-trivial bands in the $xy$-slab. \nThe goal is to find non-trivial bands whose probability distributions along $z$ match the probability distribution of chiral edge modes (or QAH surface states) in the $y$-rod {\\color{black}{shown in Figs.~4 and 5 in the main text}}.\n\nThe first step of our analysis is to compute the hybrid Wannier bands for the occupied states in the $xy$-slab. \nThis involves diagonalizing the position operator $z$ projected into the space of occupied bands at each $\\vec{k}$-point of the 2D Brillouin zone (BZ)\\cite{PythTB,Marzari2012}. \nNotice that, as we have finite size along $z$, the position operator $z$ is well-defined. \nWe thus obtain hybrid Wannier bands, which are the eigenstates of the projected $z$ operator as a function of $k_x$ and $k_y$. \nIn our calculation we find that for $q = 1\/5$, $t_{x}=-t_{y}=t_{z}=1$, $m = 2$, $2|\\Delta|=0.75$, $\\phi = 0$ and $L_{z} = 25$ the hybrid Wannier bands are non-degenerate.\nThis allows us to compute the Berry phase integrated along $k_{y}$ as a function of $k_{x}$ for each hybrid Wannier band, as shown in the first column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0}. \nWe denote such Berry phases as $\\gamma^{m}_{\\vec{G}_{2}}(k_{x})$, where $m$ indexes the hybrid Wannier band, $\\vec{G}_{2}$ indicates that we are integrating along $k_{y}$, and $k_{x}$ denotes the functional dependence on $k_{x}$. \nDue to the non-degeneracy of the hybrid Wannier bands, the Berry phases $\\gamma^{m}_{\\vec{G}_{2}}(k_{x})$ coincide with the eigenvalues of the $\\hat{y}$-directed non-Abelian slab Berry phase (Wilson loop) $\\text{arg}(\\mathcal{W})$\\cite{wieder2018axion,Wilson_2,Yu11,Alexandradinata16}. \nFrom the negative winding of $\\gamma^{m}_{\\vec{G}_{2}}(k_{x})$ shown in the first column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0}, we find that each of the $m = 2$, $7$, $12$, $17$ and $22$ hybrid Wannier bands (the first hybrid Wannier band is labelled by $m=0$) carries Chern number $C = -1$. \nImportantly, at $\\Gamma$ point, the hybrid Wannier functions for these 5 hybrid Wannier bands are localized at $z_\\Gamma \\approx -10$, $-5$, $0$, $5$ and $10$, see the second column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0} and the second column of Table~\\ref{tab:hwfc_table}. \nThis is consistent with our deduction in the main text that the CIs will be separated by $1\/q$ along $z$, which is $5$ lattice constants in this case. \nThis also supports our identification that this system can be viewed as layered stack of CIs.\nThe discrepancy between $z_\\Gamma$ and $\\langle z\\rangle$ (reported in the third column in Table~\\ref{tab:hwfc_table}), the average of the hybrid Wannier center over all momenta, can be attributed to the fact that our present model has nonzero coupling along the $z$-direction, and hence nonzero coupling between CI layers. \n\nFor those hybrid Wannier bands with non-zero winding of the Berry phase, we compute the decomposition of hybrid Wannier functions in terms of Bloch states at $\\vec{k} = \\Gamma$: \n\\begin{align}\n\t\\psi^{\\text{hybrid}}_{m,\\Gamma} = \\sum_{n=1}^{N_{\\text{occ}}} c_{m,\\Gamma}^{n} \\psi^{\\text{Bloch}}_{n,\\Gamma}, \\label{eq:wvfn_decomposition}\n\\end{align}\nwhere $N_{\\text{occ}}$ is the number of occupied bands. \nThe values of $n$ for those non-zero $ c_{m,\\Gamma}^{n} $ then directly tell us the index of the topologically non-trivial Bloch bands (not hybrid Wannier bands) at $\\vec{k} = \\Gamma$. \nHereafter, we will use $m$ to denote hybrid Wannier band indices and $n$ to denote Bloch band indices.\nIn our specific examples, the decomposition of $\\psi^{\\text{hybrid}}_{m,\\Gamma}$ in the Bloch basis where $m$ is the index of non-trivial hybrid Wannier bands are given in the third column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0}. \nWe can then identify that the Bloch band with index $n = 18,\\ldots, 22$ at $\\vec{k} = \\Gamma$ are non-trivial. \nImportantly, we notice that for the hybrid Wannier bands at $\\Gamma$ with $m \\ne 2$, $7$, $12$, $17$ and $22$, the decomposition coefficient $c^{n}_{m,\\Gamma}$ are zero for $n = 18,\\ldots, 22$, showing that the hybrid Wannier bands at $\\Gamma$ with $m = 2$, $7$, $12$, $17$ and $22$ are truly spanned only by Bloch bands with $n = 18,\\ldots, 22$. \nThese states contribute to the chiral QAH surface states. \nAlthough the band structure of $xy$-slab is complicated with various band entanglements, we can track the topologically non-trivial bands by starting from the non-trivial bands at $\\vec{k} = \\Gamma$. \nWhenever we encounter band crossing, we choose to proceed along the direction where there is no discontinuity of the bands. \nFor example, we can identify the non-trivial valence bands at $\\vec{k} = \\Gamma$ and $\\vec{k} = (0,\\pi) = Y $, as shown in Fig.~\\ref{slab_band_and_wvfn_Q_0.4pi_Lz_25_D_0.75_phi_0_and_pi} marked by orange and green respectively. \nWe also plot the probability distribution of these non-trivial states in Fig.~\\ref{slab_band_and_wvfn_Q_0.4pi_Lz_25_D_0.75_phi_0_and_pi}. \nThe probability distribution along $z$ for these non-trivial states are exactly the same as the QAH zero modes for the corresponding $y$-rod, which we have shown in the main text, and the hybrid Wannier function at $\\Gamma$ for $m = 2$, $7$, $12$, $17$ and $22$, shown in the middle column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0}.\nFurthermore, we have computed the Berry curvature of each Bloch band (not hybrid Wannier band) around the $\\Gamma$ point of the $xy$-slab in Fig.~\\ref{Berry_curvature_around_Gamma_Q_0.4pi_Lz_25_D_0.75_phi_0}. \nThose non-trivial bands with band index $n = 18,\\ldots, 22$ show negative values of Berry curvature around $\\Gamma$, which is distinct from the other Bloch bands having positive values of Berry curvature around $\\Gamma$.\nThis negative Berry curvature contributes to the total Chern number $C = -5$ of occupied bands. \nThis is consistent with $G_{xy}(\\phi = 0) = -5 e^{2}\/h$, as we stated in the main text.\n\nThe above process can be also applied to the case with $\\phi = \\pi$. \nWe also identify the non-trivial bands for the $xy$-slab, and plot their probability distribution at $\\vec{k} = \\Gamma$ and $\\vec{k} = Y$ in Fig.~\\ref{slab_band_and_wvfn_Q_0.4pi_Lz_25_D_0.75_phi_0_and_pi}. \nAgain, the probability distribution along $z$ for these non-trivial states are exactly the same as the QAH zero modes for the corresponding $y$-rod, which we have shown in the main text, and the hybrid Wannier functions at $\\Gamma$ for $m = 4$, $9$, $15$ and $20$, shown in the second column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_pi}. \nThe similar analysis of Berry phase for hybrid Wannier bands, wave function decomposition of Eq.~(\\ref{eq:wvfn_decomposition}) and Berry curvature around $\\vec{k} = \\Gamma$ are also shown in Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_pi} and Fig.~\\ref{Berry_curvature_around_Gamma_Q_0.4pi_Lz_25_D_0.75_phi_pi}. \nImportantly, we can see that in the second column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_pi} and the fifth column of Table~\\ref{tab:hwfc_table}, the hybrid Wannier functions at $\\Gamma$ corresponding to the non-trivial bands are localized at $z_\\Gamma \\approx -7.5$, $-2.5$, $2.5$ and $7.5$. \nAs above, we attribute the discrepancy between $z_\\Gamma$ and $\\langle z \\rangle$ (the average of the hybrid Wannier center over the Brillouin zone, reported in column six of Table~\\ref{tab:hwfc_table}) to the nonvanishing interlayer coupling in our model. \nThis is again consistent with our stacked layer identification. \nAs mentioned in the main text, as we vary $\\phi$ by $\\pi$, each of the CIs will be shifted by $-\\Delta \\phi \/ (2\\pi q) = -\\pi\/(2\\pi q) = -1\/(2q)$, which in our current example is equal to $-2.5$ lattice constants. \nComparing this with the second column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0}, which shows that the non-trivial CIs are at $z \\approx -10$, $-5$, $0$, $5$ and $10$, we can see that the stacked layers in Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_pi} can be identified as those in Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0} with a shift of $-2.5$ lattice constant along $z$. \nWe notice that in this case, where we have used $q = 1\/5$, $t_{x}=-t_{y}=t_{z}=1$, $m = 2$, $2|\\Delta|=0.75$, $\\phi = \\pi$ and $L_{z} = 25$, the hybrid Wannier bands are nearly degenerate at $\\Gamma$ for $m = 9$, $10$ and $m = 14$, $15$, which result in a dramatic (though still continuous) change of $\\gamma^{m}_{\\vec{G}_{2}}(k_{x})$ for $m = 9$ and $15$ around $k_{x} = 0$, as shown in the first column of Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_pi}. \nNevertheless, we are able to separate the hybrid Wannier bands.\n\n\nWe conclude this section by making several remarks. \nFirst, since the above process does not involve computing the band structure in a 3D BZ, there is no problem in generalizing the above method to 3D Weyl-CDW systems with incommensurate density waves with modulation wave vector along $z$. \nWe have carried out the same numerical analysis for $q = \\tau \/ 4$ (where $\\tau = (1+\\sqrt{5})\/2$ is the golden ratio).\nThe results are shown in the main text in which we can identify non-trivial states corresponding to CI layers and these states have exactly the same probability distribution along $z$ as the QAH zero modes in the $y$-rod at $k_{y} = 0$. \nWe shall in here briefly summarize some numerical results for $q = \\tau \/ 4$, $t_{x} = -t_{y} = t_{z} = 1$, $m=2$, $2|\\Delta|=2$: \n(1) The $xy$-slab Hall conductances with $L_{z} = 21$ are $G_{xy}(\\phi = 0)=-9 e^{2}\/h$ and $G_{xy}(\\phi = \\pi)=-8 e^{2}\/h$\\cite{dynamical_axion_insulator_BB}. \n(2) There are 9 and 8 non-trivial hybrid Wannier bands in the $xy$-slab for $\\phi = 0$ and $\\pi$, respectively. \nEach of these non-trivial hybrid Wannier bands carries Chern number $C = -1$. \n(3) Similar to Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0} and Fig.~\\ref{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_pi}, we have identified 9 and 8 non-trivial Bloch bands of the $xy$-slab using non-zero values of $c^{n}_{m,\\Gamma}$ in Eq.~(\\ref{eq:wvfn_decomposition}). \nThe Berry curvature of these 9 and 8 non-trivial Bloch bands around $\\Gamma$ are negative, which is distinct from other Bloch bands having positive values of Berry curvature around $\\Gamma$, and contribute to $G_{xy}(\\phi = 0)=-9 e^{2}\/h$ and $G_{xy}(\\phi = \\pi)=-8 e^{2}\/h$. \nThis is similar to the cases of $q = 1\/5$ in Fig.~\\ref{Berry_curvature_around_Gamma_Q_0.4pi_Lz_25_D_0.75_phi_0} and Fig.~\\ref{Berry_curvature_around_Gamma_Q_0.4pi_Lz_25_D_0.75_phi_pi}. \nSecondly, the above procedure to filter out non-trivial CI layers might break down when we have strong interlayer coupling, causing complicated hybridization between layer states. \nWe emphasize again that the hybrid Wannier center $\\left\\langle z \\right\\rangle$ averaged over the 2D Brillouin zone for a fixed non-trivial hybrid Wannier band can differ slightly from the hybrid Wannier center $z_{\\Gamma}$ at $\\vec{k} = (k_{x},k_{y})=(0,0)=\\Gamma$ where our low energy theory works. \nFor example, as shown in Table~\\ref{tab:hwfc_table}, for $\\phi = 0$, the hybrid Wannier band with index $m = 2$ has $z_{\\Gamma} = -9.9245$ while $\\left\\langle z\\right\\rangle = -9.9997 \\pm 0.0026$. \nThis can be viewed as the consequence of hybridization between layers. \nWe expect that if the layers are completely decoupled from each other, there will be no difference between $z_{\\Gamma}$ and $\\left\\langle z \\right\\rangle$. \nNotice that for $\\phi = 0$ the inversion center of the $xy$-slab is at $z = 0$, which pins both $z_{\\Gamma}$ and $\\left\\langle z \\right\\rangle$ to $z = 0$ for the $m=12$ hybrid Wannier band. \nThis is consistent with Refs.~\\onlinecite{song2017,MTQC}, where it was emphasized that the distinction between QAH and oQAH states is due to the presence or absence of a non-trivial CI layer exactly at the inversion center $\\left\\langle z \\right\\rangle = 0$, respectively. \nWe have also verified this for the incommensurate case of $q=\\tau \/4$ where $\\tau = (1+\\sqrt{5})\/2$, which indicates that the difference of a non-trivial CI layer at the inversion center $\\left\\langle z \\right\\rangle=0$ between QAH and oQAH states holds for generic CDW wave vectors. \nOur method can be viewed as a way to map a system with interlayer coupling to another system with decoupled CI layers.