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+{"text":"\\section{Introduction}\n\\label{sect:intro}\n\nThe data collected in the heavy-ion experiments at the Relativistic Heavy-Ion Collider (RHIC) are most commonly interpreted as the evidence that the matter produced in relativistic heavy-ion collisions equilibrates very fast (presumably within a fraction of 1~fm\/c) and its behavior is very well described by the perfect-fluid hydrodynamics  \\cite{Kolb:2003dz,Huovinen:2003fa,Shuryak:2004cy,Teaney:2001av,Hama:2005dz,Hirano:2007xd,Nonaka:2006yn}. Such features are naturally explained by the assumption that the produced matter is a strongly coupled quark-gluon plasma (sQGP) \\cite{Shuryak:2004kh}. Another explanation assumes that the plasma is weakly interacting, however the plasma instabilities lead to the fast isotropization of matter, which in turn helps to achieve equilibration \\cite{Mrowczynski:2005ki}. Recently, it has been also shown that the model assuming thermalization of the transverse degrees of freedom only \\cite{Bialas:2007gn} is consistent with the data describing the transverse-momentum spectra and the elliptic flow coefficient $v_2$. This result indicates  that the assumption of the fast thermalization\/isotropization might be relaxed. \n\nIn view of the problems related to the thermalization and isotropization of the plasma, it is useful to develop and analyze the models which can describe anisotropic systems. Recently, an effective model describing the anisotropic fluid\/plasma dynamics in the early stages of relativistic heavy-ion collisions has been introduced  \\cite{Florkowski:2008ag}. The model has the structure similar to the perfect-fluid relativistic hydrodynamics -- the equations of motion follow from the conservation laws. However, it admits the possibility that the longitudinal and transverse pressures are different (as usual, the longitudinal direction is defined here by the beam axis). The main characteristic feature of the proposed model is the use of the pressure relaxation function $R$ which determines the time changes of the ratio of the transverse and longitudinal pressures and, possibly, defines the way how the system becomes isotropic, i.e., how the transverse and longitudinal pressures become equilibrated. The role of the pressure relaxation function is similar and complementary to the role played by the equation of state. It characterizes the material properties of the medium whose spacetime dynamics is otherwise governed  by the conservation laws. \n\nIn this paper we develop the formulation of Ref. \\cite{Florkowski:2008ag} in two ways: i) we introduce  the microscopic interpretation for the relaxation function in the case where the considered system consists of particles whose behavior is described by the momentum-anisotropic phase-space distribution function, ii) we include the effects of the fields considering the case of anisotropic magnetohydrodynamics -- this approach may be regarded as a very crude attempt to include the effects of color fields on the particle dynamics.\n\nOur first finding is that the use of the anisotropic phase-space distribution functions leads inevitably to the pressure relaxation functions $R$ which imply that the ratio of the longitudinal and transverse pressures tends asymptotically to zero, $P_L\/P_T \\to 0$. This behavior is complementary to the recent results obtained from the analyses of the early-stage partonic free-streaming \\cite{Jas:2007rw,Broniowski:2008qk}. In fact, our approach includes the free-streaming as the special case, however, it may be also applied in the cases where the collisions are present but their effect does not change the assumed generic form of the phase-space distribution function.  The time asymptotics $P_L\/P_T \\to 0$ means that the systems with the initial prolate momentum shape, i.e., the systems that are initially elongated along the beam axis in the momentum space with \\mbox{$P_L > P_T$} (see for example Refs. \\cite{Jas:2007rw,Randrup:2003cw}), naturally pass through the transient isotropic stage where the transverse and longitudinal pressures are equal. Our second finding follows from the study of the magnetohydrodynamic model. We show that the inclusion of the fields lowers the longitudinal pressure and increases the transverse pressure, hence, for the initially prolate systems the stage when the {\\it total} longitudinal and transverse pressures become equal may be reached earlier, depending on the strength of the field.  \n\nThe two models discussed by us cannot explain the phenomenon of reaching the {\\it stable} isotropic stage. However, we indicate that the presence of the fields may have an impact on the process of isotropization, presumably restored by the effects of those collisions and\/or field instabilities that are not taken into account in the present formalism. From the practical point of view, our formalism allows for the determination of the space-time evolution of the color-neutral anisotropic distributions, which may be used, for example, as the background distributions in the analysis of the plasma instabilities. Interestingly, in the case where we have initially $P_L > P_T$, the dynamics of the background and the plasma instable behavior evolve in the same direction, i.e., the two processes restore the equality of pressures. On the other hand, if the initial configuration has $P_T > P_L$, the plasma instabilities must compete with the growing asymmetry of the background (see Ref. \\cite{Rebhan:2008uj,Rebhan:2008ry} where the growth of the instabilities was studied in the Bjorken longitudinal expansion and substantially large times of about 20 fm\/c were found for RHIC). The interplay of such competing processes may be directly related to the problem of very fast thermalization\/isotropization taking place in relativistic heavy-ion collisions. In addition, the discussed by us transformation of the longitudinal pressure into the transverse one is an interesting phenomenon analyzed in the context of the RHIC HBT puzzle -- we note that the recently proposed explanations of this puzzle suggest a very fast formation of the transverse flow \\cite{Gyulassy:2007zz,Broniowski:2008vp,Pratt:2008bc}. \n\nThe paper is organized as follows: In Sect. II we consider the anisotropic system of partons described by the momentum-anisotropic phase-space distribution function. We calculate the moments of the distribution function, determine the pressure relaxation function $R$, and argue that the general form of $R$ implies that the ratio of the longitudinal and transverse pressures tends asymptotically to zero.  In Section III we consider an example of boost-invariant magnetohydrodynamics. We analyze in detail the consistency of this approach and show that it may be treated as the special case of the formalism developed in \\cite{Florkowski:2008ag} with the appropriate relaxation function $\\hat R$. We also show how the presence of the local magnetic fields affects the dynamics of particles. We conclude in Sect. IV.\n\nBelow we assume that particles are massless and we use the following definitions for rapidity and spacetime rapidity,\n\\begin{eqnarray}\ny = \\frac{1}{2} \\ln \\frac{E_p+p_\\parallel}{E_p-p_\\parallel}, \\quad\n\\eta = \\frac{1}{2} \\ln \\frac{t+z}{t-z}, \\label{yandeta} \n\\end{eqnarray}\nwhich come from the standard parameterization of the four-momentum and spacetime coordinate of a particle,\n\\begin{eqnarray}\np^\\mu &=& \\left(E_p, {\\vec p}_\\perp, p_\\parallel \\right) =\n\\left(p_\\perp \\cosh y, {\\vec p}_\\perp, p_\\perp \\sinh y \\right), \\nonumber \\\\\nx^\\mu &=& \\left( t, {\\vec x}_\\perp, z \\right) =\n\\left(\\tau \\cosh \\eta, {\\vec x}_\\perp, \\tau \\sinh \\eta \\right). \\label{pandx}\n\\end{eqnarray} \nHere the quantity $p_\\perp$ is the transverse momentum\n\\begin{equation}\np_\\perp = \\sqrt{p_x^2 + p_y^2},\n\\label{energy}\n\\end{equation}\nand $\\tau$ is the (longitudinal) proper time\n\\begin{equation}\n\\tau = \\sqrt{t^2 - z^2}.\n\\label{tau}\n\\end{equation} \nThroughout the paper we use the natural units where $c=1$ and $\\hbar=1$.\n\n\n\\section{Anisotropic system of particles}\n\\label{sect:aniso-system}\n\nIn this Section we consider a system of particles\/partons described by the distribution function which is asymmetric in the momentum space, i.e., its dependence on the longitudinal and transverse momentum is different. We calculate the particle current, the energy-momentum tensor, and the entropy current of such a system. As the special case we consider the exponential Boltzmann-like distributions frequently used in other studies. This Section is also used to introduce general concepts of anisotropic plasma dynamics, which will be applied to the system consisting of particles and fields in the next Section. \n\n\\subsection{Anisotropic momentum distribution}\n\\label{sect:aniso-distribution}\n\nWe take into account the phase space distribution function whose dependence on the transverse and longitudinal momentum is determined by the two space-time dependent scales, $\\lambda_\\perp$ and $\\lambda_\\parallel$, namely\n\\begin{equation}\nf = f\\left( \\frac{p_\\perp}{\\lambda_\\perp},\\frac{|p_\\parallel|}{\\lambda_\\parallel}\\right).