\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c||c|c|c|}\n\\hline\n$m$ for $\\phi = 0$ & $z_{\\Gamma}$ & $\\left\\langle z\\right\\rangle$ & $m$ for $\\phi = \\pi$ & $z_{\\Gamma}$ & $\\left\\langle z\\right\\rangle$ \\\\\n\\hline\n2 & $-9.9245$ & $-9.9997 \\pm 0.0026$ & 4 & $-7.5467$ & $-7.9963 \\pm 0.0167$\\\\\n\\hline\n7 & $-4.9972$ & $-5.0 \\pm 0.0001$ & 9 & $-2.5023$ & $-2.9963 \\pm 0.0176$ \\\\\n\\hline\n12 & $0.0$ & $0.0 \\pm 0.0$ & 15 & $2.5023$ & $2.9963 \\pm 0.0176$ \\\\\n\\hline\n17 & $4.9972$ & $5.0 \\pm 0.0001$ & 20 & $7.5467$ & $7.9963 \\pm 0.0167$ \\\\\n\\hline\n22 & $9.9245$ & $9.9997 \\pm 0.0026$ & & & \\\\\n\\hline\n\\end{tabular}\n\\caption{Hybrid Wannier centers along $z$ for different hybrid Wannier bands with non-zero winding of $\\gamma^{m}_{\\vec{G}_{2}}(k_{x})$ and $q = 1\/5$.\n$m$ denotes the non-trivial hybrid Wannier band index. \nThe first and last three columns summarize the results for $\\phi = 0$ and $\\phi = \\pi$, respectively. \nThere are $5$ and $4$ non-trivial hybrid Wannier bands for $\\phi = 0$ and $\\pi$ with $m = 2$, $7$, $12$, $17$, $22$ and $m = 4$, $9$, $15$, $20$, respectively.\n$z_{\\Gamma}$ is the hybrid Wannier center along $z$ for the hybrid Wannier functions at $\\vec{k} = (k_{x},k_{y}) = (0,0) = \\Gamma$.\n$\\left\\langle z \\right\\rangle$ is the hybrid center along $z$ averaged over the 2D Brillouin zone with grid size $100 \\times 100$. \nWe also report the standard deviation of $\\left\\langle z \\right\\rangle$ in the same column.}\n\\label{tab:hwfc_table}\n\\end{table}\n\n\n\\begin{figure}[ht]\n \\includegraphics[width=\\columnwidth]{grid_100_hwf_winding_decomposition_orbitals_bloch_Q_0_4pi_Lz_25_D_0_75_phi_0.pdf}\n \\caption{Berry phase winding of non-trivial hybrid Wannier bands (first column), hybrid Wannier functions at $\\vec{k} = \\Gamma$ decomposed in terms of orbital (second column), and Bloch bases (third column) for an $xy$-slab of the Weyl-CDW model with $t_{x}=-t_{y}=t_{z}=1$, $m =2$, $2|\\Delta|=0.75$, $q = 1\/5$, $L_{z} =25$ and $\\phi = 0$. \nThe $z$-coordinate ranges from $-12, \\ldots, +12$. \nThere are 5 non-trivial hybrid Wannier bands for the occupied states, each of them displaying $-2\\pi$ winding of $\\gamma^{m}_{\\vec{G}_{2}}(k_{x})$ as $k_{x}$ goes from $0$ to $2\\pi$, such that the Chern number for each of the CI layers is $C = -1$. \nFrom the second column, we can deduce that the non-trivial states are localized at $z \\approx -10$, $-5$, $0$, $+5$, $+10$. \nFrom the third column, where we plot $|c^{n}_{m,\\Gamma}|^{2}$, we can deduce that the non-trivial Bloch band indices at $\\vec{k} = \\Gamma$ are $ n =18,\\ldots, 22$ (the first band is labelled by $n=0$).}\n \\label{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_0}\n\\end{figure}\n\n\n\n\n\\begin{figure}[ht]\n \\includegraphics[scale=0.4]{Berry_curvature_around_Gamma_Q_0_4pi_Lz_25_D_0_75_phi_0.pdf}\n \\caption{The Berry curvature around $\\vec{k} = \\Gamma$ for the 25 valance Bloch bands (not hybrid Wannier bands) of the Weyl-CDW model for an $xy$-slab with $t_{x}=-t_{y}=t_{z}=1$, $m =2$, $2|\\Delta|=0.75$, $q = 1\/5$, $L_{z} =25$ and $\\phi = 0$.\n All the figures have $k_{x}$ and $k_{y}$ in the range range $2\\pi [-0.003,0.003]$.\n We notice that the Bloch bands with indices $n = 18,\\ldots, 22$ have negative Berry curvature around $\\vec{k} = \\Gamma$ contributing to the total Chern number $C = -5$ of this $xy$-slab, leading to $G_{xy} = -5 e^{2}\/h$.}\n \\label{Berry_curvature_around_Gamma_Q_0.4pi_Lz_25_D_0.75_phi_0}\n\\end{figure}\n\n\\begin{figure}[ht]\n \\includegraphics[width=\\columnwidth]{grid_100_hwf_winding_decomposition_orbitals_bloch_Q_0_4pi_Lz_25_D_0_75_phi_pi.pdf}\n \\caption{Berry phase winding of non-trivial hybrid Wannier bands (first column), hybrid Wannier functions at $\\vec{k} = \\Gamma$ decomposed in terms of orbital (second column), and Bloch basis (third column) for an $xy$-slab of the Weyl-CDW model with $t_{x}=-t_{y}=t_{z}=1$, $m =2$, $2|\\Delta|=0.75$, $q = 1\/5$, $L_{z} =25$ and $\\phi = \\pi$.\nThe $z$-coordinate ranges from $-12, \\ldots, +12$. \nThere are 4 non-trivial hybrid Wannier bands for the occupied states, each of them displaying $-2\\pi$ winding of $\\gamma^{m}_{\\vec{G}_{2}}(k_{x})$ as $k_{x}$ goes from $0$ to $2\\pi$, such that the Chern number for each of the CI layers is $C = -1$. \nFrom the second column we can deduce that the non-trivial states are localized at $z \\approx -7.5$, $-2.5$, $+2.5$, $+7.5$. \nFrom the third column we can deduce that the non-trivial Bloch band indices at $\\vec{k} = \\Gamma$ are $n = 17$, $18$, $20$ and $21$ (the first band is labelled by $n=0$).}\n \\label{hwf_winding_decomposition_orbitals_bloch_Q_0.4pi_Lz_25_D_0.75_phi_pi}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n \\includegraphics[scale=0.4]{Berry_curvature_around_Gamma_Q_0_4pi_Lz_25_D_0_75_phi_pi.pdf}\n \\caption{The Berry curvature around $\\vec{k} = \\Gamma$ for the 25 valance Bloch bands (not hybrid Wannier bands) for the $xy$-slab of the Weyl-CDW with $t_{x}=-t_{y}=t_{z}=1$, $m =2$, $2|\\Delta|=0.75$, $q = 1\/5$, $L_{z} =25$ and $\\phi = \\pi$. \n All the figures have $k_{x}$ and $k_{y}$ in range $2\\pi [-0.003,0.003]$. \n We notice that the Bloch bands with indices $n=17$, $18$, $20$ and $21$ have negative Berry curvature around $\\vec{k} = \\Gamma$ contributing to the total Chern number $C = -4$ of this $xy$-slab, leading to $G_{xy} = -4 e^{2}\/h$.}\n \\label{Berry_curvature_around_Gamma_Q_0.4pi_Lz_25_D_0.75_phi_pi}\n\\end{figure}\n\n\n\n\\begin{figure}[ht]\n \\includegraphics[scale=0.6]{slab_band_and_wvfn_Q_0_4pi_Lz_25_D_0_75_phi_0_and_pi.pdf}\n \\caption{The $xy$-slab valence band structure of the Weyl-CDW model plotted along the path $-Y \\to \\Gamma \\to Y$ with $t_{x}=-t_{y}=t_{z}=1$, $m =2$, $2|\\Delta|=0.75$, $q = 1\/5$, $L_{z} =25$, (a) $\\phi = 0$ and (b) $\\phi = \\pi$.\n The (a) 5 and (b) 4 non-trivial bands around $\\vec{k}=\\Gamma$ and $Y$ are marked by orange and green. \n The summation of the probability distribution for the (a) 5 and (b) 4 non-trivial states at $\\vec{k}=\\Gamma$ and $Y$ are also plotted on the right of each $xy$-slab band structure for (a) $\\phi = 0$ and (b) $\\pi$. \n The $z$-coordinate ranges from $-12,\\ldots, +12$. \n As the non-trivial states at $\\vec{k} = \\Gamma$ and $\\vec{k} =Y$ have exactly same probability distribution along $z$, they lie in the same Landau level index $n$ (see Sec.~VI of the main text) subspace. \n This also confirms that our identification of non-trivial bands is consistent.}\n \\label{slab_band_and_wvfn_Q_0.4pi_Lz_25_D_0.75_phi_0_and_pi}\n\\end{figure}\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLet $g\\in L^2(\\Bbb R^d)$, and let $\\Lambda\\subset \\Bbb R^{2d}$ be a countable subset. We define the {\\it Gabor system} (also known as {\\it Weyl-Heisenberg system}) $\\mathcal G(g, \\Lambda)$ with respect to $g$ and $\\Lambda$ to be the collection of functions $\\pi(a, b)g$ defined by combined time and frequency shifts of $g$:\n$$\n\\pi(a,b)g(x)= M_{b}T_{a}g = e^{2\\pi i \\langle b, x\\rangle} g(x-a) \\quad (a,b)\\in \\Lambda .\n$$\n$\\Lambda$ is also known as {\\it time-frequency set} and\nthe frequency shift is also called modulation. We say $g$ is an {\\it orthonormal Gabor window function} with respect to $\\Lambda$, or simply a window function, if $\\mathcal G(g, \\Lambda)$ is an orthonormal basis for $L^2(\\Bbb R^d)$. See e.g. \n\\cite{OleBook} or \\cite{Groechenig-book}. \n\n\n\n\nWe call $\\Lambda$ {\\it separable} if it is of the form of $\\Lambda = {\\mathcal T}\\times \\Gamma$ for some countable subsets ${\\mathcal T}$ and $\\Gamma$ in ${\\mathbb R}^d$. Gabor systems have been introduced for the first time in 1946 by Gabor \\cite{Gabor46} and are now fundamental objects in applied and computational harmonic analysis. Moreover, for $\\mathcal G(g, \\Lambda)$ to be a Gabor orthonormal basis, the (Beurling) density of $\\Lambda$, denoted by dens$(\\Lambda)$, must be $1$ \\cite{RS}.\n\n\n\n\\medskip\n\n\nThe existence of a window function for a given lattice has been investigated for several special cases of $M$. The question of existence has been completely answered\nby Han and Wang \\cite{HanW1} for separable lattices of the form $\\Lambda= \\mathcal J\\times \\mathcal T$ with dens$(\\Lambda)=1$. They answered the question by showing the existence of a common fundamental domain for two different lattices. Later, the same authors partially answered the question for non-separable lattices (i.e. the lattices of not of the form of $\\mathcal J \\times \\mathcal T$) for special cases of matrix $M$ \\cite{HanW4}. Indeed, they proved that, when for example $M$ is a block triangular matrix, a window function $g$ exists and it can be chosen so that $|g|$ or $|\\hat g|$ is the scalar multiple of a characteristic function. They also showed the existence of a window function with compact support for rational matrices $M$.\n\n\\medskip\n\nGiven a subset $K\\subset \\Bbb R^d$,\nwe denote by $\\chi_K$ the indicator function of $K$ and by $|K|$ its Lebesgue measure.\nThe main focus of this paper is the following. Suppose that $\\Lambda = M({\\mathbb Z}^{2d})\\subset \\Bbb R^{2d}$ is a full lattice with Beurling density dens$(\\Lambda)= |\\det(M)|^{-1} =1$ and suppose that $\\mathcal G(|K|^{-1\/2} \\chi_K, \\Lambda)$ forms a Gabor orthonormal basis for $L^2(\\Bbb R^d)$. What can we say about the structure of $K$? This question is related to the study of spectral sets and translational tiles which we will call the {\\it Fuglede-Gabor Problem} later on.\n\n\\medskip\n\n\n\\begin{definition}[Spectral and tiling sets]\\label{Spectral and tiling sets} A Lebesgue measurable set\n $K\\subset \\Bbb R^d$ with positive and finite measure is a {\\it spectral set} in $\\Bbb R^d$ if there is a countable set $B\\subset \\Bbb R^d$ (not necessarily unique) such that\n exponentials $\\{e_b(x):= e^{2\\pi i \\langle b,x\\rangle} : b\\in B, x\\in K\\}$ constitute an orthogonal basis for $L^2(K)$, i.e., the exponentials are mutual orthogonal and complete in $L^2(K)$. In this case $B$ is called a {\\it spectrum} for $K$.\n\n \\medskip\n\n We say $K$ {\\it multi-tiles} $\\Bbb R^d$ by its translations if there\n is a countable set ${\\mathcal J}\\subset \\Bbb R^d$ and integer $N\\ge 1$ such that\n\n\\begin{equation}\\label{multitile-generic}\n\\sum_{t\\in{\\mathcal J}} \\chi_{K} (x+t)=N\\quad a.e. \\ x\\in \\Bbb R^d .\n\\end{equation}\nIf $N=1$, then $K$ {\\it tiles} ${\\mathbb R}^d$ and the set ${\\mathcal J}$ is called {\\it tiling set} for $K$ (For more details about multi-tiles, see e.g. \\cite{Kol}).\n \\end{definition}\n\n\\medskip\n\n Spectral sets have been studied extensively in the recent years and their study has been reduced to the study of tiling sets by the Fuglede Conjecture or Spectral Set Conjecture \\cite{Fug74} which asserts: {\\it A set $K\\subset \\Bbb R^d$ with positive and finite measure is a spectral set if and only if $K$ tiles $\\Bbb R^d$ by translations. }\n Fuglede proved the conjecture in his celebrated 1974 paper \\cite{Fug74} for the case when $K$ tiles by a lattice or $K$ has a spectrum which is a lattice. The Fuglede Conjecture led to considerable activity in the past three decades. In\n 2004, Tao \\cite{T04} disproved the Fuglede conjecture for dimension $5$ and higher, followed by Kolountzakis and Matolcsi\\rq{}s result \\cite{FMM,KM06} where they proved that the conjecture fails in dimensions $3$ and higher. For more recent results and historical comments see e.g. \\cite{BHM16,IMP17}.\n\n\n\\medskip\nSpectral sets and tiles appear naturally in the Gabor setting. Indeed, let $\\Lambda= \\mathcal J \\times \\mathcal T$ be a separable countable set (not necessarily a lattice) with dens$(\\Lambda)=1$ and let $\\Omega$ be a compact set in $\\Bbb R^d$ which tiles by $\\mathcal J$ and is spectral for $\\mathcal T$. For example, take $\\Omega = [0,1]^2$ and ${\\mathcal J} = {\\mathcal T} = {\\mathbb Z}^2$. Let $g$ be a function supported in $\\Omega$. Then an easy calculation shows that the Gabor system\n $\\mathcal G(g, \\Lambda)$ is an orthonormal basis if $|g(x)|=|\\Omega|^{-1\/2} \\chi_\\Omega(x)$\n (see also \\cite{LW03}, Lemma 3.1). We call such Gabor bases {\\it standard}. Liu and Wang \\cite{LW03} conjectured the converse of this result that for a compactly supported function $g$ and a countable separable set $\\Lambda= \\mathcal J \\times \\mathcal T$, if $\\mathcal G(g, \\Lambda)$ is an orthonormal basis for $L^2(\\Bbb R^d)$, then there is a compact set $\\Omega\\subset \\Bbb R^d$ such that\n $|g|$ is a constant multiple of $\\chi_\\Omega$, $\\Omega$ tiles by $\\mathcal J$ and is a spectral set for $\\mathcal T$. Liu and Wang proved their conjecture when the support of $g$ is an interval.\n Dutkay and the first listed author recently proved that the Liu and Wang\\rq{}s conjecture is affirmative if $g$ is non-negative \\cite[Theorem 1.8]{Dutkay-Lai}. \n\n\n \n\nHowever, the conjecture is still unsolved for general compactly supported $g$.\n\n\n\n\n\\iffalse\n\\begin{Dutkay-Lai}\\label{Dutkay-Lai}(\\cite{Dutkay-Lai}, Theorem 1.8 resp. Theorem 6.2) Suppose that $g$ is a non-negative function with bounded and measurable support $K$ in $\\Bbb R^d$. Suppose that $\\Lambda=\n \\mathcal J\\times \\mathcal T$ is a countable set with $D(\\Lambda)=1$. If $\\mathcal G(g, \\Lambda)$ is an orthonormal basis for $L^2(\\Bbb R^d)$, then the following hold.\n\\begin{itemize}\n\\item[(i)] $K$ tiles $\\Bbb R^d$ by translations by $\\mathcal J$.\n\\item[(ii)] $g= |K|^{-1\/2}\\chi_K$ a.e. on $K$.\n\\item[(iii)] $K$ is a spectral with respect to $\\mathcal T$.\n\\end{itemize}\n\\end{Dutkay-Lai}\n\n\n\nWe shall sketch the proof of Theorem A here. The completeness of the Gabor system $\\mathcal G(g, \\Lambda)$ proves that $\\cup_{j\\in \\mathcal J} K+j$ is a cover for $\\Bbb R^d$. The orthogonality implies that the collection of sets $\\{K+j: \\ j\\in \\mathcal J\\}$ are mutual disjoint up to a Lebesgue measure zero set, thus $K$ is a tiling set with respect to $\\mathcal J$. Further, the measure $d\\mu(x):=g(x)^2 dx$ has a spectrum by proving an equivalent statement that\n$$\\sum_{p\\in \\mathcal T} |\\widehat{\\mu} (x-p)|^2 =1.$$\nThis implies that the measure $\\mu$ is spectral, thus $|g|^2$ must be a constant multiple of a characteristic function by\nCorollary 1.4 of \\cite{Dutkay-Lai}. This forces $g= |K|^{-1\/2}\\chi_K$ a.e. on its support $K$, hence the conclusions of (ii) and (iii). The converse holds by a direct calculation.\n\\fi\n\n\n\n\nThe following problem links the study of window functions associated with Gabor orthonormal bases to the tiling and spectral properties of sets.\n\\medskip\n\n\\begin{problem}\\label{our conjecture1}(Fuglede-Gabor Problem)\nLet $K\\subset \\Bbb R^d$ be a measurable subset with positive and finite measure, and let $\\Lambda \\subset \\Bbb R^{2d}$ be a countable subset. If the Gabor family $\\mathcal G(|K|^{-1\/2}\\chi_K, \\Lambda)$ is an orthonormal basis for $L^2(\\Bbb R^d)$, then $K$ tiles and is a spectral set.\n \\end{problem}\n\n\n\nSince the indicator function of a set is non-negative, {\\it Problem \\ref{our conjecture1} is already affirmative if the time-frequency set is a separable countable sets} using the results of Dutkay and the first listed author.\n Therefore we only focus on the case when the time-frequency set is non-separable. Moreover, although the problem does not require any boundedness assumption of $K$, our interest will mainly be focused on the set $K$ being bounded. \n\n\n It is hard to speculate whether Problem \\ref{our conjecture1} is true or not in its full generality. But from the point of view of Fuglede's result for lattices, we still hope that the Fuglede-Gabor problem is true for non-separable lattices as well. Unfortunately, after our intensive study, we found out that, similar to many notoriously difficult problems in Gabor analysis (see e.g. \\cite{Groechenig-mystery}), \n the Fuglede-Gabor problem for lattices appears to be uneasy. This paper gives a partial answer towards the full solution together with some unexpected examples, as we explain below. \n\n\n\n\\noindent{\\bf Main Results of the paper.} Our main results will mostly be focused on the lower triangular block matrices since most matrices can be reduced to the lower triangular form:\n \\begin{equation}\\label{lower_block}\n \\Lambda = \\left(\n \\begin{array}{cc}\n A & O \\\\\n C & B \\\\\n \\end{array}\n \\right)({\\mathbb Z}^{2d}), \\ \\mbox{and} \\ |\\det(AB)|=1\n \\end{equation}\n (i.e. dens$(\\Lambda)=1$). We will use $B^{-t}$ to denote the inverse transpose of the matrix $B$. Our first general key lemma is as follows, it will serve as a key step for our further analysis.\n\n \\medskip\n\n \\begin{lemma}[Key Lemma]\\label{Th_union of FD} Let $\\Lambda = M({\\mathbb Z}^{2d})$ with $M$ an $2d\\times 2d$ invertible lower triangular block matrix of the form (\\ref{lower_block}).\nSuppose that ${\\mathcal G}\\left({|K|^{-1\/2}}\\chi_K,\\Lambda\\right)$ is a Gabor orthonormal basis. Then there exists an integer $N\\ge 1$ such that\n $$\n K = \\bigcup_{j=1}^N D_j=\\bigcup_{j=1}^N E_j\n $$\nwhere $D_j$\\rq{}s are fundamental domains of $B^{-t}(\\Bbb Z^d)$ and $E_j$\\rq{}s are almost disjoint fundamental domains of $A(\\Bbb Z^d)$ with $|D_i\\cap D_j| = 0$ and $|E_i\\cap E_j| =0$ for all $i\\ne j$. (i.e. K multi-tiles ${\\mathbb R}^d$ simultaneously by $A({\\mathbb Z}^d)$ and $B^{-t}(\\Bbb Z^d)$.)\n\\end{lemma}\n\n If we can prove that $N=1$, then $K$ will be a common fundamental domain for $A(\\Bbb Z^d)$ and $B^{-t}(\\Bbb Z^d)$ and this will imply that the Fuglede-Gabor problem holds. In particular, this is true when $A^tB$ is an integer matrix and $K$ is a bounded set, as our next result confirms.\n\n\\medskip\n\n\n\\begin{theorem}\\label{lower triangle} Let $K$ be a bounded measurable subset of $ \\Bbb R^d$ with positive measure, and let $\\Lambda\\subset \\Bbb R^{2d}$ be a lower triangular lattice in (\\ref{lower_block}). Suppose that\n ${\\mathcal G}(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a Gabor orthonormal basis for $L^2(\\Bbb R^d)$ and $A^tB$ is an integer matrix. Then $K$ tiles and is spectral. More precisely, $K$ is a common fundamental domain for $A(\\Bbb Z^d)$ and $B^{-t}(\\Bbb Z^d)$, $K$ tiles by $A(\\Bbb Z^d)$ and is spectral with spectrum $B(\\Bbb Z^d)$.\n\\end{theorem}\n\n\n\n\\medskip\n\n\n\n\nAs a consequence of Theorem \\ref{lower triangle}, we resolve the Fuglede-Gabor Problem \\ref{our conjecture1} in dimension one for rational matrices and $K$ is bounded.\n\n\\vskip.124in\n\n\\begin{theorem}\\label{rational_dim1} Suppose that $K\\subset \\Bbb R$ is a bounded set with positive Lebesgue measure. Suppose that $\\Lambda$ is a rational lattice in $\\Bbb R^{2}$ with $dens(\\Lambda)=1$. If\n ${\\mathcal G}(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a Gabor orthonormal basis for $L^2(\\Bbb R)$, then $K$ tiles and is spectral.\n\\end{theorem}\n\n\n\\medskip\n\nWe also have the following result for upper triangular block matrices using Theorem \\ref{lower triangle}.\n\n \\vskip.124in\n\n\\begin{theorem}\\label{UT} Suppose that $K\\subset \\Bbb R^d$ is a bounded set with positive Lebesgue measure. Supposes that $\\Lambda\\subset \\Bbb R^{2d}$ is a lattice such that\n $\\Lambda = \\left(\n \\begin{array}{cc}\n A & D \\\\\n O & B \\\\\n \\end{array}\n \\right)({\\mathbb Z}^{2d})$ with $dens(\\Lambda)=1$,\n $A^{-1}D$ symmetric rational matrix and $A^tB=I$. If\n ${\\mathcal G}(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a Gabor orthonormal basis for $L^2(\\Bbb R^d)$, then $K$ tiles and is spectral.\n \\end{theorem}\n\n \\medskip\n\nTheorem \\ref{lower triangle} may also be consider as a converse of \\cite[Lemma 4.1]{HanW4}, which states that if $K$ is a common fundamental domain for the lattice $A({\\mathbb Z}^d)$ and $B^{-t}({\\mathbb Z}^d)$, then for any matrix $C$, the system ${\\mathcal G}(|K|^{-1\/2}\\chi_K,\\Lambda)$ is an Gabor orthonormal basis. Therefore one may naturally expect that $N=1$ in Theorem \\ref{Th_union of FD} is always the case. However, we will show that {\\it $N>1$ can actually happen with a suitable choice of $C$ if $A^tB$ is a rational matrix (see Example \\ref{mutli-tile K})}. This poses additional difficulty to solve the Fuglede-Gabor Problem for rational matrices in higher dimension, as we shall discuss it later. Finally, for a general matrix containing irrational entries, the answer to Fuglede-Gabor problem is completely open. We will discuss this in detail in Section \\ref{Open problems}.\n\n\n\n\n %\n\n\n\n \\medskip\n\n\n\n\n\n {\\bf Outline of the paper.} We organize the paper as follows: After some definitions and recalling some known and basic facts about lattices and Gabor analysis in Section \\ref{notations}, in Section \\ref{proof of Theorem 1.3} we prove Theorem \\ref{Th_union of FD}.\n The proof of Theorems \\ref{lower triangle} are \\ref{rational_dim1} are presented in\n Section \\ref{thm:lower triangle}. In Section \\ref{thm:rational_dim1 and UT} we prove Theorem \\ref{UT}. Section \\ref{Examples} is devoted to examples illustrating the possibility for $N> 1$.\n We conclude the paper with a series of open problems in Section \\ref{Open problems} both for rational and irrational lattices as well as the full generality of the Fuglede-Gabor Problem. In our exposition, we discover that a new notion of completeness, which we will call {\\it exponential completeness}, is crucial in studying the Fuglede-Gabor problem, we will give a short study in Appendix A. In Appendix B, we will show that the octagon will not produce any Gabor orthonormal basis using rational matrices. \n\n\n\n\n\n\\iffalse\n A classical example of a set in dimension $d$ which tiles and is spectral is the unit cube $[0,1]^d$. It is known that $[0,1]^d$ tiles $\\Bbb R^d$ with integer shifts with $\\Bbb Z^d$ and $\\{e_n(x):=e^{2\\pi in \\langle n, x\\rangle } : n\\in \\Bbb Z^d\\}$ is an orthonormal basis for $L^2([0,1]^d)$, also known as Fourier basis.\n It is also well-known that the Gabor system $\\mathcal G(\\chi_{[0,1]^d}, \\alpha\\Bbb Z^d\\times\\beta\\Bbb Z^d)$ is an orthonormal basis for $L^2(\\Bbb R^d)$ when $\\beta=\\alpha^{-1}$ (see e.g. \\cite{Groechenig-book}). This simple observation suggests that there is a link between the study of the Gabor orthonormal basis and the study of tiling and spectral properties. The connection was observed in the earliest by Liu and Wang \\cite{LW03} in 2003, who conjectured the following for separable time-frequency shifts $\\Lambda$:\n \\medskip\n\n \\fi\n\n\n \\iffalse\n The link between the study of the Gabor orthonormal basis and the study of tiling and spectral properties was observed in more generality by Liu and Wang \\cite{LW03} in 2003. They conjectured that:\n\n {\\bf Liu and Wang\\rq{}s Conjecture:} (\\cite{LW03}) {\\it Given a compactly supported function $g(\\neq 0) \\in L^2(\\Bbb R^d)$ with finite and positive measure support $\\Omega$ and given a separable countable set $\\Lambda=\\mathcal J\\times \\mathcal T \\subseteq \\Bbb R^{2d}$, the Gabor family $\\mathcal G(g, \\Lambda)$ is an orthonormal basis if and only if the following hold: \\\\\n (1) $\\Omega$ tiles $\\Bbb R^d$ with translations with tiling set $\\mathcal J$. \\\\\n (2) $|g|=|\\Omega|^{-1\/2} \\chi_\\Omega$. \\\\\n (3) $\\Omega$ is a spectral set with spectrum $\\mathcal T$.}\n\nWhen (1) and (3) hold, then for any function $g$ satisfying (2) the Gabor family $\\mathcal G(g, \\Lambda)$ constitutes an orthonormal basis for $L^2(\\Bbb R^d)$ (\\cite{LW03}, Lemma 3.1).\nThe other direction of the conjecture has not been solved in general but only in some special cases. Recently,\n Dutkay and the first listed author (\\cite{Dutkay-Lai}, Theorem 6.2), proved that if for the separable countable set $\\Lambda=\\mathcal J\\times \\mathcal T$, the Gabor system $\\mathcal G(g, \\Lambda)$ is an orthonormal basis for $L^2(\\Bbb R^d)$, and if the window function $g\\neq 0$ is non-negative on its support $K$, a bounded and measurable set, then $g$ is a constant multiple of characteristic function of $K$, and $K$ tiles by $\\mathcal J$ and is spectral in $\\Bbb R^d$ with respect to $\\mathcal T$. This result provides a partial affirmative answer to the Liu and Wang\\rq{}s Conjecture in separable case.\n\n \\fi\n\n\n\n\n\n \\section{Preliminaries}\\label{notations}\n\nIn this section, we will collect several basic definitions and results required for the rest of the paper. A {\\it full-rank lattice}\n $\\Lambda\\subset \\Bbb R^d$ is a discrete and countable subgroup of $\\Bbb R^{d}$ with compact quotient group $\\Bbb R^{d}\/\\Lambda$. A full-rank lattice in $\\Bbb R^d$ is given by $\\Lambda = {M}({\\mathbb Z}^{2d})$ for some\n $2d\\times 2d$ invertible matrix $M\\in GL(2d, \\Bbb R)$. The density of $\\Lambda$ is given by dens$(\\Lambda)=|\\det(M)|^{-1}$.