\n\\label{Fxp1}\n\\end{equation}\nThe form (\\ref{Fxp1}) is valid in the local rest frame of the plasma element. For boost-invariant systems, the explicitly covariant form of the distribution function has the structure\n\\begin{equation}\nf  = f\\left( \\frac{\\sqrt{(p \\cdot U)^2 - (p \\cdot V)^2 }}{\\lambda _\\perp }, \n\\frac{|p \\cdot V|}{\\lambda _\\parallel  }\\right),\n\\label{Fxp2}\n\\end{equation}\nwhere\n\\begin{equation}\nU^{\\mu} = ( u_0 \\cosh\\eta,u_x,u_y, u_0 \\sinh\\eta),\n\\label{U}\n\\end{equation}\n\\begin{equation}\nV^{\\mu} = (\\sinh\\eta,0,0,\\cosh\\eta),\n\\label{V}\n\\end{equation}\nand $u^0, u_x, u_y$ are the components of the four vector\n\\begin{equation}\nu^\\mu = \\left(u^0, {\\vec u}_\\perp, 0 \\right) = \\left(u^0, u_x, u_y, 0 \\right).\n\\label{smallu}\n\\end{equation}\nThe four-velocity $u^\\mu$ is normalized to unity\n\\begin{equation}\n u^\\mu u_\\mu = u_0^2 - u_x^2 - u_y^2 = 1.\n\\label{normsmallu}\n\\end{equation}\nThe four-vector $U^\\mu$ describes the four-velocity of the plasma element. It may be obtained from $u^\\mu$ by the Lorentz boost along the $z$ axis with rapidity $\\eta$. The appearance of the four-vector $V^\\mu$  is a new feature related to the anisotropy -- in the rest frame of the plasma element we have $V^\\mu = (0,0,0,1)$. We note that the four vectors $U^\\mu$ and $V^\\mu$ satisfy the following normalization conditions:\n\\begin{equation}\nU^\\mu U_\\mu = 1, \\quad V^\\mu V_\\mu = -1, \\quad U^\\mu V_\\mu = 0.\n\\label{normort}\n\\end{equation}\n\nThe boost-invariant character of Eq. (\\ref{Fxp2}) is immediately seen if we write the explicit expression for $p \\cdot U$ and $p \\cdot V$ which both depend only on $y-\\eta$ and the transverse coordinates, namely \n\\begin{eqnarray}\np \\cdot U &=& p_\\perp u_0 \\cosh(y-\\eta) - {\\vec p}_\\perp \\cdot {\\vec u}_\\perp , \\nonumber \\\\\np \\cdot V &=& p_\\perp u_0 \\sinh(y-\\eta) .\n\\label{pdotUV}\n\\end{eqnarray}\nFrom Eqs. (\\ref{pdotUV}) one can also infer that in the rest frame system of the plasma element, where $\\eta = 0$ and $ {\\vec u}_\\perp = 0$, we have $p \\cdot U = p_\\perp \\cosh y$ and $p \\cdot V = p_\\perp \\sinh y$. Thus, in the local rest frame of the plasma element Eq. (\\ref{Fxp2}) is reduced to Eq. (\\ref{Fxp1}).\n\n\\subsection{Moments of anisotropic distribution}\n\\label{sect:aniso-moments}\n\nUsing the standard definitions of $N^\\mu$ and $T^{\\mu \\nu}$ as the first and the second moment of the distribution function (\\ref{Fxp2}), namely \n\\begin{eqnarray}\nN^\\mu &=& \\int \\frac{d^3p}{(2\\pi)^3 \\, E_p} \\, p^{\\mu} f,\n\\label{Nmu}\n\\end{eqnarray}\n\\begin{eqnarray}\nT^{\\mu \\nu} &=& \\int \\frac{d^3p}{(2\\pi)^3 \\, E_p} \\, p^{\\mu} p^\\nu f,\n\\label{Tmunu} \n\\end{eqnarray}\nwe obtain the following decompositions:\n\\begin{eqnarray}\nN^\\mu &=&  n \\, U^\\mu,\n\\label{Nmudec}\n\\end{eqnarray}\n\\begin{eqnarray}\nT^{\\mu \\nu} &=& \\left( \\varepsilon  + P_T\\right) U^{\\mu}U^{\\nu} \n- P_T \\, g^{\\mu\\nu} - (P_T - P_L) V^{\\mu}V^{\\nu}. \\nonumber \\\\\n\\label{Tmunudec}\n\\end{eqnarray}\nWe note that $N^\\mu$ does not have the contribution proportional to the four-vector $V^\\mu$ since such a term would be proportional to the scalar product $V^\\mu N_\\mu$ that vanishes in the local rest frame. Similarly, the energy-momentum tensor does not contain the terms proportional to the symmetric combination $V^\\mu U^\\nu + U^\\mu V^\\nu$, see Ref. \\cite{Ryblewski:2008fx} for a more explicit presentation of the analogous decompositions.  \n\nEquation (\\ref{Nmudec}) defines the particle density $n$, which may be calculated from the formula\n\\begin{eqnarray}\nn &=& \\int \\frac{d^3p}{(2\\pi)^3} \\, f\\left( \\frac{p_\\perp}{\\lambda _\\perp}, \n\\frac{| p_\\parallel |}{\\lambda _\\parallel }\\right) \\label{rho1} \\\\\n&=& \\frac{\\lambda_\\perp^2 \\,\\lambda_\\parallel}{2 \\pi^2} \\int\\limits_0^\\infty  \nd\\xi_\\perp\\,\\xi_\\perp\\, \\int\\limits_{0}^\\infty d\\xi_\\parallel \\, f\\left(\\xi_\\perp,\\xi_\\parallel \\right), \\nonumber\n\\end{eqnarray}\nwhere we have introduced the dimensionless variables \n\\begin{equation}\n\\xi_\\perp = \\frac{p_\\perp}{\\lambda_\\perp}, \\quad \\xi_\\parallel = \\frac{p_\\parallel}{\\lambda_\\parallel}.\n\\label{xis}\n\\end{equation}\nIn the similar way we calculate the energy density,\n\\begin{eqnarray}\n\\varepsilon &=&   \\int \\frac{d^3p}{ (2\\pi)^3}  \\, E_p \\, f\\left( \\frac{p_\\perp}{\\lambda _\\perp}, \\frac{| p_\\parallel |}{\\lambda _\\parallel }\\right)  \\label{epsilon1} \\\\\n&=& \\frac{\\lambda_\\perp^2 \\,\\lambda_\\parallel^2}{2\\pi^2} \\int\\limits_0^\\infty  \nd\\xi_\\perp\\,\\xi_\\perp  \\int\\limits_0^\\infty \\,d\\xi_\\parallel\\,\n\\sqrt{\\xi_\\parallel^2 + x\\, \\xi_\\perp^2 }\n\\, f\\left(\\xi_\\perp,\\xi_\\parallel \\right),\n\\nonumber \n\\end{eqnarray}\nwhere the variable $x$ is defined by the expression\n\\begin{equation}\nx = \\left( \\frac{\\lambda_\\perp}{\\lambda_\\parallel} \\right)^2.\n\\label{iks}\n\\end{equation}\nFinally, the transverse and longitudinal pressure is obtained from the equations \n\\begin{eqnarray}\nP_T &=&   \\int \\frac{d^3p}{ (2\\pi)^3}  \\, \\frac{p_\\perp^2}{2 E_p} \\, f\\left( \\frac{p_\\perp}{\\lambda _\\perp}, \\frac{| p_\\parallel |}{\\lambda _\\parallel }\\right)  \\label{PT1} \\\\\n&=& \\frac{\\lambda_\\perp^4}{2\\pi^2} \\int  \n\\frac{d\\xi_\\perp\\,\\xi_\\perp^3 \\,d\\xi_\\parallel}\n{2 \\sqrt{ \\xi_\\parallel^2 + x\\, \\xi_\\perp^2 }}\n\\, f\\left(\\xi_\\perp,\\xi_\\parallel \\right),\n\\nonumber \n\\end{eqnarray}\n\\begin{eqnarray}\nP_L &=&   \\int \\frac{d^3p}{ (2\\pi)^3}  \\, \\frac{p_\\parallel}{E_p} \\, f\\left( \\frac{p_\\perp}{\\lambda _\\perp}, \\frac{| p_\\parallel |}{\\lambda _\\parallel }\\right)  \\label{PL1} \\\\\n&=& \\frac{\\lambda_\\perp^2 \\,\\lambda_\\parallel^2}{2\\pi^2} \\int  \n\\frac{d\\xi_\\perp\\,\\xi_\\perp \\,d\\xi_\\parallel \\,\\xi_\\parallel^2}\n{\\sqrt{\\xi_\\parallel^2 + x\\, \\xi_\\perp^2 }}\n\\, f\\left(\\xi_\\perp,\\xi_\\parallel \\right).\n\\nonumber \n\\end{eqnarray}\nFrom now on the limits of the integrations over $\\xi_\\perp$ and $\\xi_\\parallel$ are always from 0 to infinity. In the local rest-frame of the fluid element, where we have $U^\\mu = (1,0,0,0)$ and $V^\\mu = (0,0,0,1)$ one finds\n\\begin{equation}\nT^{\\mu \\nu} =  \\left(\n\\begin{array}{cccc}\n\\varepsilon & 0 & 0 & 0 \\\\\n0 & P_T & 0 & 0 \\\\\n0 & 0 & P_T & 0 \\\\\n0 & 0 & 0 & P_L\n\\end{array} \\right),\n\\label{Tmunuarray}\n\\end{equation}\nhence, as expected the structure (\\ref{Fxp2}) allows for different pressures in the longitudinal and transverse directions. \n\nOne may also calculate the entropy current using the Boltzmann definition,\\footnote{The formula (\\ref{S}) assumes the classical Boltzmann statistics. It may be generalized to the case of bosons or fermions in the standard way.}\n\\begin{equation}\nS^{\\mu} = g_0\\int \\frac{d^3p}{(2 \\pi)^3} \\frac{p^\\mu}{ E_p}  \\left(\\frac{f}{g_0}\\right)\n \\, \\left[1 -  \\ln \\left(\\frac{f}{g_0}\\right) \\right],\n\\label{S}\n\\end{equation}\nhere $g_0$ is the degeneracy factor related to internal quantum numbers such as spin or color. The entropy current has the structure\n\\begin{equation}\nS^{\\mu} = \\sigma \\, U^\\mu,\n\\label{Sstr}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\sigma = \\frac{\\lambda_\\perp^2 \\,\\lambda_\\parallel}{2\\pi^2} \\int  \nd\\xi_\\perp\\,\\xi_\\perp\\,d\\xi_\\parallel \\, f\\left(\\xi_\\perp,\\xi_\\parallel \\right) \\left[1 - \\ln \\frac{f\\left(\\xi_\\perp,\\xi_\\parallel \\right)}{g_0} \\right]. \\nonumber \\\\\n\\label{sigma}\n\\end{eqnarray}\nComparison of Eqs. (\\ref{rho1}) and (\\ref{sigma}) indicates that the particle density and the entropy density are proportional, with the proportionality constant depending on the specific choice of the parton distribution function $f$.\n\n\n\\subsection{Pressure relaxation function}\n\\label{sect:rel-funct}\n\nWith the help of the variables $x = (\\lambda_\\perp\/\\lambda_\\parallel)^2$ and $n$ we may rewrite our expressions (\\ref{epsilon1}), (\\ref{PT1}), and  (\\ref{PL1}) in the concise form\n\\begin{eqnarray}\n\\varepsilon &=&  \\left(\\frac{n}{g} \\right)^{4\/3} R(x),\n\\label{epsilon2} \n\\end{eqnarray}\n\\begin{eqnarray}\nP_T &=&  \\left(\\frac{n}{g} \\right)^{4\/3}\n\\left[\\frac{R(x)}{3} + x R^\\prime(x) \\right],   \n\\label{PT2} \n\\end{eqnarray}\n\\begin{eqnarray}\nP_L &=&   \\left(\\frac{n}{g} \\right)^{4\/3} \n\\left[\\frac{R(x)}{3} - 2 x R^\\prime(x) \\right],\n\\label{PL2} \n\\end{eqnarray}\nwhere the function $R(x)$ is defined by the integral\n\\begin{equation}\nR(x) = x^{-1\/3} \\int \\frac{d\\xi_\\perp\\,\\xi_\\perp\\,d\\xi_\\parallel}{2\\pi^2}\n\\sqrt{\\xi_\\parallel^2 + x \\xi_\\perp^2} f(\\xi_\\perp,\\xi_\\parallel),\n\\label{Rofiks}\n\\end{equation}\n$R^\\prime(x) = dR(x)\/dx$, and $g$ is a constant defined by the expression\n\\begin{equation}\ng =  \\int \\frac{d\\xi_\\perp\\,\\xi_\\perp\\,d\\xi_\\parallel}{2\\pi^2}\n f(\\xi_\\perp,\\xi_\\parallel).