\n\n Let $\\Lambda$ be a lattice in $\\Bbb R^d$.\n The {\\it dual lattice} of $\\Lambda$ is defined as\n $$\\Lambda^\\perp:= \\{ x\\in \\Bbb R^{2d} : \\ \\langle \\lambda, x\\rangle\\in \\Bbb Z, \\ \\forall \\lambda\\in \\Lambda\\} .$$\nA direct calculation shows that $\\Lambda^\\perp= M^{-t}(\\Bbb Z^{d})$.\n\nA fundamental domain of a lattice $\\Lambda$ is a measurable set $\\Omega$ in $\\Bbb R^d$ which contains distinct representatives (mod $\\Lambda$) in $\\Bbb R^d$, so that the any intersection of $\\\n\\Omega$ with any coset $x+\\Lambda$ has only one element. For the existence of a fundamental domain see \\cite[Theorem 1]{Feld-Green68}. It is also evident that $\\Omega$ tiles $\\Bbb R^d$ with translations by $\\Lambda$ and\nany other tiling set differs from $\\Omega$ at most for a zero measure set.\n\n\n\\noindent{\\bf 1. A reduction lemma.} For an invertible $d\\times d$ matrix\n $A$, the operator ${\\mathcal D}_A:L^2(\\Bbb R^d)\\to L^2(\\Bbb R^d)$ defined by\n $${\\mathcal D}_Ag(x):=|\\det(A)| ^{1\/2} g(Ax).$$\n %\n is unitary, i.e, ${\\mathcal D}_A$ is onto and isometry $\\|{\\mathcal D}_Ag\\|= \\|g\\|$. The following lemma follows from a simple computation and is in general known. We will omit the detail of the proof. \n\n\\vskip.124in\n\n \\begin{lemma}\\label{lemma 3-prim}\n Let ${\\Lambda} $ be a lattice such that\n %\n\\begin{align}\\label{block lattice}\n\\Lambda= \\left(\n \\begin{array}{cc}\n A & D\\\\\n C & B \\\\\n \\end{array}\n \\right)({\\mathbb Z}^{2d})\n \\end{align}\n where $A$ is an invertible $d\\times d$ matrix. Then\n$$\\mathcal G(g, \\Lambda) = {\\mathcal D}_{A^{-1}}\\mathcal G({\\mathcal D}_Ag, \\widetilde{\\Lambda})$$\n where $$\n\\widetilde{\\Lambda} = \\left(\n \\begin{array}{cc}\n I & A^{-1}D\\\\\n A^tC & A^tB \\\\\n \\end{array}\n \\right) ({\\mathbb Z}^{2d}).\n $$\n\nConsequently, $\\mathcal G(g, \\Lambda) $ is a Gabor orthonormal basis if and only if $\\mathcal G({\\mathcal D}_Ag, \\widetilde{\\Lambda})$ is a Gabor orthonormal basis.\n \\end{lemma}\n\n\n\nIn Lemma \\ref{lemma 3-prim}, if we let\n $D=O$ and $g(x)=|K|^{-1\/2} \\chi_K$, then the conclusion of the lemma shows that ${\\mathcal G}(|K|^{-1\/2} \\chi_K, \\Lambda)$ is an orthonormal basis for $L^2(\\Bbb R^d)$ if and only if ${\\mathcal G}(|A^{-1}K|^{-1\/2} \\chi_{A^{-1}K}, \\widetilde{\\Lambda})$ is an orthonormal basis with\n$$\n\\widetilde{\\Lambda} = \\left(\n \\begin{array}{cc}\n I & O\\\\\n A^tC & A^tB \\\\\n \\end{array}\n \\right) ({\\mathbb Z}^{2d}).\n$$\n We shall use this observation later.\n\n \\iffalse\n\\begin{proof}\nLet $g = |K|^{-1\/2} \\chi_K$. For each $\\lambda = (Am,Cm+Bn),\\lambda' =(Am',Cm'+Bn')\\in \\Lambda$. Note that by a change of variable $x \\rightarrow Ax$\n$$\n\\begin{aligned}\n\\langle\\pi(\\lambda)g,\\pi(\\lambda')g\\rangle =& |K|^{-1} \\int \\chi_K (x-Am)\\chi_K (x-Am') e^{-2\\pi i \\langle C(m-m')+B(n-n'),x\\rangle}dx\\\\\n=& |K|^{-1}\\int \\chi_K (x-Am)\\chi_K (x-Am') e^{-2\\pi i \\langle C(m-m')+B(n-n'),x\\rangle}dx\\\\\n=& |\\det A||K|^{-1}\\int \\chi_K (Ax-Am)\\chi_K (Ax-Am') e^{-2\\pi i \\langle C(m-m')+B(n-n'),Ax\\rangle}dx\\\\\n=& |A^{-1}K|^{-1} \\int \\chi_{A^{-1}K} (x-m)\\chi_{A^{-1}K} (x-m')e^{-2\\pi i \\langle A^tC(m-m')+A^tB(n-n'),x\\rangle}dx.\n\\end{aligned}\n$$\nThis shows that ${\\mathcal G}(|K|^{-1\/2} \\chi_K, \\Lambda)$ is mutually orthogonal if and only if ${\\mathcal G}(|A^{-1}K|^{-1\/2} \\chi_{A^{-1}K}, \\widetilde{\\Lambda})$ is mutually orthogonal. In a similar change of variable, we see that the Parseval identities of these Gabor system are actually equivalent. Hence, the statement of the lemma follows.\n\\end{proof}\n\\fi\n\n\\medskip\n\n\\noindent{\\bf 2. Orthogonality implies completeness.} The following proposition says that completeness automatically holds for a lattice of density one if we can establish the mutually orthogonality. \n\n\n\\medskip\n\n \\begin{proposition}\\label{complt}\n Let $g\\in L^2(\\Bbb R^d)$, $\\|g\\|=1$ and $\\Lambda\\subset \\Bbb R^{2d}$ be a lattice with density $\\mbox{dens}(\\Lambda)=1$. Assume that\n $\\mathcal G(g,\\Lambda)$ is an orthonormal set. Then $\\mathcal G(g,\\Lambda)$ is complete.\n \\end{proposition}\n\n\\medskip\n For the proof of Proposition \\ref{complt} we require the following lemma.\n Note that for a positive Borel measure $\\mu$,\n $$\n f\\ast\\mu (x) = \\int f(x-y)d\\mu(y),\n $$\n given that the integral is well-defined. If\n $\\mu = \\sum_{\\lambda\\in\\Lambda}\\delta_{\\lambda}$, then $\\chi_K\\ast\\mu = 1$ ($\\le 1$) if and only if $K$ tiles (packs) ${\\mathbb R}^d$ by $\\Lambda$\\footnote{Recall that $K$ packs ${\\mathbb R}^d$ by ${\\mathcal J}$ if $\\sum_{t\\in{\\mathcal J}}\\chi_{K}(x-t)\\le1$, a.e. $x\\in \\Bbb R^d$}. With this introduction we recall the following result.\n\n\\medskip\n\n \\begin{lemma}\\label{GLW}(\\cite[Theorem 2.1]{GLW})\n Suppose that $f,g\\in L^1({\\mathbb R}^d)$ are non-negative functions such that $\\int f(x) dx = \\int g(x) dx= 1$. Suppose that for positive Borel measure $\\mu$ on $\\Bbb R^d$\n$$\nf\\ast\\mu \\le 1 \\ \\mbox{and} \\ g\\ast\\mu\\le 1.\n$$\nThen $f\\ast\\mu = 1$ if and only if $ g\\ast \\mu= 1$.\n\\end{lemma}\n\n\n Given $f,g\\in L^2({\\mathbb R}^d)$, the {\\it short time Fourier transform} is defined by\n \\begin{align}\\label{STFT}\nV_gf(t,\\xi)=\\int f(x)\\overline{g(x-t)}e^{-2\\pi i \\langle\\xi,x\\rangle}dx, \\quad (t,\\xi)\\in \\Bbb R^{2d}\n \\end{align}\nand it is a continuous function on ${\\mathbb R}^{2d}$ \\cite{Groechenig-book}.\n\n\n \\begin{proof}[Proof of Proposition \\ref{complt}]\n The mutual orthogonality of $\\mathcal G(g, \\Lambda)$ implies the Bessel inequality of the system:\n\\begin{align}\\label{Bessel}\n\\sum_{(t,\\xi)\\in \\Lambda} |V_gf(t, \\xi)|^2\\leq \\|f\\|^2 \\ , \\quad \\forall f\\in L^2(\\Bbb R^d).\n\\end{align}\nLet $ s,\\xi \\in \\Bbb R^d$. The inequality (\\ref{Bessel}) for $e^{2\\pi i \\langle \\nu, x\\rangle} f(x-s)$ in the place of $f$ yields\n\\begin{align}\\notag\n\\sum_{(t,\\xi)\\in \\Lambda} |V_gf(t-s, \\xi-\\nu)|^2\\leq \\|f\\|^2 \\quad \\forall \\ (s, \\nu)\\in \\Bbb R^{2d}.\n\\end{align}\nHence, $|V_gf|^2\\ast \\delta_\\Lambda \\leq \\| f\\|^2$. Take $G:= \\| f\\|^{-2}|V_gf|^2$. Then $\\int_{\\Bbb R^{2d}} G(z) dz=1$ and $G\\ast \\delta_\\Lambda\\leq 1$. On the other hand, $\\Lambda$ is a lattice with density $1$. Let $\\Omega\\subset \\Bbb R^{2d}$ be any fundamental domain for $\\Lambda$. Then $|\\Omega|=1$ and it tiles $\\Bbb R^{2d}$ by $\\Lambda$. Therefore $\\chi_\\Omega\\ast \\delta_\\Lambda=1$. Now Lemma \\ref{GLW} implies that $G\\ast \\delta_\\Lambda =1$. But this is equivalent to the completeness of the system $\\mathcal G(g, \\Lambda)$ and we are done.\n \\end{proof}\n\n\\medskip\n\n\n\n\\noindent{\\bf 3. Some reduction to lower triangular block matrices.} The following result is due to Han and Wang which states that any invertible integer matrix can be converted into a lower triangular integer matrix. We will need it in later sections. An integer matrix $P$ is called {\\it unimodular} if $\\det P=1$.\n\n\n\\medskip\n\n\\begin{lemma}\\label{HanW1}(\\cite{HanW4}, Lemma 4.4) Let $M$ be an $d\\times d$ invertible integer matrix. Then there is an $d\\times d$ unimodular matrix $P$ such that $MP$ is a lower triangular integer matrix.\n\\end{lemma}\n\n\n\n\nAs a corollary of Lemma \\ref{HanW1} we can show that any rational matrix can be represented as a lower triangular rational matrix.\n\n\\medskip\n\n\\begin{corollary}\\label{M=N}\nLet $M$ be an $d\\times d$ invertible rational matrix. Then there is a lower triangular rational matrix $N$ such that $M(\\Bbb Z^d) = N(\\Bbb Z^d)$.\n\\end{corollary}\n\nHenceforth, we shall say matrix $M$ is {\\it equivalent} to $N$ if $M(\\Bbb Z^d) = N(\\Bbb Z^d)$. \n\n\n\\medskip\n\n\\noindent{\\bf 4. Exponential Completeness.}\\label{exponential completeness} Recall that a collection of functions $\\{\\varphi_n\\}$ is said to be {\\it complete} in $L^2(\\Omega)$ if $\\langle f, \\varphi_n\\rangle =0$ for all $n$ implies that $f=0$ a.e. on $L^2(\\Omega)$. Given $f\\in L^2({\\mathbb R}^d)$, the Fourier transform of $f$ is defined to be $\\widehat{f}(\\xi)= \\int_{{\\mathbb R}^d}f(x)e^{-2\\pi i \\langle\\xi, x\\rangle}dx$. In our study, we will need to following weaker notion of the completeness property. \n\n\\medskip\n \n\\begin{definition}\\label{exponential completeness} \n Let $\\Lambda$ be a countable set and let $\\Omega$ be a Lebesgue measurable set with positive finite measure. We say that the set of exponentials $\\{e^{2\\pi i \\langle\\lambda,x\\rangle}:\\lambda\\in\\Lambda\\}$ (or $\\Lambda$) is {\\it exponentially complete} for $L^2(\\Omega)$ if there does not exist any $\\xi\\in{\\mathbb R}^d$ such that \n\n $$\n \\widehat{\\chi_{\\Omega}}(\\lambda-\\xi)=\\int_{\\Omega} e^{2\\pi i \\langle \\xi ,x\\rangle} e^{-2\\pi i \\langle \\lambda, x\\rangle} dx =0, \\ \\forall \\lambda\\in\\Lambda.\n $$\n \\end{definition}\n\n\\medskip\n\n\n\\begin{remark}\\label{remark2}\nThroughout the paper, we will see that exponential completeness plays an important role in constructing Gabor orthonormal basis using non-separable lattices. If a countable set of exponentials is complete for $L^2(\\Omega)$, then it must be exponentially complete (otherwise $e^{2\\pi i \\langle \\xi,x\\rangle}$ will be orthogonal to all $e^{2\\pi i \\langle\\lambda,x\\rangle}$ contradicting completeness). However, the converse is not true.\n For example, the set of exponentials associated to the lattice $\\Lambda=\\sqrt{2}{\\mathbb Z}$ is exponentially complete in $L^2([0,1])$, but it is not complete in it (see Lemma \\ref{lemma 4}). In Appendix 1, we will give a short study about the exponential completeness for lattices in $L^2[0,1]^d$.\n\\end{remark}\n\n\n\n\n\\section{Proof of Lemma \\ref{Th_union of FD} - Union of fundamental domains}\\label{proof of Theorem 1.3} \n We now prove our Theorem \\ref{Th_union of FD}. It follows from two theorems in two separate fields. The first one is taken from the study of Fuglede's problems. It was first proved by Jorgensen and Pedersen \\cite[Theorem 6.2 (b)]{JoPe} and then Lagarias and Wang \\cite[Theorem 2.1]{LW97} gave a simpler proof.\n\n\\medskip\n\n\\begin{theorem}\\label{thJoPeLgWa}\nLet $\\Omega\\subset{\\mathbb R}^d$ be a Lebesgue measurable set with positive finite measure. Suppose $\\Gamma$ is a full-rank lattice such that $\\Gamma \\subseteq \\{\\xi: \\widehat{\\chi_{\\Omega}}(\\xi) = 0\\} \\cup \\{0\\}$. Then\n$\n\\Omega = \\bigcup_{j=1}^N D_j$, up to measure zero, where $D_j$ are fundamental domains for the dual lattice $\\Gamma^\\perp$, $|D_i\\cap D_j|=0, \\ i\\neq j$ ~ and $N=|\\Omega|\/|D_j|$.\n \\end{theorem}\n\n\n\n\\medskip\n\nGiven a lattice $\\Lambda=M(\\Bbb Z^{2d})$, the {\\it adjoint lattice} $\\Lambda^\\circ$ is a lattice such that\n $$J(\\Lambda^\\circ)= \\Lambda^\\perp $$\n where\n $J= \\left(\n \\begin{array}{cc}\n O & -I \\\\\n I & O \\\\\n \\end{array}\n \\right)$. In other words, $\\Lambda^\\circ= J^{-1}M^{-t}(\\Bbb Z^d)$.\n\nThe Ron-Shen duality theorem \\cite{RS-duality97} is well-known in Gabor analysis. It was first proved over symplectic lattices, it is known to be true over any lattice (see e.g \\cite[Theorem 2.3]{Groechenig-mystery} for a proof by Poisson Summation Formula). We will need the following version of duality theorem.\n\n\\medskip\n\n\\begin{theorem}\\label{thRS} ${\\mathcal G}(g,\\Lambda)$ is a Gabor orthonormal basis if and only if ${\\mathcal G}(g,\\Lambda^{\\circ})$ is a Gabor orthonormal basis.\n\\end{theorem}\n\n\\begin{proof}[Sketch of Proof.] This statement is well-known. Here we provide a simple proof based on \\cite[Theorem 2.3]{Groechenig-mystery} and Proposition \\ref{complt}. Since $(\\Lambda^{\\circ})^{\\circ} =\\Lambda$, both sides of the statements are symmetric and we just need to prove one side of the equivalence. Suppose that ${\\mathcal G}(g,\\Lambda)$ is a Gabor orthonormal basis. Then $g$ is the only dual window with the property that \n$$\n\\langle g, \\pi (\\mu)(g)\\rangle = 0 \\ \\forall \\mu\\in\\Lambda^{\\circ}\\setminus \\{0\\}\n$$\n(by \\cite[Theorem 2.3]{Groechenig-mystery}). This means that for all distinct $\\mu,\\mu'\\in\\Lambda^{\\circ}$, $\\langle \\pi(\\mu)g, \\pi (\\mu')(g)\\rangle = c \\langle g, \\pi (\\mu-\\mu')(g)\\rangle = 0$ ($c$ is some unimodular constant). Thus ${\\mathcal G}(g,\\Lambda^{\\circ})$ is mutually orthogonal. As ${\\mathcal G}(g,\\Lambda)$ is a Gabor orthonormal basis, $\\|g\\|=1$ and dens$(\\Lambda^{\\circ})$ = dens$(\\Lambda)$ =1, by Proposition \\ref{complt}, ${\\mathcal G}(g,\\Lambda^{\\circ})$ is complete and is thus an orthonormal basis.