\n\\label{gconst}\n\\end{equation}\n\nIt is quite interesting to observe that the structure of Eqs. (\\ref{epsilon2}) - (\\ref{PL2}) agrees with the structure derived in \\cite{Florkowski:2008ag}, where no reference to the underlying microscopic picture was made but only the general consistency of the approach based on the anisotropic energy-momentum tensor (\\ref{Tmunudec}) and the conservation laws was studied. The only difference is that the entropy density $\\sigma$ used in Ref. \\cite{Florkowski:2008ag} is now replaced by the particle density $n$. \n\nIn fact, one may repeat the arguments presented in \\cite{Florkowski:2008ag} replacing the assumption of the conservation of entropy by the assumption of the particle-number conservation  (note that we have shown above that $n$ and $\\sigma$ are proportional if one uses the ansatz (\\ref{Fxp1})). In such a case we end up with the structure which exactly matches Eqs. (\\ref{epsilon2}) - (\\ref{PL2}) and $R$ may be identified with the pressure relaxation function. Moreover, the results of Ref. \\cite{Florkowski:2008ag} allow us to relate the variable $x = \\lambda_\\perp^2\/\\lambda_\\parallel^2$ with the quantity $n \\tau^3$ -- we shall come back to the discussion of this point below, after the analysis of some special cases of the anisotropic distribution functions.\n\n\\subsection{Boltzmann-like anisotropic distribution}\n\\label{sect:aBoltz}\n\nAs the special case of the anisotropic distribution function we may consider the exponential distribution of the form \n\\begin{equation}\nf_1 = g_0 \\exp \\left( -\\sqrt{\\frac{p_\\perp ^2}{\\lambda_\\perp  ^2} + \n\\frac{p_\\parallel^2}{\\lambda_\\parallel^2} } \\,  \\right),\n\\label{aBoltz1}\n\\end{equation}\nwhich may be regarded as the generalization of the Boltzmann equilibrium distribution where $\\lambda_\\perp = \\lambda_\\parallel = T$ (as explained above, $g_0$ is the degeneracy factor connected with internal quantum numbers). In this case we recover the structure (\\ref{epsilon2}) - (\\ref{PL2}) with the relaxation function of the form \\footnote{Note that for $x < 1$ the function $(\\arctan\\sqrt{x-1})\/\\sqrt{x-1}$ should be replaced by $(\\hbox{arctanh}\\sqrt{1-x})\/\\sqrt{1-x}$}\n\\begin{equation}\nR_1(x) = \\frac{3\\, g_0\\, x^{-\\frac{1}{3}}}{2 \\pi^2} \\left[ 1 + \\frac{x \\arctan\\sqrt{x-1}}{\\sqrt{x-1}}\\right]\n\\end{equation}\nand the constant (\\ref{gconst}) is simply \n\\begin{equation}\ng_1 = \\frac{g_0}{\\pi^2}.\n\\label{g1const}\n\\end{equation}\n\nAnother interesting anisotropic distribution function has the factorized form\n\\begin{equation}\nf_2 = g_0 \\exp\\left( -\\frac{p_\\perp}{\\lambda_\\perp} \\right)\n\\exp\\left( - \\frac{|p_\\parallel|}{\\lambda_\\parallel} \\right).\n\\label{aBoltz2}\n\\end{equation}\nIn this case we obtain\n\\begin{eqnarray}\nR_2(x) &=& \\frac{g_0 x^{-1\/3}}{2 \\pi^2 (1+x)^2} \\left[\\,\n1 + 5 x \\sqrt{x} + 2 x^2 \\sqrt{x} - 2 x\n\\vphantom{\\frac{1+\\sqrt{x}+\\sqrt{1+x}}{1+\\sqrt{x}-\\sqrt{1+x}}}\n\\right. \\nonumber \\\\\n& & \\left. \\quad + \\frac{3 x}{\\sqrt{x+1}} \n\\ln \\frac{1+\\sqrt{x}+\\sqrt{1+x}}{1+\\sqrt{x}-\\sqrt{1+x}} \\,\\,\n\\right]\n\\end{eqnarray}\nand \n\\begin{equation}\ng_2 = \\frac{g_0}{2 \\pi^2}.\n\\label{g1const}\n\\end{equation}\nThe calculation of the entropy density gives $\\sigma = 4 n$ and $\\sigma = 8 n$, for the cases $f=f_1$ and $f=f_2$, respectively.\n\nThe structure of Eq. (\\ref{Rofiks}) implies that for $x \\ll 1$ \\mbox{($\\lambda_\\perp \\ll \\lambda_\\parallel$)} the function $R(x)$ behaves like $x^{-1\/3}$. In this limit $P_T = 0$ and $\\varepsilon = P_L$. Similarly, for $x \\gg 1$ \\mbox{($\\lambda_\\perp \\gg \\lambda_\\parallel$)} the function $R(x)$ behaves like $x^{1\/6}$, implying that $P_L = 0$ and $\\varepsilon = 2 P_T$. This behavior is expected if we interpret the parameters $\\lambda_\\perp$ and $\\lambda_\\parallel$ as the transverse and longitudinal temperatures, respectively. In agreement with those general properties we find \n\\begin{eqnarray}\nR_1(x) &\\approx &  \\frac{3 g_0 }{2 \\pi^2} \n\\left[ x^{-1\/3} + \\frac{1}{2}(\\ln 4 - \\ln x) x^{2\/3} \\right] , \n\\nonumber \\\\\nR_2(x) &\\approx & \\frac{g_0 }{2 \\pi^2} \n\\left[ x^{-1\/3} +\\frac{1 }{2} (6 \\ln 2 - 8 - 3 \\ln x) x^{2\/3}\\right] , \n\\nonumber \\\\\n\\label{smalliks}\n\\end{eqnarray}\nfor $x \\ll 1$, and\n\\begin{eqnarray}\nR_1(x) &\\approx & \\frac{3 g_0 }{4 \\pi}\n\\left( x^{1\/6} + \\frac{1}{2} x^{-5\/6}\\right),\n\\nonumber \\\\\nR_2(x) &\\approx & \\frac{g_0 }{\\pi^2}\n\\left( x^{1\/6} + \\frac{1}{2} x^{-5\/6}\\right),\n\\label{bigiks}\n\\end{eqnarray}\nfor $x \\gg 1$. \n\nIn Fig. \\ref{fig:ratios} we plot the ratios: $P_L\/P_T$ (solid line), $P_L\/\\varepsilon$ (decreasing dashed line), and $P_T\/\\varepsilon$ (increasing dashed line) for the two cases: $f = f_1$ (a) and $f=f_2$ (b). The considered ratios are functions of the $x$ parameter only. In agreement with the remarks given above we see that $\\varepsilon = P_L$ for $x = 0$, and $\\varepsilon = 2 P_T$ in the limit $x \\to \\infty$. For $f = f_1$ the two pressures become equal if  $x=1$, since in this case the distribution function $f_1$ becomes exactly isotropic. For $f = f_2$ the equality of pressures is reached for $x \\approx 0.7$. Except for such small quantitative differences, the behavior of the pressures is very much similar in the two cases, as can be seen from the comparison of the upper and lower part of Fig. \\ref{fig:ratios}.\n\n\\begin{figure}[t]\n\\begin{center}\n\\subfigure{\\includegraphics[angle=0,width=0.45\\textwidth]{RATIOS1.eps}} \\\\\n\\subfigure{\\includegraphics[angle=0,width=0.45\\textwidth]{RATIOS2.eps}} \n\\end{center}\n\\caption{(Color online) The ratios $P_L\/P_T$ (solid red lines), $P_L\/\\varepsilon$ (decreasing blue dashed lines), and $P_T\/\\varepsilon$ (increasing blue dashed lines) shown as functions of the variable $x$, {\\bf a)} the results for the distribution function (\\ref{aBoltz1}), {\\bf b)} the same for the distribution function (\\ref{aBoltz2}). }\n\\label{fig:ratios}\n\\end{figure}\n\nWe note that the choice $R(x) = x^{1\/6}$ corresponds to the case of transverse hydrodynamics, see Ref. \\cite{Florkowski:2008ag}. In the transverse-hydrodynamic approach the matter forms non-interacting transverse clusters which do not interact with each other yielding $P_L=0$. The concept of transverse hydrodynamics was initiated in Refs. \\cite{Heinz:2002rs,Heinz:2002xf} and recently reformulated in Refs. \\cite{Bialas:2007gn,Chojnacki:2007fi,Ryblewski:2008fx}.\n\n\\subsection{Time dependence of pressure anisotropy}\n\\label{sect:PToverPL}\n\nIn this Section we briefly recall the arguments of Ref. \\cite{Florkowski:2008ag} concerning the consistency of the anisotropic plasma dynamics. The basic assumptions are the particle number conservation,\n\\begin{equation}\n\\partial_\\mu N^\\mu = \\partial \\left( n U^\\mu \\right) = 0,\n\\label{partcons}\n\\end{equation}\nand the energy-momentum conservation law,\n\\begin{equation}\n\\partial_\\mu T^{\\mu \\nu} = 0,\n\\label{enmomcon}\n\\end{equation}\nwith the energy-momentum tensor of the form (\\ref{Tmunudec}). As shown in the previous Sections, the entropy conservation used in \\cite{Florkowski:2008ag} may be typically identified with the particle number conservation. In view of the further development of the model discussed in the next Section we shall turn to the particle number conservation as the basic input. We note that the assumption (\\ref{partcons}) means that our description may be valid only after the time when most of the particles is produced. \n\nThe projection of the energy-momentum conservation law (\\ref{enmomcon}) on the four-velocity $U_\\nu$ indicates that the energy density is generally a function of two variables, \\mbox{$\\varepsilon = \\varepsilon(n,\\tau)$}. The mathematical consistency of this approach, i.e., the requirement that $d\\varepsilon$ is a total differential, implies directly that the functions $\\varepsilon(n,\\tau)$, $P_T(n,\\tau)$, and $P_L(n,\\tau)$ must be of the form (\\ref{epsilon2}) - ({\\ref{PL2}) where\n\\begin{equation}\nx = x_0 \\frac{n \\tau^3}{n_0 \\tau_0^3}\n\\label{oldiks}\n\\end{equation}\nwith $x_0, n_0$ and $\\tau_0$ being constants that may be used to fix the initial conditions. In particular, it is convenient to regard $\\tau_0$ as the initial time, and $n_0$ as the maximal initial density (at the very center of the system). Then, $x_0$ is the maximal initial value of $x$. Note that the particle density $n$ is very small at the edge of the system, hence the initial transverse pressure is always zero in this region. On the other hand, at the center of the system at the initial time $\\tau=\\tau_0$ we may have $P_T < P_L$ or $P_T > P_L$ depending on the value of $x_0$.  \n\nCombing Eqs. (\\ref{iks}) and (\\ref{oldiks}) we are coming to the main conclusion reached so far: For the microscopic phase-space distribution function of the form (\\ref{Fxp1}), the pressure relaxation function is completely determined by Eq. (\\ref{Rofiks}) where \n\\begin{equation}\nx = \\frac{\\lambda_\\perp^2}{\\lambda_\\parallel^2} =  x_0 \\frac{n \\tau^3}{n_0 \\tau_0^3}.\n\\label{newiks}\n\\end{equation}\nIn the region where the matter is initially formed we have $0 < n \\leq n_0$ and the right-hand-side of Eq. (\\ref{newiks}) grows with time -- the particle density $n$ cannot decrease faster than $1\/\\tau^3$, since this would require a three-dimensional expansion at the speed of light. We thus conclude that the ratio of the parameters $\\lambda_\\parallel\/\\lambda_\\perp$ tends asymptotically in time to zero. Consequently, for sufficiently large evolution times the ratio of the longitudinal and transverse pressures becomes negligible. As mentioned above, even if the initial conditions require that $P_T$ is larger than $P_L$ at the center of the system, at the edges we have very small density which means that $P_L >> P_T$ in this region. In the case where the longitudinal expansion dominates, $n = n_0 \\tau_0\/\\tau$ and $x = x_0 \\tau^2\/\\tau_0^2$, hence $x$ and $\\tau$ are simply related. \n\n\n\n\n\\subsection{Longitudinal free-streaming}\n\\label{sect:PToverPL}\n\nThe anisotropic distribution functions considered in the previous Sections should satisfy the Boltzmann kinetic equation (in some reasonable approximation). In this respect we assume that the effects of both the free-streaming and the parton collisions do not change the generic structure (\\ref{Fxp2}), while the time changes of the parameters $\\lambda_\\perp$,$\\lambda_\\parallel$, and $u^\\mu$ are determined by the conservation laws. The spirit of this approach is very similar to that used in the perfect-fluid hydrodynamics, where the collisions maintain the equilibrium shape of the distribution function, whereas the conservation laws determine the time changes of the parameters such as temperature or the fluid velocity. \n\nClearly the relation of our framework to the underlying kinetic theory should be elaborated in more detail in further investigations, which may possibly determine the microscopic conditions which validate our approximations. Here, we may easily analyze the case of pure free-streaming where the distribution function satisfies the collisionless kinetic equation\n\\begin{equation}\np^\\mu \\partial_\\mu f(x,p) = 0.\n\\label{kineq}\n\\end{equation}\nFor the pure longitudinal expansion (with vanishing transverse flow, ${\\vec u}_\\perp=0$, and the parameters $\\lambda_\\perp,\\lambda_\\parallel$ depending only on the proper time $\\tau$) we rewrite Eq. (\\ref{kineq}) in the form \n\\begin{equation}\n\\left[ \\cosh(y-\\eta) \\frac{\\partial}{\\partial\\tau}\n+ \\frac{\\sinh(y-\\eta)}{\\tau} \\frac{\\partial}{\\partial \\eta} \\right] f\\left(w,v\\right) = 0,\n\\label{kineq1}\n\\end{equation}\nwhere $w = p_\\perp\/\\lambda_\\perp(\\tau)$ and $v = p_\\perp \\sinh(y-\\eta)\/\\lambda_\\parallel(\\tau)$, see Eqs. (\\ref{Fxp2}) and (\\ref{pdotUV}). By direct differentiation we obtain \\footnote{For simplicity we consider here the functions depending on $v^2$ and disregard the sign of the absolute value.}\n\\begin{equation}\n\\frac{\\partial f}{\\partial w} \\frac{d\\lambda_\\perp}{\\lambda_\\perp^2 d\\tau}\n+ \\frac{\\sinh(y-\\eta)}{\\lambda_\\parallel^2} \\frac{\\partial f}{\\partial v}\n\\left[\\frac{d\\lambda_\\parallel}{d\\tau} + \\frac{\\lambda_\\parallel}{\\tau}  \\right] =0.\n\\end{equation}\nThe solution to this equation exists for any form of the function $f$ provided $\\lambda_\\perp$ is a constant and $\\lambda_\\parallel \\sim 1\/\\tau$. Thus, we may write  \n\\begin{equation}\nf = f\\left( \\frac{p_\\perp}{\\lambda_\\perp^0}, \\frac{\\tau p_\\perp \\sinh(y-\\eta)}{\\tau_0 \\lambda_\\parallel^0} \\right),\n\\label{freestreamsol}\n\\end{equation}\nwhere $\\tau_0$, $\\lambda_\\perp^0$ and $\\lambda_\\parallel^0$ are constants. \n\nIn the considered case the variable $x$ equals\n\\begin{equation}\nx = \\left( \\frac{\\lambda_\\perp}{\\lambda_\\parallel} \\right)^2 =\n\\left( \\frac{\\lambda_\\perp^0 }{\\lambda_\\parallel^0 \\tau_0} \\right)^2 \\, \\tau^2\n\\end{equation}\nhence, it is consistent with Eq. (\\ref{newiks}), where for the boost-invariant longitudinal expansion we may substitute $n = n_0 \\tau_0\/\\tau$. We thus see that our approach includes the boost-invariant free-streaming as the special case. In particular, Eq. (\\ref{freestreamsol}) agrees with the form of the color neutral background used in Refs. \\cite{Rebhan:2008uj,Rebhan:2008ry,Martinez:2008di}. In the next Section we show that our framework includes also the case where the partons interact with local magnetic fields. \n\n\\section{Locally anisotropic magnetohydrodynamics}\n\\label{sect:AMHD}\n\nIn this Section we generalize the formulation discussed in Sect. \\ref{sect:aniso-system}. We analyze in detail magnetohydrodynamics as an example of the physical system consisting of particles and fields, which is also known to exhibit strong anisotropic behavior. Of course, the magnetohydrodynamics by itself cannot be directly applied to modeling of the early stages of heavy-ion collisions. However, several phenomena analyzed in its framework show similarities with the color field dynamics discussed in the context of Color Glass Condensate \\cite{McLerran:1993ni,Kharzeev:2001yq} and Glasma \\cite{Lappi:2006fp}, hence we think that the elaboration of this example may shed light on more complicated color-hydrodynamics which may be the right description of the early stages of heavy-ion collisions. \n\nOur analysis of the boost-invariant magnetohydrodynamics, where the initial magnetic field is parallel to the collision axis,  shows that in the considered system a similar phenomena take place as in the system consisting of particles only. The presence of the fields lowers the longitudinal pressure (which eventually may be negative) and increases the transverse pressure, see a related analysis in \\cite{Vredevoogd:2008id}.\n\n\\subsection{General formulation}\n\\label{sect:binv}\n\nAt first let us recapitulate the main physical assumptions of locally anisotropic magnetohydrodynamics (for recent formulation see for example \\cite{PhysRevE.47.4354,PhysRevE.51.4901}). Let $U^\\mu$ be the plasma four-velocity and $F^{\\mu \\nu}$ be the electromagnetic-field tensor. We define the rest-frame electric and magnetic field by the following equations \n\\begin{equation}\nE^\\mu = F^{\\mu \\nu} U_\\nu, \n\\label{Emu}\n\\end{equation}\n\\begin{equation}\nB^\\mu = \\frac{1}{2} \\epsilon^{\\mu \\alpha \\beta \\gamma} U_\\alpha F_{\\beta \\gamma},\n\\label{Bmu}\n\\end{equation}\nwhere $\\epsilon^{\\alpha \\beta \\gamma \\delta}$ is a completely antisymmetric tensor with $\\epsilon^{0123} = 1$. Eqs. (\\ref{Emu}) and (\\ref{Bmu}) yield\n\\begin{equation}\nF^{\\mu \\nu} = E^\\mu U^\\nu - E^\\nu U^\\mu + \\frac{1}{2} \\epsilon^{\\mu \\nu \\alpha \\beta} \\left(B_\\alpha U_\\beta - B_\\beta U_\\alpha \\right).\n\\label{Fmunu1}\n\\end{equation}\nWe note that both $E^\\mu$ and $B^\\mu$ are spacelike and orthogonal to $U^\\mu$,\n\\begin{eqnarray}\nE_\\mu E^\\mu & \\leq & 0, \\quad E_\\mu U^\\mu = 0, \\label{EU} \\\\\nB_\\mu B^\\mu & \\leq & 0, \\quad B_\\mu U^\\mu = 0. \n\\label{BU} \n\\end{eqnarray}\n\nThe picture  of anisotropic magnetohydrodynamics requires that $U^\\mu$ corresponds to the frame where the electric field is absent, \n\\begin{equation}\nE^\\mu = 0.\n\\label{Emu0}\n\\end{equation}\nIn this case the Maxwell equations may be written in the form\n\\begin{equation}\n\\partial_\\mu F^{\\mu \\nu} = 4 \\pi j^\\nu, \n\\label{maxwell1}\n\\end{equation}\n\\begin{equation}\n\\partial_\\mu {}^* F^{\\mu \\nu} = 0,\n\\label{maxwell2}\n\\end{equation}\nwhere\n\\begin{equation}\nF^{\\mu \\nu} = \\frac{1}{2} \\epsilon^{\\mu \\nu \\alpha \\beta} (B_\\alpha U_\\beta - B_\\beta U_\\alpha)\n\\label{Fmunu2}\n\\end{equation}\nand ${}^* F^{\\mu \\nu}$ is the dual electromagnetic tensor\n\\begin{equation}\n{}^*F^{\\mu \\nu} =  B^\\mu U^\\nu - B^\\nu U^\\mu.\n\\label{Fdual}\n\\end{equation}\nBesides Eqs. (\\ref{maxwell1}) -- (\\ref{Fdual}) the plasma dynamics is determined by the particle conservation law, see Eq. (\\ref{partcons}), the electromagnetic current conservation (the consequence of Eq. (\\ref{maxwell1})), and the energy-momentum conservation law for matter and fields. Before we analyze the role of the conservation laws we shall discuss, however, the constraints coming from the boost-invariance. \n\n\\subsection{Imposing longitudinal boost-invariance}\n\\label{sect:binv}\n\nOur aim is to construct the boost-invariant field tensors $F^{\\mu \\nu}(x)$ and ${}^*F^{\\mu \\nu}(x)$. The condition of boost-invariance requires that the transformed fields at the new spacetime positions are equal to the original fields in the new positions. Formally, this condition may be written in the form\n\\begin{equation}\nF^{\\mu \\nu \\, \\prime}(x^\\prime) = L^\\mu_{\\,\\,\\,\\alpha} L^\\nu_{\\,\\,\\,\\beta }   \nF^{\\alpha \\beta}(x) = F^{\\mu \\nu}(x^\\prime).\n\\end{equation}\nwhere $L$ describes the longitudinal Lorentz boost. Similarly, for a boost-invariant four-vector field $A^\\mu(x)$ we have\n\\begin{equation}\nA^{\\mu \\,\\prime}(x^\\prime) = L^\\mu_{\\,\\,\\,\\alpha} \nA^{\\alpha}(x) = A^{\\mu}(x^\\prime).