\n\\end{proof}\n\nFor a lower triangular lattice $\\Lambda= \\left(\n \\begin{array}{cc}\n A & O \\\\\n C & B \\\\\n \\end{array}\n \\right){\\mathbb Z}^{2d}$, the adjoint lattice $\\Lambda^\\circ$ is also a lower triangular and it can be calculated as follows:\n %\n$$\n\\Lambda^{\\circ} = \\left(\n \\begin{array}{cc}\n O & I \\\\\n -I & O \\\\\n \\end{array}\n \\right)\\left(\n \\begin{array}{cc}\n A^{-t} & -A^{-t}C^tB^{-t} \\\\\n O & B^{-t} \\\\\n \\end{array}\n \\right) ({\\mathbb Z}^{2d}) = \\left(\n \\begin{array}{cc}\n O & B^{-t} \\\\\n -A^{-t} & A^{-t}C^tB^{-t} \\\\\n \\end{array}\n \\right) ({\\mathbb Z}^{2d}).\n$$\n\nFrom the other hand we can write $$\\left(\n \\begin{array}{cc}\n O & B^{-t}\\\\\n -A^{-t}& A^{-t}C^tB^{-t} \\\\\n \\end{array}\n \\right) = \\left(\n \\begin{array}{cc}\n B^{-t} & O\\\\\n A^{-t}C^tB^{-t} & A^{-t} \\\\\n \\end{array}\n \\right)\n \\left(\n \\begin{array}{cc}\n O & I \\\\\n -I & O\\\\\n \\end{array}\n \\right)({\\mathbb Z}^{2d}).\n$$\nBut $\\left(\n \\begin{array}{cc}\n O & I \\\\\n -I & O\\\\\n \\end{array}\n \\right) ({\\mathbb Z}^{2d})={\\mathbb Z}^{2d}$, therefore we have\n$$\n\\Lambda^{\\circ}= \\left(\n \\begin{array}{cc}\n B^{-t} & O\\\\\n A^{-t}C^tB^{-t} & A^{-t} \\\\\n \\end{array}\n \\right) ({\\mathbb Z}^{2d}).\n$$\n\n\n\\begin{proof}[Proof of Theorem \\ref{Th_union of FD}]\nThe orthogonality of the Gabor system implies that\n \\begin{align}\\label{ortho}\n \\int_K e^{-2\\pi i \\langle Bn, x\\rangle} dx = 0 \\quad \\forall \\ n\\in \\Bbb Z^d\\setminus\\{0\\} .\n \\end{align}\n By Theorem \\ref{thJoPeLgWa}, (\\ref{ortho}) implies that $K$ can be written as\n $\n K = \\bigcup_{j=1}^N D_j,\n $\n where $D_j$ is a fundamental domain for $B^{-t}({\\mathbb Z}^d)$. On the other hand, by the duality Theorem \\ref{thRS}, ${\\mathcal G}({|K|^{-1\/2}}\\chi_K,\\Lambda^{\\circ})$ is a Gabor orthonormal basis, too. Similarly, the exponentials $\\{e^{2\\pi i \\langle A^{-t}n, x\\rangle}: n\\in {\\mathbb Z}^d\\}$ are mutually orthogonal in $L^2(K)$. Hence, by Theorem \\ref{thJoPeLgWa}, we have\n $\n K = \\bigcup_{j=1}^M E_j\n $\n where $E_j$\\rq{}s are fundamental domains for $A({\\mathbb Z}^d)$. Since $\\det(AB)=1$ we conclude that $|D_i|=|E_j|, \\forall \\ i, j$. Since $|K|<\\infty$, the latter forces that $M=N$, hence the proof of the theorem is completed.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof of Theorems \\ref{lower triangle} and \\ref{rational_dim1} - Lower triangular matrices}\\label{thm:lower triangle}\n\nTo prove Theorem \\ref{lower triangle}, first we shall apply some preliminary reductions to the theorem, as follows. Due to Lemma \\ref{lemma 3-prim} and the hypothesis of the theorem on the matrices $A$ and $B$, for the proof it is sufficient\n to assume that\n$\n\\Lambda = \\left(\n \\begin{array}{cc}\n I & O\\\\\n C & B\\\\\n \\end{array}\n \\right) ({\\mathbb Z}^{2d}) ,\n $\n where $B$ is an invertible matrix with integer entries (since originally $A^tB$ has integer entries by the assumption of Theorem \\ref{lower triangle}). Notice by the density condition dens$(\\Lambda) = 1$, we have $|\\det(B)|=1$. Thus $B^{-1}$ is also an integral matrix with determinant 1 and we have\n $\\Bbb Z^{2d} = \\left(\n \\begin{array}{cc}\n I & O\\\\\n O & B^{-1}\\\\\n \\end{array}\n \\right) ({\\mathbb Z}^{2d})$. Thus\n we can rewrite $\\Lambda$ as follows:\n $$\n\\Lambda = \\left(\n \\begin{array}{cc}\n I & O\\\\\n C & B\\\\\n \\end{array}\n \\right) \\left(\n \\begin{array}{cc}\n I & O\\\\\n O & B^{-1}\\\\\n \\end{array}\n \\right) ({\\mathbb Z}^{2d}) = \\left(\n \\begin{array}{cc}\n I & O\\\\\n C & I\\\\\n \\end{array}\n \\right) ({\\mathbb Z}^{2d}).\n $$\nTherefore to prove Theorem \\ref{lower triangle} it suffices to consider $\\Lambda = \\left(\n \\begin{array}{cc}\n I & O\\\\\n C & I\\\\\n \\end{array}\n \\right) ({\\mathbb Z}^{2d})$. In this case if ${\\mathcal G}\\left(|K|^{-1\/2}\\chi_K,\\Lambda\\right)$ is an orthonormal basis, then by Theorem \\ref{Th_union of FD} we must have\n $\nK = \\bigcup_{j=1}^N E_j\n $\nwhere $E_j$\\rq{}s are disjoint fundamental domains of ${\\mathbb Z}^d$. In particular, $K$ is a multi-tile for $\\Bbb R^d$ with respect to ${\\mathbb Z}^d$, i.e.,\n\\begin{equation}\\label{multitile}\n\\sum_{k\\in{\\Bbb Z}^d} \\chi_K(x+k) = N \\ \\mbox{a.e.} \\ x\\in [0,1)^d.\n\\end{equation}\n Our goal is to show that $N=1$. For this, the following proposition will serve a key role.\n \\medskip\n\n\n\\begin{proposition}\\label{prop2} Suppose $K$ is a bounded set which multi-tiles $\\Bbb R^d$ with respect to $\\Bbb Z^d$ at level $N$, i.e. (\\ref{multitile}) holds. Suppose that $N>1$. Then there exists $m \\in{\\mathbb Z}^d$ such that\n\\begin{enumerate}\n\\item $K\\cap (K+m)$ has positive Lebesgue measure.\n\\item $ K\\cap (K+m)$ consists of distinct representative (mod ${\\mathbb Z}^d$).\n\\item $K\\cap (K+m)$ is a packing by ${\\mathbb Z}^d$, i.e.\n$$\n\\sum_{n\\in {\\mathbb Z}^d} \\chi_{K\\cap (K+m)} (x+n)\\le 1. \\ a.e.\n$$\n\\end{enumerate}\n\\end{proposition}\n \\begin{proof}\nPut $Q := [0,1)^d$. The identity (\\ref{multitile}) means that for a.e. $x\\in Q$, there are exactly $N$ integers $n_1,...,n_N$ such that $x+n_i\\in K$ $i=1,...,N$. Using this observation we will decompose $K$ as follows. For each $x\\in Q$, put\n$$\nK(x) := \\{k\\in \\Bbb Z^d: x+ k\\in K\\}.\n$$\nThen $|K(x)|=N$.\n Let $S \\subset{\\mathbb Z}^d$ and $|S| = N$. Define\n$$\nK_S = \\{x\\in Q: K(x) = S\\}.\n$$\nSince $K$ is a multi-tile of level $N$, we have\n$$\nK = \\bigcup_{|S|=N} (K_S+S), \\ \\mbox{and} \\ \\bigcup_{|S|=N} K_S = Q = [0,1)^d,\n$$\nwhere the union runs through all possible subsets $S\\in{\\mathbb Z}^d$ of cardinality $N$. Furthermore,\n $K_S \\cap K_{S'} =\\emptyset, \\forall S\\ne S'\n$, since there are exactly $N$ integers $n$ for which $x+n\\in K$. By the boundedness of $K$, there are only finitely many possible $S\\in{\\mathbb Z}^d$ with $|S|=N$ such that $|K_S|>0$. Thus, we can enumerate those $S$ as $S_1,...,S_r$ so that\n\\begin{equation}\\label{decomp}\nK = \\bigcup_{i=1}^r (K_{S_i}+S_i).\n\\end{equation}\n\n(Notice the decomposition (\\ref{decomp}) also holds for any multi-tile bounded set $K$ with respect to any lattice $\\Gamma$ in place of $\\Bbb Z^d$ and\n any bounded fundamental set of $\\Gamma$ in the place of $Q$.)\n\n\n\nWe order ${\\mathbb Z}^d$ by the natural lexicographical ordering. We then enumerate all possible elements in $S_i$, $1\\le i\\leq r$, by\n$$\n n_1^{S_i}<....1$. By Proposition \\ref{prop2}, there exists ${m}\\in{\\mathbb Z}^d$ such that $K\\cap (K+{m})$ has positive Lebesgue measure and $K\\cap (K+{ m})$ is a packing in $\\Bbb R^d$ by ${\\mathbb Z}^d$. Thus by Lemma \\ref{lemma 4}, the set $\\{e^{2\\pi i \\langle n,x\\rangle}: n\\in{\\mathbb Z}^d\\}$ is complete in $K\\cap (K+{m})$. By Remark \\ref{remark2}, ${\\mathbb Z}^d$ is exponentially complete in $L^2(K\\cap (K+{m}))$. Obviously, ${ m}=0$ cannot satisfy the packing property (3) in Proposition \\ref{prop2} since we have assumed $N>1$. Thus ${m}\\ne 0$. On the other hand, the orthogonality of the Gabor system implies that for any $n\\in \\Bbb Z^d$ we must have\n\\begin{align}\\label{eq}\n \\widehat{\\chi_{K\\cap (K+{m})}} (C{m}+n)= \\int_{\\Bbb R^d} \\chi_K(x)\\chi_{K}(x-{m}) e^{-2\\pi i \\langle C { m}, x\\rangle}e^{-2\\pi i \\langle n, x\\rangle}dx =0\n \\end{align}\n for all $n\\in {\\mathbb Z}^d$. \nThis contradicts the exponential completeness of ${\\mathbb Z}^d$. Therefore, the assumption $N>1$ cannot hold, and $K$ is thus a fundamental domain of ${\\mathbb Z}^d$. This completes the proof.\n\\end{proof}\n\n\n\\begin{remark} In proving Theorem \\ref{lower triangle}, once $\\Lambda$ is reduced to the lattice \n$${\\mathcal A}({\\mathbb Z}^{2d}), \\ \\mbox{where} \\ {\\mathcal A}= \\left(\n \\begin{array}{cc}\n I & O\\\\\n C & I \\\\\n \\end{array}\n \\right),\n$$\none may suspect that we can apply a metaplectic transformation to further reduce the lattice to the separable lattice ${\\mathbb Z}^{d}\\times {\\mathbb Z}^d$ and hence the problem is trivially solved. Unfortunately, this approach does not seem to work. To be precise, one can look up \\cite{Groechenig-book} for the metaplectic transformation in this case. We have that ${\\mathcal G}(|K|^{-1\/2}\\chi_K,{\\mathcal A}({\\mathbb Z}^{2d}))$ is a Gabor orthonormal basis if and only if ${\\mathcal G}( \\tilde{g},{\\mathbb Z}^{d}\\times{\\mathbb Z}^d)$ is a Gabor orthonormal basis, where \n$$\n\\tilde{g} = e^{-\\pi i\\langle x,Cx\\rangle}|K|^{-1\/2}\\chi_K.\n$$\nAlthough the lattice is separable, the window funciton $\\tilde{g}$ is now complex-valued, we cannot conclude that $K$ has no overlap in the time domain as it were the case when $C=O$. \n\n\\smallskip\n\nNonetheless, metaplectic transformation and symplectic matrices seems to provide some strong tools that may lead to a progess in the Fuglede-Gabor problem, readers may refer to \\cite{Folland,deGosson,Groechenig-book} for details about these tools.\n\\end{remark}\n\n\\medskip\n\n\n\n Theorem \\ref{rational_dim1} is now straightforward. We prove it here for the sake of completeness. \n\n\\begin{proof}[Proof of Theorem \\ref{rational_dim1}]\nLet $\\Lambda= M(\\Bbb Z^2)$ where $M=\\left(\n \\begin{array}{cc}\n a & d \\\\\n c & b \\\\\n \\end{array}\n \\right)$ is a rational matrix with $\\det M=1$. Let $q$ be the least common multiple of $a, b, c $ and $d$. Then we can write $\\Lambda$ as $\\Lambda= q^{-1} \\tilde M(\\Bbb Z^2)$ where $\\tilde M$ is an integer matrix. By Lemma \\ref{HanW1}, we can find a unimodular integer matrix $P$ such that $\\tilde M P$ is the lower triangular integer matrix. By the unimodularity of the matrix $P$ we have\n $q^{-1} \\tilde M(\\Bbb Z^2) = q^{-1} \\tilde M P(\\Bbb Z^2)$. Therefore, $\\Lambda = q^{-1} \\tilde M P(\\Bbb Z^2)$ and $q^{-1} \\tilde M P$ is a lower triangular rational matrix. We can therefore write\n $$\n \\Lambda = \\left(\n \\begin{array}{cc}\n \\alpha & 0 \\\\\n \\gamma & \\beta \\\\\n \\end{array}\n \\right)(\\mathbb Z^2)$$\n for some $\\alpha, \\beta, \\gamma\\in \\Bbb Q$.\n Notice that the density of $\\Lambda$ equals $1$, meaning that $\\alpha\\beta = 1$. Thus, all assumptions of Theorem \\ref{lower triangle} are satisfied. Hence, $K$ is a translational tile with tiling set $\\Bbb Z$ and is a spectral set. \n\\end{proof}\n\n\n \\section{ Proof of Theorem \\ref{UT} - Upper triangular matrices}\\label{thm:rational_dim1 and UT}\nWe will discuss a case of upper triangular matrices which can be converted into the lower one. Then we will use Theorem \\ref{lower triangle} to prove the theorem. First we need few lemmas.\n\\medskip\n\n\\begin{lemma}\\label{Gamma-D}\n Let $D$ be a $d\\times d$ rational matrix.\n Then there is an integer lattice $\\Gamma$ such that $D\\gamma\\in \\Bbb Z^d$ for $\\gamma\\in \\Gamma$.\n \\end{lemma}\n \\begin{proof}\n Define\n$$\n\\Gamma := \\{k\\in{\\mathbb Z}^d: Dk\\in{\\mathbb Z}^d\\}.\n$$\nIt is easy to check that $\\Gamma$ is a lattice contained in $\\Bbb Z^d$. Moreover, $\\Gamma$ contains $p\\Bbb Z^d$ where\n $p$ is the least common multiple of the denominators of entries of $D$. Thus, $\\Gamma$ is a full-rank lattice and has the form of\n$\n\\Gamma = M({\\mathbb Z}^d),\n $\nfor some invertible $d\\times d$ matrix $M$ with integer entries.\n\\end{proof}\nObserve that according to the Lemma \\ref{Gamma-D}, for any given rational matrix $D$, there is an integer $M$ such that $DM$ is integer.\n With this observation, we have the following result.\n\n \\medskip\n\n\\begin{lemma}\\label{complete residue classes} Let $D$ be a rational matrix, and let $\\Gamma$ and $M$ be given as in Lemma \\ref{Gamma-D} and $\\det M=n$.\nSuppose that $\\{\\gamma_1,\\cdots, \\gamma_n\\}$ be a complete representative (mod $M({\\mathbb Z}^d)$) in ${\\mathbb Z}^d$. If $D$ is symmetric, then $\\{M^tD\\gamma_1,\\cdots,$ $ M^tD\\gamma_n\\}$ is a complete representative (mod $M^t{\\mathbb Z}^d$) in ${\\mathbb Z}^d$.\n\\end{lemma}\n\\begin{proof} We saw above that by\n the structure of $M$ and $\\Gamma$, $DM$ is an integer matrix, therefore $DM(\\mathbb Z^d)\\subseteq{\\mathbb Z}^d$. We also have $(DM)^t{\\mathbb Z}^d\\subseteq{\\mathbb Z}^d$. This implies that\n$$M^tD({\\mathbb Z}^d) = (D^tM)^t({\\mathbb Z}^d )\\subseteq{\\mathbb Z}^d.