\n\\end{equation}\nIt is easy to check that the four-vectors $U^\\mu$ and $V^\\mu$ defined by Eqs. (\\ref{U}) and (\\ref{V}) are invariant under Lorentz boosts with rapidity $\\alpha$ along the longitudinal axis, defined by the matrix\n\\begin{equation}\nL^{\\mu}_{\\,\\, \\nu}(\\alpha) =\n\\left(\n\\begin{array}{cccc}\n\\cosh \\alpha & 0 & 0 & \\sinh \\alpha \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n\\sinh \\alpha & 0 & 0 & \\cosh \\alpha\n\\end{array} \n\\right).\n\\label{Lmunu}\n\\end{equation}\nSince $U^\\mu$ and $V^\\mu$ are boost-invariant, the structure of Eqs. (\\ref{Fmunu2}) and (\\ref{Fdual}) suggests that the boost-invariant formalism follows from the ansatz \n\\begin{equation}\nB^\\mu = B V^\\mu.\n\\label{BmuVmu}\n\\end{equation}\nwhere $B$ is a scalar function depending on $\\tau$ and transverse coordinates ${\\vec x}_\\perp$. Equation (\\ref{BmuVmu}) defines the field tensors as the tensor products of the boost-invariant four-vectors, hence, by construction the field tensors are boost-invariant. In addition, we observe that in this case Eq. (\\ref{BU}) is automatically fulfilled. \n\n\\subsection{Homogeneous dual field equations}\n\\label{sect:conlaw}\n\nProjection of the homogeneous dual field equations (\\ref{maxwell2}) on the four-velocity $U_\\nu$ gives\n\\begin{equation}\nV^\\mu \\partial_\\mu B + B \\partial_\\mu V^\\mu  - B U_\\nu U^\\mu \\partial_\\mu V^\\nu = 0.\n\\label{dualeq1}\n\\end{equation}\nFor the boost-invariant systems all terms in (\\ref{dualeq1}) are identically zero, hence it is automatically fulfilled. On the other hand, the projection of (\\ref{maxwell2}) on the four-vector $V_\\nu$ gives\n\\begin{equation}\nU^\\mu \\partial_\\mu \\ln \\left( \\frac{n \\tau}{B} \\right) = 0,\n\\label{dualeq2}\n\\end{equation}\nhence $B$ is related to the particle density $n$ and the proper time $\\tau$ by the expression\n\\begin{equation}\nB = B_0 \\frac{n \\tau}{n_0 \\tau_0}.\n\\label{Bntau}\n\\end{equation}\nOne may check that with the ansatz (\\ref{Bntau}) all four equations in (\\ref{maxwell2}) are automatically satisfied, hence Eq. (\\ref{Bntau}) is the main piece of information delivered by the homogeneous dual field equations. In particular the equation ${\\vec \\nabla} \\cdot {\\vec B} = 0$ turns out to be equivalent with the continuity equation for the particle number. \n\nCollecting now Eqs. (\\ref{BmuVmu}) and (\\ref{Bntau}) we find the explicit form of the dual field tensor ${}^* F^{\\mu \\nu} $,\n\\begin{widetext}\n\\begin{eqnarray}\n{}^* F^{\\mu \\nu} =  \\frac{B_0 n \\tau}{n_0 \\tau_0} \\left(\n\\begin{array}{cccc}\n0              & u_x \\sinh\\eta  & u_y \\sinh\\eta  & -u^0 \\\\\n-u_x \\sinh\\eta & 0             & 0 & -u_x \\cosh\\eta  \\\\\n-u_y \\sinh\\eta & 0             & 0 & -u_y \\cosh\\eta  \\\\\nu^0 & u_x \\cosh\\eta  & u_y \\cosh\\eta  & 0\n\\end{array} \\right). \\nonumber \\\\\n\\end{eqnarray}\n\\end{widetext}\n\nThe structure of the dual tensor allows us to infer the form of the electric and magnetic fields, \n\\begin{eqnarray}\n{\\vec B} &=&  \\frac{B_0 n \\tau}{n_0 \\tau_0}\\left(- u_x \\sinh\\eta  , -u_y \\sinh\\eta  , u^0  \\right),\n\\label{vecB1} \\\\\n{\\vec E} &=&  \\frac{B_0 n \\tau}{n_0 \\tau_0}\\left(- u_y \\cosh\\eta  , u_x \\cosh\\eta  , 0  \\right),\n\\label{vecE1}\n\\end{eqnarray}\nThe above structure implies directly that with no transverse expansion, i.e., for $u_x=u_y=0$ only the longitudinal magnetic field is present in the system and, in view of the relation $n=n_0 \\tau_0\/\\tau$, it should be a constant, $B=B_0$.  \n\nIn our general approach the situation $u_x=u_y=0$ corresponds  to the initial condition for the evolution of the system. It resembles the case of the Glasma \\cite{Lappi:2006fp} where also the longitudinal chromo-magnetic field is present, however, in the case of Glasma the direction of the field is random with the coherence transverse length set by the saturation scale (another difference is the presence of the longitudinal chromo-electric field in the Glasma). When the transverse expansion starts, due to the presence of the transverse pressure, it initiates the formation of the transverse magnetic and electric fields which are always perpendicular to each other, ${\\vec B} \\cdot {\\vec E} = 0$. We note, however, that in the local rest frame of the plasma element, the only non-zero component is $B_z$.\n\nA more compact form representing the fields ${\\vec B}$ and ${\\vec E}$ may be achieved if we use the following parameterization of the particle current \n\\begin{eqnarray}\nN^\\mu &=& n \\left(u^0 \\cosh\\eta, u_x, u_y, u^0 \\sinh\\eta \\right) \\nonumber \\\\\n&=& \\left(n \\, u^0 \\cosh\\eta, n_x, n_y, n u^0 \\sinh\\eta \\right).\n\\label{Nmun}\n\\end{eqnarray}\nUsing the quantities $n_x$ and $n_y$ we write\n\\begin{eqnarray}\n{\\vec B} &=&  \\frac{B_0}{n_0 \\tau_0}\\left(- z\\, n_x   , - z\\, n_y  , n\\, \\tau\\, u^0  \\right),\n\\label{vecB2} \\\\\n{\\vec E} &=&  \\frac{B_0}{n_0 \\tau_0}\\left(- t\\, n_y   , t\\, n_x   , 0  \\right).\n\\label{vecE2}\n\\end{eqnarray}\n\n\n\\subsection{Inhomogeneous field equations}\n\\label{sect:conlaw}\n\nWe turn now to the inhomogeneous field equations (\\ref{maxwell1}). In our approach those equations may be used to determine  the electromagnetic current of the system, $j^\\mu = (\\rho, j_x, j_y, j_z)$. The straightforward calculation, where the form of the magnetic and electric fields given by Eqs.  (\\ref{vecB2}) and (\\ref{vecE2}) is used, leads us to the expressions \n\\begin{eqnarray}\nj^0 = \\rho &=& \\frac{B_0\\,t}{n_0 \\tau_0} \\left(\\partial_y n_x - \\partial_x n_y \\right), \\nonumber \\\\\nj^1 = j_x &=&  \\frac{B_0}{n_0 \\tau_0} \\left[\\tau \\partial_y (n u^0) + 2 n_y + \\tau \\partial_\\tau n_y \\right], \\nonumber \\\\\nj^2 = j_y &=&  \\frac{B_0}{n_0 \\tau_0} \\left[-\\tau \\partial_x (n u^0) - 2 n_x - \\tau \\partial_\\tau n_x \\right], \\nonumber \\\\\nj^3 = j_z &=&  \\frac{B_0\\,z}{n_0 \\tau_0} \\left(\\partial_y n_x - \\partial_x n_y \\right).\n\\label{jmu}\n\\end{eqnarray}\nOne may check by the explicit calculation that the electromagnetic four-current $j^\\mu$ defined by Eq. (\\ref{jmu}) is conserved, as required by the equation (\\ref{maxwell1}). \n\nIn the magnetohydrodynamic appoach one usually assumes that matter is neutral. In our case, the neutrality condition $\\rho = 0$ implies that the flow must be rotationless, i.e.,  the following equation should be satisfied \n\\begin{equation}\n\\partial_y n_x - \\partial_x n_y = 0.\n\\label{rotless}\n\\end{equation}\nIn this case also the longitudinal component of the electromagnetic current vanishes, which means that the non-zero current circulates around the $z$-axis. \n\nThe explicit calculation with the magnetic and electric fields given by Eqs.  (\\ref{vecB2}) and (\\ref{vecE2}) shows also that\n\\begin{equation}\n{\\vec E} + {\\vec v} \\times {\\vec B} = 0.\n\\label{Ohm1}\n\\end{equation}\nThis is nothing else but the non-covariant version of the condition (\\ref{Emu0}).\n\n\n\\subsection{Conservation laws}\n\\label{sect:conlaw}\n\nBesides the Maxwell equations, the equations of magnetohydrodynamics include the conservation laws for: the particle number, the electromagnetic current (following directly from Eq. (\\ref{maxwell1})), and the energy-momentum of the combined system consisting of matter and fields. The total energy-momentum conservation law may be written in the form \n\\begin{equation}\n\\partial_\\mu {\\hat T}^{\\mu \\nu} = 0.\n\\label{enmomconhat}\n\\end{equation}\nwhere the energy-momentum tensor, ${\\hat T}^{\\mu \\nu}$, including the contributions from matter and fields has the structure\n\\begin{eqnarray}\n{\\hat T}^{\\mu \\nu} &=& \\left(\\varepsilon + P_T + \\frac{B^2}{4\\pi} \\right) U^\\mu U^\\nu\n-\\left(P_T + \\frac{B^2}{8\\pi} \\right) g^{\\mu \\nu} \\nonumber \\\\\n&+& \\left(P_L - P_T - \\frac{B^2}{4\\pi} \\right) V^\\mu V^\\nu,\n\\label{Thatmunu}\n\\end{eqnarray}\nOne may reduce the tensor (\\ref{Thatmunu}) to the form (\\ref{Tmunudec}) if we introduce the following variables\n\\begin{eqnarray}\n{\\hat \\varepsilon} &=& \\varepsilon + \\frac{B^2}{8 \\pi} \n= \\varepsilon + {\\bar \\varepsilon}, \\nonumber \\\\\n{\\hat P_T} &=& P_T + \\frac{B^2}{8 \\pi} = P_T + {\\bar P}_T, \\nonumber \\\\\n{\\hat P_L} &=& P_L - \\frac{B^2}{8 \\pi} = P_L + {\\bar P}_L.\n\\label{hatvariables}\n\\end{eqnarray}\nClearly, the variables with a hat describe the sum of the matter and field contributions to the total energy density and transverse\/longitudinal pressures (the field contributions are marked with a bar).\n\n\\begin{figure}[t]\n\\begin{center}\n\\subfigure{\\includegraphics[angle=0,width=0.45\\textwidth]{RATIOS1c10.eps}} \\\\\n\\subfigure{\\includegraphics[angle=0,width=0.45\\textwidth]{RATIOS1c50.eps}} \n\\end{center}\n\\caption{(Color online) The ratios: ${\\hat P}_L\/{\\hat P}_T$ (solid red lines), ${\\hat P}_L\/{\\hat \\varepsilon}$ (decreasing blue dashed lines), and ${\\hat P}_T\/{\\hat \\varepsilon}$ (increasing blue dashed lines) shown as functions of the variable $x$, \\mbox{{\\bf a)} the results} for the distribution function (\\ref{aBoltz1}) and $c=0.1$, {\\bf b)} the same for $c=0.5$.}\n\\label{fig:ratiosc}\n\\end{figure}\n\nFollowing the same method as that introduced in Ref. \\cite{Florkowski:2008ag} we find that the energy-momentum conservation leads to the differential equation\n\\begin{equation}\nd{\\hat \\varepsilon} = \\frac{{\\hat \\varepsilon}+{\\hat P_T} }{n} dn + \n\\frac{{\\hat P_T} -  {\\hat P_L} }{\\tau} d\\tau.\n\\label{dhateps}\n\\end{equation}\nExactly this structure implies that the energy density and pressures are of the form (\\ref{epsilon2}) -- (\\ref{PL2}). So we may immediately write\n\\begin{eqnarray}\n{\\hat \\varepsilon} &=&  \\left(\\frac{n}{g} \\right)^{4\/3} {\\hat R}(x),\n\\label{epsilon3} \n\\end{eqnarray}\n\\begin{eqnarray}\n{\\hat P}_T &=&  \\left(\\frac{n}{g} \\right)^{4\/3}\n\\left[\\frac{{\\hat R}(x)}{3} + x {\\hat R}^\\prime(x) \\right],   \n\\label{PT3} \n\\end{eqnarray}\n\\begin{eqnarray}\n{\\hat P}_L &=&   \\left(\\frac{n}{g} \\right)^{4\/3} \n\\left[\\frac{{\\hat R}(x)}{3} - 2 x {\\hat R}^\\prime(x) \\right], \n\\label{PL3} \n\\end{eqnarray}\nwhere the complete relaxation function for matter and fields equals\n\\begin{equation}\n{\\hat R}(x) = R(x) + c_0 x^{2\/3},\n\\label{hatR}\n\\end{equation}\nwith the parameter $c_0$ defined by the equation\n\\begin{equation}\nc_0 = \\frac{B_0^2}{8\\pi} \\left( \\frac{g}{n_0}\\right)^{4\/3} x_0^{-2\/3}.\n\\label{c0}\n\\end{equation}\n\nIn Fig. \\ref{fig:ratiosc} we show the ratios: ${\\hat P}_L\/{\\hat P}_T$ (solid red lines), ${\\hat P}_L\/{\\hat \\varepsilon}$ (decreasing blue dashed lines), and ${\\hat P}_T\/{\\hat \\varepsilon}$ (increasing blue dashed lines) shown as functions of the variable $x$. Similarly to the case without the magnetic field we observe that the ratio of the longitudinal and transverse pressures decreases with $x$. The new feature of the case with the field is, however, that this ratio may become negative. This behavior reflects the negative contribution of the field pressure $ {\\bar P}_L = -B^2\/(8\\pi)$ to the total pressure ${\\hat P}_L$. It becomes dominant for the large values of $x$, where the matter  contribution, growing as $x^{1\/6}$, may be neglected with the field contribution, growing as $x^{2\/3}$.\n\nIn view of our discussion in Sect. II, the variable $x$ depends monotonically on time, hence the $x$ dependence reflects to large extent the time evolution of the studied ratios. If the initial conditions assume very small value of $x_0$ (and consequently of the initial $x$) the system has initially larger total longitudinal pressure than the transverse pressure ${\\hat P}_T$. The time evolution tends to equilibrate and then to invert the ratio of the two pressures. The time scale for this process is determined by the initial value of the field, $B_0$, as can be noticed by the comparison of the upper and lower part of Fig. \\ref{fig:ratiosc}. \n\nWe close this section with the following remark. Since, $B$ is a function of $n$ and $\\tau$, we may rewrite Eq. (\\ref{dhateps}) in the equivalent form as\n\\begin{equation}\nd{\\varepsilon} = \\frac{{\\varepsilon}+{P_L} }{n} dn + \n\\frac{{P_T} -{P_L} }{B} dB.\n\\label{dhatepsnew}\n\\end{equation}\nThis equation displays the dependence of the energy density $\\varepsilon$ on the particle density $n$ and the magnetic field $B$. The equation of the form  $\\varepsilon = \\varepsilon(n,B)$ plays a role of the equation of state. For the boost-invariant systems the functional dependence $\\varepsilon(n,B)$ may be changed to the non-trivial dependence of $\\varepsilon$ on $n$ and $\\tau$, as introduced in Ref. \\cite{Florkowski:2008ag}.\n\n\\section{Conclusions}\n\nIn this paper we have developed the formalism introduced in Ref. \\cite{Florkowski:2008ag} discussing i) the system described by the anisotropic distribution function and ii) the system of partons interacting with local magnetic fields. The presented results may be used to analyze anisotropic systems formed in relativistic heavy-ion collisions. In particular, they may be used to find anisotropic neutral distribution functions which form the background for the field instabilities possibly responsible for the genuine thermalization\/isotropization. In addition, our analysis indicates that the process of stable isotropization may require that the assumption concerning boost-invariance and\/or entropy conservation should be relaxed.   \n \n\\medskip\nAcknowledgements: We thank W. Broniowski and \\mbox{St. Mr\\'owczy\\'nski} for helpful discussions and critical comments.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nReal numbers (${\\mathbb R}$), complex numbers (${\\mathbb C}$) and quaternions (${\\mathbb H}$) are the $3$\nassociative division algebras over the reals. By allowing a ${\\mathbb Z}_2$-grading, $7$ further associative superdivision algebras are encountered (in a graded superdivision algebra, each homogeneous element admits inverse). The total number of $10=3+7$ is known as the ``$10$-fold way\", see \\cite{bae} for a short introduction on the topic of superdivision algebras.\\par\nThis purely mathematical property has striking connections with physical applications. The $10$-fold way appears in the construction of the so-called ``periodic table of topological insulators and superconductors\", which presents features like the $mod~ 8$ Bott's periodicity. The physical significance of the periodic table is discussed in \\cite{kit}.\nIn \\cite{{zir},{alzi}} the Cartan's classification of symmetric spaces was related to the $10$ classes of symmetries\nof random-matrix theories. The $10$-fold way classification of generic Hamiltonians and topological insulators and superconductors has been discussed in \\cite{rsfl} (see also the references therein). On the mathematical side, different implementations of the $10$-fold way have been analyzed in \\cite{frmo}; the one which is relevant for our paper (besides the Morita's equivalence of Clifford modules and the graded extension of the Dyson's threefold way \\cite{dys}) is the direct relation with  the \\cite{wal} classification of graded Brauer groups, i.e. superdivision algebras.\\par\nSince there is no need to stop at ${\\mathbb Z}_2$, the notion of superdivision algebra can be immediately extended to accommodate a ${\\mathbb Z}_2\\times {\\mathbb Z}_2$-grading. The aim of this paper is to classify the inequivalent, associative, ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras. We obtain $13$ inequivalent cases. Before presenting the mathematical results we mention that the motivation of this work lies in the recent surge of interest in ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  physics. We present a succinct state of the art on this topic. ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  Lie algebras and superalgebras were introduced in \\cite{{rw1},{rw2}} (see also \\cite{sch}) and investigated by mathematicians ever since.\nThe attempts to use them for physical applications were rather sporadic. This situation changed when it was pointed out in \\cite{{aktt1},{aktt2}} that ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  Lie superalgebras naturally appear as dynamical symmetries of Partial Differential Equations describing  L\\'evy-Leblond nonrelativistic spinors. An ongoing intense activity followed.  A model of ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  \ninvariant quantum mechanics was discussed in \\cite{brdu}, while the systematic construction of ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  classical mechanics \nwas introduced in \\cite{akt1} and the canonical quantization of the models presented in \\cite{akt2}.  Further developments include the construction of two-dimensional models \\cite{{bru},{kuto},{bru2}}, graded superconformal quantum mechanics \\cite{aad}, superspace \\cite{{brdu2}, {kuto},{aido}}, extensions and bosonization of double-graded supersymmetric quantum mechanics \\cite{{aad2},{que}}, and so forth. The construction of models with ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  parabosons was discussed  in \\cite{kuto}.\nThe theoretical detectability of the parafermions implied by the ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  Lie superalgebras was proved in\\cite{top1}, while in \\cite{top2} the result was extended to the parabosons implied by ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  Lie algebras (for previous works on ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  parastatistics see \\cite{{tol},{stvdj}} and the references therein).