$$\nThus, $M^tD\\gamma_i$ are all integer vectors for all $i = 1,\\cdots,n$.\n\n\\medskip\n\nNext, we show that $\\{M^tD\\gamma_1,...,M^tD\\gamma_n\\}$ consists of distinct representative (mod $M^t({\\mathbb Z}^d)$) in $\\Bbb Z^d$. Suppose that this is not the case. Then for some $i\\neq j$ we must have $M^tD\\gamma_i - M^tD\\gamma_j \\in M^t({\\mathbb Z}^d)$. This implies that $D\\gamma_i-D\\gamma_j\\in {\\mathbb Z}^d$, which means that $\\gamma_i$ and $\\gamma_j$ belong to the same representative class (mod $M({\\mathbb Z}^d)$) in ${\\mathbb Z}^d$ which is a contradiction.\n\n\\medskip\n\nFinally, we show that $\\{M^tD\\gamma_1,...M^tD\\gamma_n\\}$ is complete. This follows immediately by counting the number of\ncosets present in ${\\mathbb Z}^d\/M^t({\\mathbb Z}^d)$ which is $|\\det(M^t)| = |\\det(M)|= n$. Hence, $\\{M^tD\\gamma_1,...M^tD\\gamma_n\\}$ is complete.\n\\end{proof}\n\nIn the following constructive lemma we shall present a class of upper triangle lattice which can be converted into a lower triangular lattice.\n\n \\begin{lemma}\\label{a technical lemma}\n For $\\Lambda=\n \\left(\n \\begin{array}{cc}\n I & D \\\\\n O & I \\\\\n \\end{array}\n \\right)(\\Bbb Z^{2d})$ with $D$ rational and symmetric, there are integer matrices $E$ and $X$ such that\n $\\Lambda = \\left(\n \\begin{array}{cc}\n E^{-t} & O \\\\\n X & E\\\\\n \\end{array}\n \\right)({\\mathbb Z}^{2d})$.\n \\end{lemma}\n\n \\begin{proof}\n Let $\\{{\\bf e_i}: \\ i=1, \\cdots, d\\}$ denote the standard basis in $\\Bbb Z^d$. Associated to this $D$ let\n $\\Gamma$ and $M$ be given as in Lemma \\ref{Gamma-D}.\n Then by Lemma \\ref{complete residue classes}, for any $i$ there exists $z_i\\in \\Bbb Z^d$ and $\\gamma^i\\in \\{\\gamma_1, \\cdots, \\gamma_n\\}$ such that ${\\bf e_i}= M^tz_i + M^tD\\gamma^i$. Put $Z:=[z_1 \\cdots z_d]$\n and $X=[\\gamma_1\\cdots \\gamma_d]$. It is clear that $Z$ and $X$ are integer matrices.\n\n\n A direct calculation shows that $(Z+DX)(\\Bbb Z^d)= M^{-t}(\\Bbb Z^d)$.\n\n \\medskip\n\nPut $P:=\\left(\n \\begin{array}{cc}\n Z& -DM \\\\\n X & M\\\\\n \\end{array}\n \\right)$. Then $P$ is an integer matrix with $\\det P=1$ and $P(\\Bbb Z^{2d}) = \\Bbb Z^{2d}$. Recall that\n $DM$ is an integer matrix. So we can write\n $$\\Lambda=\n \\left(\n \\begin{array}{cc}\n I & D \\\\\n O & I \\\\\n \\end{array}\n \\right)(\\Bbb Z^{2d}) = \\left(\n \\begin{array}{cc}\n I & D \\\\\n O & I \\\\\n \\end{array}\n \\right)P(\\Bbb Z^{2d}) = \\left(\n \\begin{array}{cc}\n M^{-t} & O \\\\\n X & M\\\\\n \\end{array}\n \\right)(\\Bbb Z^{2d}). $$ Now take $E=M$, and we are done.\n\\end{proof}\n\\medskip\n\nAt this point we are ready to complete the proof of Theorem \\ref{UT}.\n\n\\begin{proof}[Proof of Theorem \\ref{UT}]\n Let $\\Lambda=\n \\left(\n \\begin{array}{cc}\n A & D \\\\\n O & B \\\\\n \\end{array}\n \\right)(\\Bbb Z^{2d})$ where $B= A^{-t}$ and $\\widetilde{D}: = A^{-1}D$ is rational and symmetric. By Lemma \\ref{lemma 3-prim}, the associated matrix can be reduced to a block matrix of the form $\\left(\n \\begin{array}{cc}\n I & \\widetilde{D} \\\\\n O & I \\\\\n \\end{array}\n \\right)$ where $\\widetilde{D}$ is rational and symmetric. Therefore, it is sufficient to prove the theorem for a lattice of the form\n $\\Lambda=\n \\left(\n \\begin{array}{cc}\n I &D \\\\\n O & I \\\\\n \\end{array}\n \\right)(\\Bbb Z^{2d})$ where $D$ is rational and symmetric. By Lemma \\ref{a technical lemma}, there are integer matrices $E$ and $X$ such that $\\Lambda= \\left(\n \\begin{array}{cc}\n E^{-t} & O \\\\\n X & E\\\\\n \\end{array}\n \\right)({\\mathbb Z}^{2d})$. The rest of the proof is now a conclusion of Theorem \\ref{lower triangle} and we have completed the proof.\n \\end{proof}\n\n \\medskip\n\n In \\cite{HanW4}, the authors proved that for any lattice $\\Lambda$ formed by an upper triangular matrix exists a window $g$ such that ${\\mathcal G}(g,\\Lambda)$ is a Gabor orthonormal basis and such $g$ satisfies $\\widehat{g} = \\chi_K$, where $K$ is the common fundamental domain for the diagonal block matrices $A$ and $B^{-t}$. However, their proof does not provide any constructive technique for producing a compactly supported window. The proof of Theorem \\ref{UT} provides a technique to construct a large class of examples of sets $K$ forming a Gabor orthonormal basis with respect to the lattice generated by upper triangular matrices. We explain this next.\n\n\\medskip\n\n\\begin{proposition}\\label{converse of UT} Assume that the lattice $\\Lambda$ and matrices $A$, $B$ and $D$ are given as in Theorem \\ref{UT} and satisfying\n the hypotheses of the theorem. Then there is a set $K$ such that ${\\mathcal G}(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a Gabor orthonormal basis for $L^2(\\Bbb R^d)$. \n\\end{proposition}\n\n \\begin{proof}\n Let $A$, $B$ and $D$ be given. Notice that \n by Lemma \\ref{lemma 3-prim} and the hypotheses of the proposition we know that for any given set $K$, the system ${\\mathcal G}(|K|^{-1\/2}\\chi_K, \\Lambda)$ is an orthonormal basis if and only if ${\\mathcal G}(|A^{-1}(K)|^{-1\/2}\\chi_{A^{-1}(K)}, \\tilde \\Lambda)$ is an orthonormal basis where $\\tilde\\Lambda= \\left(\\begin{array}{cc} I & A^{-1}D\\\\ O & I\\end{array}\\right)(\\Bbb Z^{2d})$. And, by Lemma \\ref{a technical lemma} we also know that for $\\tilde\\Lambda$ there are integer matrices $X$ and $E$ such\n $$E(\\Bbb Z^d) = \\{n\\in \\Bbb Z^d: ~ A^{-1}Dn\\in \\Bbb Z^d\\}$$ \n and \n $\\widetilde\\Lambda= \\left(\\begin{array}{cc} {E}^{-t} & O\\\\ X & E \\end{array}\\right)(\\Bbb Z^{2d})$.\n By appealing to Lemma \\ref{lemma 3-prim} one more time,\n ${\\mathcal G}(|K|^{-1\/2}\\chi_K, \\Lambda)$ is an orthonormal basis if and only if\n %\n $${\\mathcal G}\\left(|K|^{-1\/2}\\chi_{K}, \\left(\\begin{array}{cc} A{E}^{-t} & O\\\\ A^{-t} X & A^{-t} E \\end{array}\\right)(\\Bbb Z^{2d})\\right)$$\n is an orthonormal basis. Now take $K$ as a fundamental domain of $AE^{-t}(\\Bbb Z^d)$ and we are done. \n \\end{proof}\n\n\\medskip\n\nThe following gives a more explicit example.\n\n\\begin{example}\\label{D not I} Let $A = \\left(\n \\begin{array}{cc}\n \\frac{1}{2} & 0\\\\\n 0 & 2 \\\\\n \\end{array}\n \\right)$, $B = \\left(\n \\begin{array}{cc}\n 2 & 0 \\\\\n 0 & \\frac{1}{2} \\\\\n \\end{array}\n \\right)$. Take $D= \\left(\n \\begin{array}{cc}\n 1 & 0\\\\\n 0 & \\frac{1}{3} \\\\\n \\end{array}\n \\right)$. Then $A^{-1}D$ is rational and symmetric. Let $\\Gamma= \\{n\\in \\Bbb Z^2: \\ A^{-1}Dn\\in \\Bbb Z^2\\}$. $\\Gamma$ is a full lattice and a simple calculation shows that $\\Gamma= E(\\Bbb Z^2)$ where $E= \\left(\n \\begin{array}{cc}\n 1& 0\\\\\n 0 & 6 \\\\\n \\end{array}\n \\right)$.\n Now let $K$ be any fundamental domain of the lattice $AE^{-t} ({\\mathbb Z}^2)= \\left(\n \\begin{array}{cc}\n 1\/2& 0\\\\\n 0 & 1\/3 \\\\\n \\end{array}\n \\right)(\\Bbb Z^2)$. Then by Proposition \\ref{converse of UT} the system\n $\\mathcal G(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a Gabor orthonormal basis for $L^2(\\Bbb R^2)$ for the lower triangular lattice $\\Lambda$ with diagonal matrices $AE^{-t}$ and $A^{-t}E$ and any matrix $C$.\n \\end{example}\n\n\n\\iffalse\n \\begin{example}\\label{D=I} Let $A = \\left(\n \\begin{array}{cc}\n \\frac{1}{2} & 0\\\\\n 0 & 2 \\\\\n \\end{array}\n \\right)$, $B = \\left(\n \\begin{array}{cc}\n 2 & 0 \\\\\n 0 & \\frac{1}{2} \\\\\n \\end{array}\n \\right)$. Take $D=I$. Then $A^{-1}D=A^{-1}$ is rational and symmetric. Let $\\Gamma= \\{n\\in \\Bbb Z^2: \\ A^{-1}n\\in \\Bbb Z^2\\}$. $\\Gamma$ is a full lattice and we can see that $\\Gamma= E(\\Bbb Z^2)$ where $E= \\left(\n \\begin{array}{cc}\n 1& 0\\\\\n 0 & 2 \\\\\n \\end{array}\n \\right)$.\n Now take $K=[0,1\/2]\\times [0,1]$. Then $K$ is a fundamental domain for the lattice $AE^{-t} ({\\mathbb Z}^2)$\nand $\\mathcal G(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a Gabor orthonormal basis for $L^2(\\Bbb R^2)$ by Proposition \\ref{converse of UT}.\n \\end{example}\n\\fi\n\n \\section{Examples}\\label{Examples}\nConsider $\\Lambda=L(\\Bbb Z^{2d})$ where $L= \\left(\\begin{array}{cc}\n A & O \\\\\n C & B \\\\\n \\end{array}\n \\right)$ is rational. In Theorem \\ref{lower triangle} we proved that for a set $K$ if $\\mathcal G(|K|^{-1\/2}\\chi_K, \\Lambda)$ is an orthonormal basis for $L^2(\\Bbb R^d)$ and $A^tB$ is an integer matrix, then $K$ is a fundamental domain of $A(\\Bbb Z^d)$. By Theorem \\ref{Th_union of FD} this means $N=1$. Thus $K$ tiles by $A({\\mathbb Z}^d)$ and is spectral with spectrum $B^{-t}(\\mathbb Z^d)$. In this section we will provide examples showing that $N>1$ can also happen in Theorem \\ref{Th_union of FD} if $A^tB$ is not an integer matrix and yet the system $\\mathcal G(|K|^{-1\/2}\\chi_K, \\Lambda)$ is an orthonormal basis for $L^2(\\Bbb R^d)$. \n\n\n\n\n\n\n \n \n \n %\n %\n \n\n\n\n\n \\medskip\n\n We are now ready to present our example of a set $K$ which is the union of fundamental domains of lattice $A(\\Bbb Z^d)$ and the union of fundamental domains of lattice $B^{-t}(\\Bbb Z^d)$, $\\chi_K$ is a window function for a possible Gabor orthonormal basis, and $A^tB$ is {\\it not an integer matrix}. \n\n\n\\begin{example}\\label{mutli-tile K}\n\nLet $K = [0,2]\\times [0,1]$ and\n$\\Lambda = \\left(\n \\begin{array}{cccc}\n I& O \\\\\n C & B \\\\\n \\end{array}\n \\right)({\\mathbb Z}^{4})$\nwhere\n $B=\n \\left(\n \\begin{array}{cc}\n 1\/2 & 0 \\\\\n 0& 2\n \\end{array}\n \\right)$ and $C= \\left(\n \\begin{array}{cc}\n c_{11}& c_{12} \\\\\n c_{21}& c_{22}\n \\end{array}\n \\right)$. The system $\\mathcal G(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a Gabor orthonormal basis if\n $c_{21}$ is an odd number.\n \\end{example}\n\n \\begin{proof}\n Observe that\n\\begin{align}\\notag\n[0,2]\\times [0,1] &= [0,1]^2 + \\{(0,0), (1,0)\\} \\\\\\notag\n &= [0,2]\\times[0,1\/2]+ \\{(0,0), (0,1\/2)\\}.\n \\end{align}\n\nThis shows that $K$ is union of two fundamental domains of $\\Bbb Z^2$ and union of two fundamental domains of $B^{-t}(\\Bbb Z^2)$, respectively.\n\n Let $({m}, n) \\in {\\mathbb Z}^4$ with ${m},{ n} \\in{\\mathbb Z}^2$ and $({m},{ n})\\neq (0,0)$. Put\n %\n\\begin{align}\\notag\n\\mathcal I: = &\\int \\chi_{K}(x-{m})\\chi_{K}(x) e^{-2\\pi i \\langle C({m}), x\\rangle} e^{-2\\pi i \\langle B({n}), x\\rangle}dx\\\\\\notag\n=&\\int \\chi_{(K+{\\bf m})\\cap K}(x) e^{-2\\pi i \\langle C({ m}), x\\rangle} e^{-2\\pi i \\langle B({n}), x\\rangle}dx .\n\\end{align}\n Mutual orthogonality of the Gabor system will follow if we can show that $\\mathcal I=0$ for any $( m, n)\\neq (0,0)$. Note that\n$$\n(K+m)\\cap K \\simeq \\left\\{\n \\begin{array}{ll}\n K, & \\hbox{${m} = 0$;} \\\\\n {[0,1]}^2, & \\hbox{${m} = (\\pm 1,0)$;} \\\\\n \\emptyset , & \\hbox{otherwise.}\n \\end{array}\n \\right.\n$$\n\n(Here, we write $A\\simeq B$ if $A = B+k$ for some $k\\in{\\mathbb Z}^d$.)\n\n\n Since $K$ is a union of fundamental domains for $B^{-t}(\\Bbb Z^2)$, then if ${m}=0$, we automatically have $\\mathcal I=0. $\n\n\n If ${ m}= (\\pm 1,0)$, for any ${ n} = (n_{1},n_{2})$, $\\mathcal I$ is equal to the following integral up to a unimodular constant:\n\\begin{equation}\\label{eq6.3}\n\\begin{aligned}\n\\int_{[0,1]^2} e^{-2\\pi i \\langle C((\\pm 1, 0)^t), x\\rangle} e^{-2\\pi i \\langle B({n}), x\\rangle}dx=&\\int_{[0,1]}\\int_{[0,1]} e^{-2\\pi i (\\pm c_{11}x_1\\pm c_{21}x_2)} e^{-2\\pi i (\\frac{1}{2}n_{1}x_1+2n_{2}x_2)}dx_1dx_2 \\\\\n=& \\widehat{\\chi_{[0,1]}}(\\pm c_{11}+\\frac{1}{2} n_1) \\widehat{\\chi_{[0,1]}}(\\pm c_{21}+2 n_2).\n\\end{aligned}\n\\end{equation}\n It is obvious that the last line equals to zero for all $(n_1, n_2)$ only if $c_{21}$ is an odd number. For other cases of $m$ it is trivial that $\\mathcal I=0$. Thus the orthogonality is obtained and\n the completeness\n is a direct conclusion of Proposition \\ref{complt}.\n\\end{proof}\n\nWe notice that the previous example exploited the fact that $B({\\mathbb Z}^2)$ is exponentially incomplete for $L^2[0,1]^2$ in (\\ref{eq6.3}). The following example illustrates a case where, contrary to the previous example, the finite union of fundamental domains cannot form a Gabor orthonormal basis for any choice of matrix $C$.