\nAll this ongoing activity makes reasonable to expect that the classification of ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras would not just be a mathematical curiosity, but could find a way to concrete physical applications. \\par\nAs an efficient tool to classify ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras we apply the so-called ``alphabetic presentation of Clifford algebras\" introduced in \\cite{tove}, which is here extended to superdivision algebras. In this framework the generators are expressed as matrices defined by equal-length words in a $4$-letter alphabet (the letters encode a basis of invertible $2\\times 2$ real matrices and in each word the symbol of tensor product is skipped). Each one of the $7$\ninequivalent ${\\mathbb Z}_2$-graded division algebra admits an alphabetic presentation; it follows from this property that any ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebra can be alphabetically presented. Within this scheme, spotting the inequivalent classes of ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras is reduced to a simple exercise in combinatorics. We obtain $13$ inequivalent cases which are split into $4$ real, $5$ complex and $4$ quaternionic subcases. The number of generators, in each subcase, is respectively $4$, $8$ and $16$. Correspondingly, the faithful representations  are given by $4\\times 4$ matrices with real entries for the real subcases,  $8\\times 8$ matrices with real entries for the complex subcases and $16\\times 16$ matrices with real entries for the quaternionic subcases. \\par\nFor each inequivalent case the multiplication table of the superdivision algebras generators can be directly read from a given matrix representation.\n\\par\nThe scheme of the paper is the following. The efficient tool of the ``alphabetic presentation of Clifford algebras\" to express real representations of Clifford algebras is briefly recalled in Section {\\bf 2}. In Section {\\bf 3} the notion of graded superdivision algebra is explained. In Section {\\bf 4} we apply the alphabetic framework to recover the $10$-fold way classification of division\nand ${\\mathbb Z}_2$-graded superdivision algebras.  This paves the way for presenting, in Section {\\bf 5}, the core of the paper, namely the classification of the associative ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras.  A brief commentary on the obtained results is given in the Conclusions. The alphabetic presentation of ${\\mathbb Z}_2$-graded and ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras is introduced in Appendix {\\bf A}. \n\n\\section{The alphabetic presentation of Clifford algebras}\n\nIt was pointed out in \\cite{oku} that real, almost complex and quaternionic Clifford algebras are recovered from real matrix representations by taking into account the respective properties concerning the Schur's lemma. This implies that, while almost complex and quaternionic Clifford algebras can be represented by complex matrices, nothing is lost when using only real matrices. The real representations discussed in \\cite{oku} are particularly useful for classification purposes, since they allow to consider at once all three cases of real, complex and quaternionic structures.\\par\n\nThe irreducible representations of Clifford algebras have been classified in \\cite{abs}.\nIt is well known, see e.g. \\cite{oku}, that Clifford algebra generators can be expressed as tensor products of the (complex) Pauli matrices and the $2\\times 2$ identity.\nIn order to get the real representations one should drop the imaginary unit $i$. It follows that real representations of Clifford algebras can be obtained by tensoring the $4$ basis elements of the $2\\times 2$ real matrices. One can associate a letter to each one of these $4$ matrices. By dropping for convenience the unnecessary symbol ``$\\otimes$\" of tensor product, which is understood, a matrix representing a Clifford algebra generator can be expressed by a word in a $4$-letter alphabet. This is the ``alphabetic presentation of Clifford algebras\" introduced in \\cite{tove}. Any irreducible representation of a Clifford algebra is then expressed as a set of equal-length words written in this alphabet, the constructive algorithms presented in \\cite{oku} and \\cite{crt} being recasted \\cite{tove} in this language.\\par\nWe are now briefly detailing this construction since it will be applied in the following, at first to recover the $10$-fold way and later to classify the ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras. \\par\nOne associates the four letters with a basis of invertible matrices spanning the vector space of  $2\\times 2$ real matrices. In this paper we use the following conventions:\n{\\small{\\bea\\label{letters}\n&I=\\left(\\begin{array}{cc}1&0\\\\0&1\\end{array}\\right),\\qquad X=\\left(\\begin{array}{cc}1&0\\\\0&-1\\end{array}\\right),\\qquad\nY=\\left(\\begin{array}{cc}0&1\\\\1&0\\end{array}\\right),\\qquad A=\\left(\\begin{array}{cc}0&1\\\\-1&0\\end{array}\\right).\\qquad&\n\\eea\n}}\nThe letter $I$ is chosen because it stands for ``the identity matrix\", while $A$ stands for ``the antisymmetric matrix\". An $n$-character word written in this $4$-letter alphabet represents a $2^n\\times 2^n$ real matrix. There are some useful tips for detecting the matrix structure. For instance:\n\\\\\n{\\it i})  a word whose initial letter is $I$ or $X$  represents a block-diagonal matrix, while if it starts with $Y$ or $A$ the matrix is block-antidiagonal;\\\\\n{\\it ii}) a word containing an even (odd) number of letters $A$ represents a symmetric (antisymmetric) matrix;\\\\\n{\\it iii}) two different $X,Y,A$ letters represent mutually anticommuting matrices, making easier to check whether two matrices defined by equal-length\nwords either commute or anticommute.\\par\nFor illustrative purposes and later convenience we discuss the alphabetic presentation of the quaternions. \nA faithful representation of the three imaginary quaternions $q_i$ ($i=1,2,3$) satisfying\n\\bea\\label{quatmult}\nq_iq_j &=&-\\delta_{ij} +\\varepsilon_{ijk}q_k\n\\eea\n(where $\\varepsilon_{ijk}$ is the totally antisymmetric tensor normalized as $\\varepsilon_{123}=1$) \nis given by either\n\\bea\\label{quat1}\n&{\\overline q}_1= IA,\\qquad {\\overline q}_2=AY,\\qquad {\\overline q}_3= AX,&\n\\eea\nor by\n\\bea\\label{quat2}\n&{\\widetilde q}_1= AI,\\qquad {\\widetilde q}_2=YA,\\qquad {\\widetilde q}_3= XA.&\n\\eea\nFollowing the rules mentioned above the $4\\times 4$ real matrices expressed by (\\ref{quat1}) are given by\n{\\footnotesize{\\bea\n&{\\overline q}_1=\\left(\\begin{array}{cccc}0&1&0&0\\\\-1&0&0&0\\\\0&0&0&1\\\\0&0&-1&0\\end{array}\\right),\\qquad {\\overline q}_2=\\left(\\begin{array}{cccc}0&0&0&1\\\\0&0&1&0\\\\0&-1&0&0\\\\-1&0&0&0\\end{array}\\right),\\qquad {\\overline q}_3=\\left(\\begin{array}{cccc}0&0&1&0\\\\0&0&0&-1\\\\-1&0&0&0\\\\0&1&0&0\\end{array}\\right).&\n\\eea }}\nThe three imaginary quaternions ${\\overline q}_i$ defined by (\\ref{quat1}) satisfy the\n\\bea\n\\{{\\overline q}_i,{\\overline q}_j\\}&=& - 2\\delta_{ij}\\cdot {\\mathbb I}_4 \\qquad\\quad ({\\mathbb I}_4:= II)\n\\eea\nrelations, making them the gamma matrices (see below) of the $Cl(0,3)$ Clifford algebra. Their quaternionic structure is encoded, see \\cite{oku}, in the Schur's lemma, which states that the most general matrix $S$ commuting with the ${\\overline q}_i$'s is of quaternionic form, being expressed by\n\\bea\nS= \\lambda_0\\cdot II+\\lambda_1\\cdot AI +\\lambda_2\\cdot YA+\\lambda_3\\cdot XA \\quad &\\Rightarrow& ~\\quad [S,{\\overline q}_i]=0,\n\\eea\nfor arbitrary real values $\\lambda_J\\in {\\mathbb R}$, with $J=0,1,2,3$. \\par\n$S$ is defined in terms of the ${\\widetilde q}_j$ matrices entering (\\ref{quat2}).\nObviously, the identification of (\\ref{quat1}) as defining the imaginary quaternions and of (\\ref{quat2}) as defining the associated quaternionic structure can be switched.\\par\nThe multiplication table of the $4$ letters, obtained from the (\\ref{letters}) identifications,  is\n\n\\bea\\label{multi}\n&\\begin{array}{|c||c|c|c|c|}\\hline~_r\\backslash^{c}&I&X&Y&A \\\\\\hline \\hline I&I&X&Y&A \\\\ \\hline X&X&I&A&Y\\\\ \\hline Y&Y&-A&I&-X \\\\ \\hline  A&A&-Y&X&-I\\\\ \\hline\n\\end{array}&\n\\eea\nThe entries of the table denote the result of the multiplication of the row letters ``$r$\" (acting on the left) with the column letters ``$c$\".\\par\nOur nomenclature of Clifford algebras follows \\cite{oku,crt}.  The Clifford algebra $Cl(p,q)$ is the Enveloping algebra, over the ${\\mathbb R}$ field, of a set of $n\\times n$ gamma matrices $\\gamma_I$ $(I=1,2,\\ldots , p+q)$ satisfying, for any $I,J$ pair:\n\\bea\\label{gammagamma}\n\\gamma_I\\gamma_J+\\gamma_J\\gamma_I &=& 2\\eta_{IJ} \\cdot{\\mathbb I}_n.\n\\eea \n${\\mathbb I}_n$ denotes the $n\\times n $ identity matrix, while $\\eta_{IJ}$ is a pseudo-Euclidean diagonal metric\nwith signature $+1$ for $I,J=1,\\ldots, p$ and $-1$ for $I,J=p+1,\\ldots p+q$.\\par\nThe irreducible representation of $Cl(p,q)$ is recovered when $n$ is the minimal integer which allows solutions of (\\ref{gammagamma}) (sometimes it is useful, as in the applications to superdivision algebras, to skip the irreducibility requirement).\\par\nThe three $2\\times 2$ matrices associated with the letters $X,Y,A$ are the gamma matrices of the $Cl(2,1)$ Clifford algebra; the whole set of four letters (\\ref{letters}) is a two-dimensional faithful representation of ${\\widetilde{\\mathbb H}}$, the algebra of the split-quaternions (see \\cite{mcc} for the definition of the split versions of the division algebras). \n\n\\section{Graded superdivision algebras}\n\nIn this Section we recall the notion of graded superdivision algebra.