\n\n\\medskip\n\n \\begin{example}\\label{example2}\n Let $K=[0,2]^2$ and let $B$ be the matrix as in Example \\ref{mutli-tile K}. Then $K$ is union of $4$ fundamental domains of the lattice $\\Bbb Z^2$ and union of $4$ fundamental domains of $B^{-t}(\\Bbb Z^2)$. However, there is no matrix $C$ for which $\\chi_K$ is a window function yielding an orthonormal basis for the lower triangular lattice $\\Lambda=\\left(\\begin{array}{cc} I & O\\\\ C & B\\end{array}\\right)(\\Bbb Z^{4})$. \\end{example}\n\n \n\n\\begin{proof}\nThe fact that $K$ is union of fundamental domains of $\\Bbb Z^d$ and $B^{-t}(\\Bbb Z^d)$ is straight forward. In short,\n$$\n\\begin{aligned}\nK =& [0,1]^2 + \\{(0,0), (1,0), (0,1), (1,1)\\}\\\\\n =& [0,2]\\times [0,1\/2] + \\{(0,0), (0,1\/2), (0,1), (0,3\/2)\\}.\n\\end{aligned}\n$$\nSuppose that there exists a matrix $C$ such that $\\chi_K$ is a window function for the lower triangular lattice $\\Lambda=\\left(\\begin{array}{cc} I & O\\\\ C & B\\end{array}\\right)(\\Bbb Z^{4})$. Thus, for any $(0,0)\\neq (m, n)\\in \\Bbb Z^{4}$ we must have\n$$\n 0 = {\\mathcal I}: =\\int \\chi_{(K+{m})\\cap K}(x) e^{-2\\pi i \\langle C({ m}), x\\rangle} e^{-2\\pi i \\langle B({n}), x\\rangle}dx.\n$$\nFrom the other side, the only non-empty intersection sets $(K+ m)\\cap K$ are\n$$\n(K+m)\\cap K \\simeq \\left\\{\n \\begin{array}{ll}\n K, & \\hbox{if }{m}=0 \\\\\n {[0,1]\\times[0,2]}, & \\hbox{if } {m} = (\\pm1,0) \\\\\n {[0,2]\\times[0,1]}, & \\hbox{if } m = (0,\\pm1) \\\\\n {[0,1]}^2, & \\hbox{if } m = (\\pm1,\\pm1).\n \\end{array}\n\\right.\n$$\nFrom the last three intersections, we obtain that if ${\\mathcal I}=0$, then the following three equations must hold for all integer vectors $(n_1,n_2)$, respectively:\n\n\n \\begin{align}\\notag\n \\widehat{\\chi_{[0,1]}}(\\pm c_{11}+\\frac{1}{2} n_1) \\widehat{\\chi_{[0,2]}}(\\pm c_{21}+2 n_2) &=0 \\\\\\notag\n \\widehat{\\chi_{[0,2]}}(\\pm c_{12}+\\frac{1}{2}n_1) \\widehat{\\chi_{[0,1]}}(\\pm c_{22}+2 n_2)\n &=0\\\\\\notag\n \\widehat{\\chi_{[0,1]}}(\\pm c_{11}\\pm c_{12}+ \\frac{1}{2} n_1) \\widehat{\\chi_{[0,1]}}(\\pm c_{21}\\pm c_{22}+2 n_2) &=0\n \\end{align}\n\n\\medskip\n\nWe claim that if the first equation holds, then $\\widehat{\\chi_{[0,2]}}(\\pm c_{21}+2 n_2) =0$ for all integers $n_2$ and $c_{21}\\in 2{\\mathbb Z}+\\{\\frac{1}{2} ,1,\\frac{3}{2} \\}$.\n\n\\medskip\n\nTo justify the claim, suppose that there exists an integer $n_2$ such that $\\widehat{\\chi_{[0,2]}}(\\pm c_{21}+2 n_2) \\ne0$. Then $\\widehat{\\chi_{[0,1]}}(\\pm c_{11}+\\frac{1}{2} n_1)=0$ for all integers $n_1$. However, this would imply the existence of an exponentials $e^{-2\\pi i c_{11} x}$ such that it is orthogonal to all $\\{e^{2\\pi i (\\frac{1}{2} n) x}: n\\in{\\mathbb Z}\\}$ in $L^2[0,1]$. This is impossible since the exponentials set $\\{e^{2\\pi i (\\frac{1}{2} n) x}: n\\in{\\mathbb Z}\\}$ is complete in $L^2[0,1]$. Hence, we have only $\\widehat{\\chi_{[0,2]}}(\\pm c_{21}+2 n_2) =0$ for all integers $n_2$. Finally, since the zero set for $\\widehat{\\chi_{[0,2]}}$ is $\\frac{1}{2} {\\mathbb Z}$ except zero, then we must have $c_{21}\\in 2{\\mathbb Z}+\\{\\frac{1}{2} ,1, \\frac{3}{2} \\}$, as desired.\n\n\\medskip\n\nSimilarly, the second equation implies that $\\widehat{\\chi_{[0,1]}}(\\pm c_{22}+2 n_2) =0$ for all integers $n_2$ and thus $c_{22}\\in 2{\\mathbb Z}+1$ must be an odd integer.\n\n\\medskip\n\nThe third equation implies that\n\\begin{equation}\\label{eq5.4}\n\\widehat{\\chi_{[0,1]}}(\\pm c_{21}\\pm c_{22}+2 n_2) =0\n\\end{equation}\nfor any integers $n_2$. Now, we write $c_{21} = 2k_1+ \\frac{1}{2} j$, for some $j = 1,2,3$ and $c_{22} = 2k_2+1$. Then $\\pm c_{21}\\pm c_{22}\\in 2{\\mathbb Z}+ \\{\\frac{3}{2},2,\\frac{5}{2}\\}$. In the case of fraction, (\\ref{eq5.4}) cannot be zero. If $\\pm c_{21}\\pm c_{22} = 2m+2$, we take $n_2 = -m-1$. Then (\\ref{eq5.4}) will imply $\\widehat{\\chi_{[0,1]}}(0) = 1$, which is impossible. Thus, the third equation can never be zero. This implies that such $C$ does not exist.\n\\end{proof}\n\n\\medskip\nThe following example proves that the boundedness property for the set $K$ is necessary in Proposition \\ref{prop2}. \n\n \\begin{example}\\label{example3} Let $I_0 = [0,1)$ and $I_n = \\left[1-\\frac{1}{2^{n-1}},1-\\frac{1}{2^{n}}\\right)$ for $n\\ge1$. Define \n $$\n K = \\bigcup_{k\\in{\\mathbb Z}} (k+ I_{|k|}) \n $$\n The set $K$ is unbounded and we have the following. \n \n\\begin{enumerate}\\item $K$ multi-tiles ${\\mathbb R}$ by ${\\mathbb Z}$ at level 3. However, for any $m\\ne 0$, $K\\cap( K+m)$ is not a packing of ${\\mathbb R}$. Therefore, Proposition \\ref{prop2} does not hold if $K$ is unbounded.\n \\item Nonetheless, $K$ cannot form a Gabor orthonormal basis using lattices of the form $\n \\left(\n \\begin{array}{cc}\n 1 & 0 \\\\\n c& 1\n \\end{array}\n \\right) ({\\mathbb Z}^2)$. \\end{enumerate}\n\\end{example}\n\n\n\\begin{proof}\nThe fact that $K$ mult-tiles ${\\mathbb R}$ by ${\\mathbb Z}$ at level 3 follows from a direct observation, so we omit the details here. We note that for any $m\\ne 0$,\n$$\nK\\cap (K+m)\\supset I_{|m|} \\cup (I_{|m|}+m).\n$$\nHence, for all $x\\in I_{|m|}$, $\\sum_{n\\in{\\mathbb Z}} \\chi_{K\\cap(K+m)}(x+n)\\ge 2$. Therefore, it is never a packing for any $m\\ne 0$. To show the last statement, notice \n\n$$\nK\\cap (K+1) = \\left[0,\\frac12\\right)\\cup\\left[1,\\frac32\\right), \\ K\\cap (K+2) = \\left[\\frac12,\\frac34\\right)\\cup\\left[1,\\frac32\\right)\\cup\\left[\\frac52,\\frac{11}{4}\\right).\n$$\nSuppose that $K$ forms a Gabor orthonormal basis using some lattice of the form $\n \\left(\n \\begin{array}{cc}\n 1 & 0 \\\\\n c& 1\n \\end{array}\n \\right) ({\\mathbb Z}^2)$. Then $\\widehat{\\chi_{K\\cap(K+1)}}(c+m) = 0$ and $\\widehat{\\chi_{K\\cap(K+2)}}(c+m) = 0$ for all $m\\in{\\mathbb Z}$. In particular, \n $$\n \\widehat{\\chi_{K\\cap(K+1)}} (c+m) = (1+e^{2\\pi i (c+m)})\\widehat{\\chi_{[0,1\/2]}}(c+m) =0, \\forall m\\in{\\mathbb Z}.\n $$\nWe see that the only possibility of the above is that $c\\in\\frac12+{\\mathbb Z}$. We now consider \n$$\n\\widehat{\\chi_{K\\cap(K+2)}} (\\xi) = e^{2\\pi i \\frac12\\xi}\\left(1+e^{2\\pi i \\frac12\\xi}+e^{2\\pi i \\frac34\\xi}+e^{2\\pi i \\frac2\\xi}\\right)\\widehat{\\chi_{[0,1\/4)}}(\\xi).\n$$\nIf $c \\in\\frac12+{\\mathbb Z}$ and $\\widehat{\\chi_{K\\cap(K+2)}} (c) =0$, we must have \n$$\n0=1+e^{2\\pi i \\frac12c}+e^{2\\pi i \\frac34c}+e^{2\\pi i 2c} = 2+e^{2\\pi i \\frac12c}+e^{2\\pi i \\frac34c}.\n$$ \nThis forces $e^{2\\pi i \\frac12c} = e^{2\\pi i \\frac34c}=-1$. Thus, \n$$c\\in 2(1\/2+{\\mathbb Z})\\cap \\frac43(1\/2+{\\mathbb Z}) = (1+2{\\mathbb Z})\\cap \\left(\\frac23+\\frac43{\\mathbb Z}\\right)\\subset 1+2{\\mathbb Z}$$\nThis is a contradiction since $c\\in\\frac12+{\\mathbb Z}$ is never an integer. This completes the proof.\n \\end{proof}\n\n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n\n\n\n\n\n\n\n\\section{Discussions and Open problems}\\label{Open problems}\n\nThis paper investigates the Fuglede-Gabor Problem \\ref{our conjecture1} over the lattices. We believe that this problem should be true for all lattices. We solved the problem completely in dimension one when the lattice is rational and in higher dimensions when the lattice is integer. In what follows we shall explain how to resolve Problem \\ref{our conjecture1} in full generality for any lattices $\\Lambda = \\left(\n \\begin{array}{cc}\n A & D \\\\\n C & B \\\\\n \\end{array}\n \\right)({\\mathbb Z}^{2d})$. In fact,\n it is sufficient to solve the following two cases.\n\\begin{enumerate}\n\\item[(1)]\\label{item 1} {\\bf Rational case:} After converting a rational lattice into a lower triangular rational matrix by Corollary \\ref{M=N} and reducing the matrix where $A=I$ by Lemma \\ref{lemma 3-prim}, $\\Lambda$ is a lattice of the form of $ \\left(\n \\begin{array}{cc}\n I & O \\\\\n C & B \\\\\n \\end{array}\n \\right)({\\mathbb Z}^{2d})$, where $B$ and $C$ are $d\\times d$ rational matrices but $B$ is not necessarily an integral matrix. \n\n\n \\medskip\n\n \n\n\n\n\\item[(2)] {\\bf Irrational case:} After a reduction process by Lemma \\ref{lemma 3-prim}, $\\Lambda$ is a lattice of the form of \n$$\\left(\n \\begin{array}{cc}\n I & D\\\\\n C & B \\\\\n \\end{array}\n \\right) ({\\mathbb Z}^{2d})$$ which contains irrational entries.\n\\end{enumerate}\n\n\n\\medskip\nIn what follows, we shall discuss these two cases in more details.\n\n\n\n\\subsection{Rational Case}\nLet $\\Lambda= \\left(\n\\begin{array}{cc}\nA & O\\\\\nC & B\n\\end{array}\\right)(\\Bbb Z^{2d})$ be a lower triangular rational matrix.\n Example \\ref{mutli-tile K} tells us that when $A^tB$ is a non-integer matrix, $K$ is not a fundamental domain for the lattices $A(\\Bbb Z^d)$ and $B^{-t}(\\Bbb Z^d)$ but the union of their fundamental domains. However, in Example \\ref{example2} we see that there exists no $C$ such that $K$, as the union of fundamental domains, is a set whose characteristic function generates a Gabor orthonormal basis associated to the given matrices $A$ and $B$. We predict that this failure is due to the number of decompositions of $K$ into fundamental domains of $B^{-t}(\\Bbb Z^d)$.\n In this concern and in relation to the examples illustrated in Section \\ref{Examples},\n we conjecture the following problem for the lattices\n$\\Lambda= \\left(\n \\begin{array}{cc}\n I & D\\\\\n C & B \\\\\n \\end{array}\n \\right) ({\\mathbb Z}^{2d})$.\n \\medskip\n\n\n\n \\medskip\n\n \\begin{conjecture}\\label{our conjecture} Let $K\\subset \\Bbb R^d$ and $B$ be a rational matrix with $\\det(B)=1$. Let\n $s$ be the least common multiple of the denominators of the matrix entries $(b_{ij})$.\n There is a matrix $C$ such that $\\mathcal G(|K|^{-1\/2}\\chi_K, \\Lambda)$ is a Gabor orthonormal basis for $L^2(\\Bbb R^d)$ if and only if $K$ tiles and\n $$K= \\bigcup_{i=1}^sE_i= \\bigcup_{i=1}^sF_i $$\n where $E_i$ and $F_i$ are\n fundamental domains of $\\Bbb Z^d$ and $B^{-t}(\\Bbb Z^d)$, respectively.\n \\end{conjecture}\n\nIt is obvious that the conjecture automatically holds when $B$ is an integer matrix which means $s=1$. Indeed, this is the result of Theorem \\ref{lower triangle}.\n\n\n\n\n\n\n \\medskip\n\n Observe that if $K$ is as in Conjecture \\ref{our conjecture}, then for any non-zero $n\\in \\Bbb Z^d$ we have $\\int_K e^{2\\pi i \\langle Bn, x\\rangle} dx=0$. And, there are only finitely many $m\\in \\Bbb Z^d$ such that $|K\\cap (K+m)|\\neq 0$, as $K$ is a finite union of fundamental domains of $\\Bbb Z^d$. To prove the orthogonality, one must first show the existence of a matrix $C$ such that for all $n\\in \\Bbb Z^d$\n\\begin{equation}\\label{eq6}\n\\int_{K\\cap K+m} e^{-2\\pi i \\langle Cm, x\\rangle} e^{-2\\pi i \\langle Bn, x\\rangle} dx=0,\n\\end{equation}\nwhich is equivalent to the exponential incompleteness of the lattices $B(\\Bbb Z^d)$ for $L^2(K\\cap K+m)$. It appears that we need to study the exponential completeness of the lattices over different domains. In fact, the following problem has not yet had a definite answer.\n \n \\medskip\n\n\\begin{problem} Given a set $K$ with positive and finite measure, classify all invertible $d\\times d$ matrices $B$ for which \n the exponents $\\{ e^{-2\\pi i \\langle Bn, x\\rangle} : n\\in \\Bbb Z^d\\}$ are exponentially complete in $L^2(K)$. \n\\end{problem}\n\n\\medskip\n\nProposition \\ref{prop_G} provides a sufficient condition for matrices $B$ when $K=[0,1]^d$. Unfortunately, the converse of the proposition does not hold in dimensions $d=2$ and higher, as the counterexample following Proposition \\ref{prop_G} shows. However, in Proposition \\ref{dimension one} we obtain a full characterization of exponential completeness for $[0,1]$ in dimension $d=1$ for integer lattices in $\\Bbb Z$. \n\n\n\\medskip \n\nAs mentioned earlier, the exponential completeness of a set does not imply the completeness of the set in general. For a recent developments in study of \n the completeness, frame and Riesz bases properties of exponentials we refer the reader to the paper by De Carli and her co-authors \\cite{De Carli}. \n\n\n\n \n\n\n\n\n\\subsection{Irrational Cases} The case for irrational lattices is more challenging and complicated. It appears that the lower and upper triangular case is asymmetric. We have seen that in Theorem \\ref{lower triangle}, the lower triangular block matrix $C$ is not involved in the statement. Thus, irrational entries are allowed for lower triangular matrices. On the other hand, Han and Wang \\cite{HanW4} implicitly conjectured the following problem in their paper \\cite{HanW1}:\n\n\\medskip\n\n {\\bf Han and Wang\\rq{}s Conjecture:} {\\it Let $\\Lambda = \\left(\n \\begin{array}{cc}\n 1 & \\alpha \\\\\n 0 & 1 \\\\\n \\end{array}\n \\right)({\\mathbb Z}^{2})$ where $\\alpha$ is irrational. Then there doesn't exist compactly supported window $g$ such that ${\\mathcal G}(g,\\Lambda)$ forms a Gabor orthonormal basis for $L^2({\\mathbb R})$.}\n\n \\medskip\n\n Observe that if $\\Lambda= \\left(\n \\begin{array}{cc}\n 1& \\alpha \\\\\n 0 & 1 \\\\\n \\end{array}\n \\right)$, with $\\alpha$ irrational, our method applied to construct Example \\ref{D not I} would not work for the existence of $K$ since in that case $\\Gamma=\\{0\\}$. This observation predicts that Han and Wang Conjecture \\cite{HanW1} might be true, although we do not have a proof for it now. However, a simple calculation shows that the function $\\chi_{[0,1]}$ can not be a window function for this lattice. \n\n \\medskip\n\n\n \n\n\\subsection{Full generality of Fuglede-Gabor Problem \\ref{our conjecture1}} \nIt is known that non-symmetric convex bodies as well as convex sets with a point of non-vanishing Gaussian curvature have no basis of exponentials and yet they do not tile \\cite{K99,IKT01}. Recently, similar results were also proved for Gabor bases. Indeed, the authors in \\cite{IM17} proved that in dimensions $d\\neq1$ (mod 4), convex sets with a point of non-vanishing Gaussian curvature cannot generate any Gabor orthonormal basis with respect to any countable time-frequency set $\\Lambda$. Also, the authors in \\cite{CL} proved that non-symmetric convex polytopes do not produce any Gabor orthonormal basis with respect to any countable time-frequency set. However, the result for non-symmetric convex domains is not known yet. The existent results predict that Fuglede-Gabor problem will still be true to some extent.\n\n\n \n\n\n\\medskip\n\n\n\n One may notice that there are also \n examples of spectral sets which do not tile by translations (see e.g.\n \\cite{T04}, \\cite{Matolcsi} and \\cite{KM06}). Therefore, by a result of Dutkay and the first listed author \\cite{Dutkay-Lai}, it is known that the indicator function of these sets can not serve as Gabor orthonormal window with resepct to any separable time-frequency set. However, they may still produce a Gabor orthonormal basis using some non-separable and countable time-frequency sets. \n This requires some input of new ideas.\n\n\n\\medskip\n\nFinally, it is known that octagon does not tile ${\\mathbb R}^2$ by translations, but it is a multi-tile by ${\\mathbb Z}^2$. The following example tells us that it does not form Gabor orthonormal basis using any lattices, confirming that the Fuglede-Gabor problem holds up to some extent.\n\n\\medskip\n\n\n\n\n\n \\begin{example}\\label{Octagon} Let ${\\mathcal O}_8\\subset \\Bbb R^2$ be the octagon symmetrically centred at the origin with integer vertices $\\{(\\pm 1, \\pm 2), (\\pm 2, \\pm 1)\\}$. ${\\mathcal O}_8$ multi-tiles $\\Bbb R^2$ with $\\Bbb Z^2$ and it is the union of $s=14$ fundamental domains of $\\Bbb Z^2$. Yet there doesn't exist any $\\Lambda= \\left(\n \\begin{array}{cc}\n I & O \\\\\n C & B \\\\\n \\end{array}\\right) (\\Bbb Z^4)$ with rational matrix $B$ for which $\\mathcal G(|{\\mathcal O}_8|^{-1\/2}\\chi_{{\\mathcal O}_8}, \\Lambda)$ forms an orthonormal basis for $L^2(\\Bbb R^2)$ .\n \\end{example}\n \n \n The proof of this example is involved and so it will be provided in the next section. \n \n \n\n\n \n \n \n \n\\iffalse\n \\begin{lemma}\nLet $K$ be a measurable set in $\\Bbb R^d$ with positive and finite measure. Let $\\Lambda$ be the lower triangular lattice with diagonal block matrices $A=I$ and $B$ and $C\\neq 0$.\n\n Assume that there is a non-zero $m\\in \\Bbb Z^2$ such that $K\\cap K+m$ is a packing by $B^{-t}(\\Bbb Z^d)$. Then $\\mathcal G(\\chi_K, \\Lambda)$ is not an orthogonal set.\n\\end{lemma}\n\n\\begin{proof}\nThe proof is a direct consequence of Lemma \\ref{lemma 4}.\n\\end{proof}\n\\fi\n\n\\section{Gabor orthonormal bases on Octagon and Proof of Example \\ref{Octagon}}\\label{Appendix Octagon}\n\nIn this appendix, we will prove that the octagon symmetrically centered at the origin with integer vertices $\\{(\\pm 1, \\pm 2), (\\pm 2, \\pm 1)\\}$ cannot admit any Gabor orthonormal basis with $\\Lambda= \\left(\n \\begin{array}{cc}\n I & O \\\\\n C & B \\\\\n \\end{array}\\right) (\\Bbb Z^4)$, where $B$ is a rational matrix. For this we need the following lemma.\n \n \n\\begin{lemma}\\label{lemma_shear}\nLet $M = \\left(\n \\begin{array}{cc}\n \\alpha & 0 \\\\\n 0 & 1\/\\alpha \\\\\n \\end{array}\\right)$ and $M_{\\beta} = \\left(\n \\begin{array}{cc}\n \\alpha & 0 \\\\\n \\beta & 1\/\\alpha \\\\\n \\end{array}\\right)$, $\\alpha\\neq 0$. Then $\\Omega$ is a multi-tile by $M_{\\beta}({\\mathbb Z}^2)$ if and only if $\\Omega$ is a multi-tile by $M({\\mathbb Z}^2)$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\alpha>0$ and put $Q_{\\alpha} = [0,\\alpha)\\times[0,1\/\\alpha)$. Then $Q_{\\alpha}$ is a fundamental domain for the lattice $M(\\Bbb Z^2)$. We claim that $Q_{\\alpha}$ is a fundamental domain for $M_{\\beta}(\\Bbb Z^2)$ for any $\\beta\\in\\Bbb R$. Indeed, this follows from a direct calculation: For almost every $(x,y)$, we have \n$$\n\\begin{aligned}\n\\sum_{m,n\\in{\\mathbb Z}}\\chi_{Q_{\\alpha}}(x-\\alpha m,y-\\beta m-n\/\\alpha) =& \\sum_{m\\in{\\mathbb Z}}\\chi_{[0,\\alpha)}(x-\\alpha m)\\left(\\sum_{n\\in{\\mathbb Z}}\\chi_{[0,1\/\\alpha)}(y-\\beta m-n\/\\alpha)\\right) \\\\\n=&\\sum_{m\\in{\\mathbb Z}}\\chi_{[0,\\alpha)}(x-\\alpha m)=1.\\\\\n\\end{aligned}\n$$\n \n\nLet $\\Omega$ be any set which is a tile by $M({\\mathbb Z}^2)$. Let $\\{E_{(u,v)} : (u,v)\\in \\Bbb Z^2\\}$ be a partition of $Q_{\\alpha}$ such that \n %\n$$\n\\Omega = \\bigcup_{(u,v)\\in{\\mathbb Z}^2} (E_{(u,v)}+(\\alpha u,v\/\\alpha)) .$$ \n \n \n Then set $\\Omega$ is a tile by $M_\\beta(\\Bbb Z^2)$. Indeed, \n$$\n\\begin{aligned}\n\\sum_{m,n\\in{\\mathbb Z}}\\chi_{\\Omega}(x-\\alpha m,y-\\beta m-n\/\\alpha)=&\\sum_{m,n\\in{\\mathbb Z}} \\sum_{(u,v)\\in{\\mathbb Z}^2}\\chi_{E_{(u,v)+(\\alpha u,v\/\\alpha)}} (x-\\alpha m,y-\\beta m-n\/\\alpha)\\\\\n=&\\sum_{(u,v)\\in{\\mathbb Z}^2}\\sum_{m,n\\in{\\mathbb Z}} \\chi_{E_{(u,v)}} (x-\\alpha (m+u),y-\\beta m-(n+v)\/\\alpha)\\\\\n=&\\sum_{(u,v)\\in{\\mathbb Z}^2}\\sum_{m,n\\in{\\mathbb Z}} \\chi_{E_{(u,v)}} (x-\\alpha m,y-\\beta m-n\/\\alpha)\\\\\n=&\\sum_{m,n\\in{\\mathbb Z}}\\sum_{(u,v)\\in{\\mathbb Z}^2} \\chi_{E_{(u,v)}} (x-\\alpha m,y-\\beta m-n\/\\alpha)\\\\\n=&\\sum_{m,n\\in{\\mathbb Z}} \\chi_{Q_{\\alpha}} (x-\\alpha m,y-\\beta m-n\/\\alpha)=1.\\\\\n\\end{aligned}\n$$\nNote that above we used an application of Fubini\\rq{}s theorem. \nNow let \n $\\Omega$ be a multi-tile with respect to the lattice $M(\\Bbb Z^2)$. Then $\\Omega = \\bigcup_{i=1}^N\\Omega_i$, where $\\Omega_i$ are tiles by $M({\\mathbb Z}^2)$. Thus, by the previous results above, each $\\Omega_i$ is a tile by $M_\\beta(\\Bbb Z^2)$, hence we have the following. \n$$ \n\\sum_{m,n\\in{\\mathbb Z}}\\chi_{\\Omega}(x-\\alpha m,y-\\beta m-n\/\\alpha)=\\sum_{i=1}^N\\sum_{m,n\\in{\\mathbb Z}}\\chi_{\\Omega_i}(x-\\alpha m,y-\\beta m-n\/\\alpha)=N.\n$$\n\n The converse can be obtained by a similar calculation. \n \n\\end{proof}\n\n \\begin{proof}[Proof of Example \\ref{Octagon}]\n Before we prove our claim, note that if the family\n ${\\mathcal G}(|{\\mathcal O}_8|^{-1\/2}\\chi_{{\\mathcal O}_8}, \\Lambda)$ is a Gabor basis and $B$ is an integer matrix, then according to Theorem \\ref{lower triangle} the set ${\\mathcal O}_8$ must tile which is impossible. Thus, we assume that $B$ has some non-integer rational entries. Let ${\\bf m}_0=(3,2)$. Then ${\\mathcal P}:=K\\cap (K+{\\bf m}_0)$, $K=\\mathcal O_8$, is the parallelogram with vertices\n $\\{(1,2), (2,1), (1,1), (2,0)\\}$. Hence,\n $$\n {\\mathcal P} = Q[0,1]^2+(1,1)^t, \\ \\mbox{where} \\ Q = \\left(\n \\begin{array}{cc}\n 1 & 0 \\\\\n -1 & 1 \\\\\n \\end{array}\\right).\n $$\n The Fourier transform of $\\chi_{\\mathcal P}$ at $\\xi = (\\xi_1,\\xi_2)$ is given by \n $$\n \\widehat{\\chi_{{\\mathcal P}}}(\\xi) = c ~ \\widehat{\\chi_{[0,1]^2}}(Q^{T}\\xi)\n $$\n where $c:=c(\\xi)$ is some unimodular constant.\n \n \n \\medskip\n \n \n By the mutual orthogonality of the element $( m_0, n)$ with $(0, 0)$ for all $ n\\in{\\mathbb Z}^2$, we obtain \n \n \\begin{align}\\label{integral-octagon}\n 0= I:= & \\int_{K\\cap (K+{m}_0)} e^{-2\\pi i \\langle C m_0, x\\rangle} e^{-2\\pi i \\langle Bn,x\\rangle} dx \n = c\\cdot\\widehat{\\chi_{[0,1]^2}}(Q^TCm_0+Q^TB n) . \n \\end{align}\n \n \n If $Cm_0=0$, by putting ${n}=0$, then one must have $c=0$, which is a contradiction . Otherwise, (\\ref{integral-octagon}) shows that $Q^TB({\\mathbb Z}^2)$ is exponentially incomplete for $L^2[0,1]^2$. By Theorem \\ref{Theorem_Appendix1} in the appendix, $Q^TB$ is equivalent to one of the following two forms.\n %\n %\n \\begin{align}\\label{matrices forms}\n \\left(\n \\begin{array}{cc}\n \\frac{1}{q'} & 0 \\\\\n r' & q' \\\\\n \\end{array}\\right) \\ \\ \\mbox{or} \\ \\ \\left(\n \\begin{array}{cc}\n p' & 0 \\\\\n \\frac{r''}{s''} & \\frac{1}{p'} \\\\\n \\end{array}\\right)\n \\end{align}\n for some integers $p',q', r'>1$, gcd of $r'$ and $q'$ is strictly greater than 1 and $(r'',s'')$ is relatively prime. \n \n \\medskip\n \n \n We now prove that $B$ also is equivalent to the same desired form in (\\ref{matrices forms}). We are going to establish the first case, the second case is similar. Recall that $Q = \\left(\n \\begin{array}{cc}\n 1& 0 \\\\\n -1 & 1 \\\\\n \\end{array}\\right)$. If $Q^TB\\tilde U = \\left(\n \\begin{array}{cc}\n \\frac{1}{q} & 0 \\\\\n r & q \\\\\n \\end{array}\\right)$ for some unimodular integer matrix $\\tilde U$, then $B\\tilde U= \\left(\n \\begin{array}{cc}\n \\frac{1+rq}{q} & q \\\\\n r & q \\\\\n \\end{array}\\right)$. Note that $1+rq$ and $q^2$ is relatively prime. Thus there exist co-prime integers $u,v$ such that $(1+rq)u+q^2v = 1$. Define $ U = \\left(\n \\begin{array}{cc}\n u & -q^2 \\\\\n v & 1+rq \\\\\n \\end{array}\\right)$. Then $\\det(U)=1$ and the lattice $B\\tilde U(\\mathbb Z^2) = B\\tilde UU({\\mathbb Z}^2)$. Note that \n $$\n B\\tilde UU = \\left(\n \\begin{array}{cc}\n \\frac{1+rq}{q} & q \\\\\n r & q \\\\\n \\end{array}\\right)\\left(\n \\begin{array}{cc}\n u & -q^2 \\\\\n v & 1+rq \\\\\n \\end{array}\\right) = \\left(\n \\begin{array}{cc}\n \\frac{1}{q} & 0 \\\\\n ru+vq & q\\\\\n \\end{array}\\right).\n $$ \n This shows that $B$ is equivalent to the desired form in (\\ref{matrices forms}). \n For the rest, we prove that the neither of these forms can form an Gabor orthonormal basis. By the Lemma \\ref{lemma_shear} and Theorem \\ref{Th_union of FD}, we just need to prove that the octagon ${\\mathcal O}_8$ is not a multi-tile for the matrices $\\left(\\begin{array}{cc}\n p& 0 \\\\\n 0& 1\/p \\\\\n \\end{array}\\right)$ and $\\left(\\begin{array}{cc}\n 1\/p& 0 \\\\\n 0 & p \\\\\n \\end{array}\\right)$ where $p>1$ is an integer. By the symmetry of the octagon, we just need to prove that ${\\mathcal O}_8$ is not a multi-tile for the first one. Let $B_p:=\\left(\\begin{array}{cc}\n p& 0 \\\\\n 0& 1\/p \\\\\n \\end{array}\\right)$. To prove this, we first note that an elementary calculation shows that the area of ${\\mathcal O}_8$ is 14. If it multi-tiles by $B_p({\\mathbb Z}^2)$, then for almost every $x\\in{\\mathbb R}^2$, the cardinality of the set $(x+B_p({\\mathbb Z}^2))\\cap {\\mathcal O}_8$ is $l=14$.\n To obtain a contradiction \n consider the rectangle $R_p:=[0,1)\\times[0,1\/p)$ for $p>1$. It is a simple observation that $R_p$ can be covered $12p$ by translations of $\\mathcal O_8$ by the matrix $B_p$. This is a contradiction to the level of multi-tiling $l=14$, thus we have completed the proof.\n \n \n \\iffalse \n \n \n \\medskip\n \n If $p=2$, we consider the upper corner of triangle formed by the vertices $(1,2), (1,3\/2)$ and $(3\/2,3\/2)$ and denote it by $\\Delta$, then for any $x\\in \\Delta$, \n $$ \n x+\\left\\{(k,j\/2): k\\in\\{0,-2\\},j \\in\\{0,-1,-2,-3,-4,-5,-6,-7\\}\\right\\}\\in (x+B_2({\\mathbb Z^2}))\\cap {\\mathcal O}_8.\n$$\nThere are 16 elements and $\\Delta$ has positive measure. This shows that ${\\mathcal O}_8$ cannot be a multi-tile by $B_2({\\mathbb Z}^2)$.\n\n\\medskip\n\nIf $p\\ge3$, observe that the rectangle $[0,1)\\times[0,1\/p)$ is covered $12p$ times by $B_p(\\Bbb Z^2)$, and $p\\geq 3$. \n\n $ \\{(0,j\/p): -2p\\leq j\\leq 2p-1\\}$ exactely $4p$ times. Since $4$ does not divide $14$, thus we have completed the proof.\n \\fi \n\n\\end{proof}\n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}