\nIn a ${\\mathbb Z}_2$-graded superdivision algebra the generators are split into even (belonging to the $0$-graded sector) and odd (belonging to the $1$-graded sector). In a ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded   superdivision algebra the generators are accommodated into $2$ bits of information ($00, 10, 01, 11$ graded sectors). A graded superdivision algebra can\nbe denoted as ${\\mathbb D}^{[p]}$, where the non-negative integer $p$ indicates the ${\\mathbb Z}_2^p$-grading. \nIn this work we limit to consider the $p=0,1,2$ values. The three ordinary associative division algebras over the reals (${\\mathbb R}, {\\mathbb C}, {\\mathbb H}$) are obtained from $p=0$. For later purposes we can set\n\\bea\\label{d0conventions}\n&{\\mathbb R} := D^{[0]}_{{\\mathbb R};1},\\qquad {\\mathbb C} := D^{[0]}_{{\\mathbb R};2},\\qquad {\\mathbb H} := D^{[0]}_{{\\mathbb R};3}. &\n\\eea\nFor $p=1,2$ we respectively have\n\\bea\n{\\mathbb D}^{[1]} &=& {\\mathbb D}^{[1]}_0\\oplus {\\mathbb D}^{[1]}_1,\\nonumber\\\\\n{\\mathbb D}^{[2]} &=& {\\mathbb D}^{[2]}_{00}\\oplus {\\mathbb D}^{[2]}_{01 }\\oplus {\\mathbb D}^{[2]}_{10}\\oplus {\\mathbb D}^{[2]}_{11}.\n\\eea   \nA homogeneous element $g$ belongs to a given graded sector ($g\\in{\\mathbb D}^{[1]}_i$ or $g\\in{\\mathbb D}^{[2]}_{ij}$,\nwhere $i,j$ take values $0,1$). A multiplication, which respects the grading, is defined. \nLet $g_A, g_B\\in {\\mathbb D}^{[p]}$ be two homogeneous elements, whose respective gradings are\n$i_A,i_B$ for ${\\mathbb Z}_2$ and the pairs $(i_A,j_A)$, $(i_B,j_B)$ for ${\\mathbb Z}_2\\times {\\mathbb Z}_2$.\nThe multiplied element $g_A\\cdot g_B\\in {\\mathbb D}^{[p]}$ is homogeneous and its graded sector,\neither $i_{A+B}$ or $(i_{A+B}, j_{A+B})$, is obtained from $mod~ 2$ arithmetics:\n\\bea\\label{mod2sum}\n{\\mathbb Z}_2:\\quad i_{A+B} = i_A+i_B; &\\quad&  {\\mathbb Z}_2\\times{\\mathbb Z_2}: \\quad (i_{A+B}, j_{A+B}) = (i_A+i_B, j_A+j_B).\n\\eea \nThe unit element $1$ will also be denoted as ``$e_0$\" (it belongs to the $0$ sector or, respectively, the $00$ sector). In this paper we assume the multiplication to be associative.\\par\nIn a graded superdivision algebra each nonvanishing homogeneous element is invertible. As a consequence, each graded sector is isomorphic, as vector space, to one of the three vector spaces of real numbers, complex numbers or quaternions. It easily follows that the common dimension (in real counting) of each graded sector of a given superdivision algebra is $1$, $2$ or $4$, depending on the case.\\par\nEach graded sector is spanned by the respective set of basis vectors $e_J$, $f_J$, $g_J$, $h_J$, that can be assigned according to\n\\bea\\label{sdivconventions}\n{\\textrm{for}}\\quad {\\mathbb Z}_2\\qquad~&:& \\quad e_J\\in {\\mathbb D}^{[1]}_0,\\quad f_J\\in {\\mathbb D}^{[1]}_1;\n\\nonumber\\\\\n{\\textrm{for}}\\quad {\\mathbb Z}_2\\otimes{\\mathbb Z}_2&:&\\quad e_J\\in {\\mathbb D}^{[2]}_{00},\\quad f_J\\in {\\mathbb D}^{[2]}_{01},\\quad g_J\\in {\\mathbb D}^{[2]}_{10},\\quad h_J\\in {\\mathbb D}^{[2]}_{11}.\n\\eea\nDepending on the spanning vector space, the suffix $J$ is restricted to be\n\\bea\\label{sdivspan}\n&{\\mathbb R}: \\quad J=0;\\qquad {\\mathbb C}:\\quad J=0,1; \\qquad {\\mathbb H}: \\quad J=0,1,2,3.&\n\\eea\nA generic homogeneous element $e$ belonging to either the $0$ or the $00$ graded sector is expressed by the\nlinear combination $e=\\sum_J \\lambda^J e_J$, where the $\\lambda^J$'s ($\\lambda^J\\in {\\mathbb R}$) are real\nparameters. This formula is extended in obvious way to homogeneous elements belonging to the other graded sectors. \nAs mentioned before, in our conventions $e_0$ will denote the unit element.\\par\n~\\par\n{\\it Remark}: the following remark will be used later on. In a ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebra the three graded sectors $01,10,11$ are on equal footing and can be interchanged by permutation. Indeed, by setting\n$\\alpha=01$, $\\beta= 10$, $\\gamma=11$, the table of the $mod~2$ sums given in (\\ref{mod2sum}) of the ${\\mathbb Z}_2\\times{\\mathbb Z}_2$ grading reads as follows:\n\\bea\\label{alphabetagamma}\n&\\begin{array}{|c||c|c|c|c|}\\hline &00&\\alpha&\\beta&\\gamma\\\\\\hline \\hline 00&00&\\alpha&\\beta&\\gamma\\\\  \\hline \\alpha&\\alpha&00&\\gamma&\\beta\\\\ \\hline\n\\beta&\\beta&\\gamma&00&\\alpha\\\\  \\hline \\gamma&\\gamma&\\beta&\\alpha&00\\\\ \\hline\n\\end{array}&\n\\eea\nThe table is invariant under permutations of $\\alpha,\\beta,\\gamma$.\\par\n~\\par\nEach graded superdivision algebra over ${\\mathbb R}$ can be presented by normalizing any given generator $g\\in {\\mathbb D}^{[p]}$ so that its square is, up to a sign, the identity ($g^2=\\pm e_0$). This leaves a sign ambiguity ($\\pm g$) in the normalization of each generator $g\\neq e_0$. The graded superdivision algebras are divided into classes of equivalence based on the following set of transformations:\\\\\n{\\it i}) the sign flippings $g\\mapsto \\pm g$ for the generators $g\\neq e_0$,\\\\\n{\\it ii}) the permutation of the generators inside each graded sector and\\\\\n{\\it iii}) for the ${\\mathbb Z}_2\\times{\\mathbb Z}_2$ grading ($p=2$),  the permutation of the equal footing sectors $10,01,11$.\\par\nThis set of transformations defines the $7$ (and respectively $13$) inequivalent classes of ${\\mathbb Z}_2$-graded (${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded) superdivision algebras.\\par\nIn each class of equivalence, a given multiplication table produced among generators induces equivalent multiplication tables obtained by applying the above three transformations. In the case of the quaternions, for instance, this is tantamount to flip the sign ($\\epsilon_{ijk} \\mapsto -\\epsilon_{ijk}$) of the totally antisymmetric tensor entering (\\ref{quatmult}).\\par   \nA multiplication table is straightforwardly recovered from a faithful matrix representation of the graded superdivision algebra.\n\n\\section{The  10-fold way revisited}\nEach ${{\\mathbb{Z}}_2}$-graded superdivision algebra admits an alphabetic presentation; \nthe generators (see Appendix {\\bf A}) are expressed by equal-length words in a $4$-letter alphabet and the (\\ref{letters}) identification of letters and $2\\times 2$ matrices holds. \\par\nIn this Section we present the alphabetic derivation of the $10$-fold way for division and ${\\mathbb{Z}}_2$-graded superdivision algebras.  We start at first with the three ordinary division algebras.\\par\nThe field of the real numbers ${\\mathbb R}$ is the only exception with the (\\ref{letters}) identification of letters with $2\\times 2$ matrices, since the unit element $e_0$ is the number $1$; the real numbers can, nevertheless, be accommodated into the scheme by setting $e_0=I$, the two-dimensional identity matrix. \\par\nThe complex numbers ${\\mathbb C}$ are alphabetically expressed by the two single-character  words $I$ and $A$ (respectively, the identity element and the imaginary unit). \\par\n The quaternions ${\\mathbb H}$ can be expressed by four two-character words:\neither $II, IA, AY, AX$, see (\\ref{quat1}) or $II, AI, YA, XA$, see (\\ref{quat2}). The $4\\times 4$ matrices corresponding to these two choices are related by a similarity transformation.\\par\nIn terms of the (\\ref{d0conventions}) positions we can express the sets of generators as\n\\bea\nI~~\\quad\\quad ~~&\\in&{\\mathbb D}^{[0]}_{{\\mathbb R};1},\\nonumber\\\\\nI, ~A\\quad~~~~~ &\\in &{\\mathbb D}^{[0]}_{{\\mathbb R};2},\\nonumber\\\\\nII, ~IA,~AX,AY&\\in&{\\mathbb D}^{[0]}_{{\\mathbb R};3}.\n\\eea\nThe seven ${\\mathbb Z}_2$-graded superdivision algebras ($7=2+3+2$) will be denoted as \\par\n${\\mathbb D}^{[1]}_{{\\mathbb R};\\ast}~$ for the real series (with $\\ast=1,2$),\\par\n ${\\mathbb D}^{[1]}_{{\\mathbb C};\\ast}~$ for the complex series (with $\\ast=1,2,3$) and \\par ${\\mathbb D}^{[1]}_{{\\mathbb H};\\ast}~$ for the quaternionic series (with $\\ast=1,2$).\n\n\\subsection{The real  Z2-graded superdivision algebras}\n\nTwo inequivalent ${\\mathbb Z}_2$-graded superdivision algebras are obtained from the real series. On the basis of \nformulas (\\ref{mat22}, \\ref{z2sectors}, {\\ref{z2even}) in Appendix ${\\bf A}$ they can be expressed as\n\\bea\\label{z2real}\n {\\mathbb D}^{[1]}_{{\\mathbb R};1} &:& \\qquad I \\in {\\mathbb D}^{[1]}_0, \\qquad A\\in {\\mathbb D}^{[1]}_1,\\nonumber\\\\\n {\\mathbb D}^{[1]}_{{\\mathbb R};2} &:& \\qquad I \\in {\\mathbb D}^{[1]}_0, \\qquad Y \\in {\\mathbb D}^{[1]}_1,\n\\eea\nwhere ${\\mathbb D}^{[1]}_0$ (${\\mathbb D}^{[1]}_1$) denote the respective even and odd sectors.\\par\n~\\par\nSince $A^2=-I$, $Y^2=I$ we have that:\\\\\n~\\\\\n$~$ {\\it i}) ${\\mathbb D}^{[1]}_{{\\mathbb R};1}~$ corresponds to the complex numbers ${\\mathbb C}$ endowed with a ${\\mathbb Z}_2$-grading;\\\\\n{\\it ii}) ${\\mathbb D}^{[1]}_{{\\mathbb R};2}~$ corresponds to the  split-complex numbers ${\\widetilde{{\\mathbb C}}}$ endowed with a ${\\mathbb Z}_2$-grading.\\par\n\n\n\\subsection{The complex Z2-graded superdivision algebras}\n\nThree inequivalent ${\\mathbb Z}_2$-graded superdivision algebras are obtained from the complex series. On the basis of \nformulas (\\ref{mat22}, \\ref{z2sectors}, \\ref{z2even}) in Appendix ${\\bf A}$ they can be expressed as\n\\bea\\label{z2complex}\n {\\mathbb D}^{[1]}_{{\\mathbb C};1} &:& \\qquad II,~IA \\in {\\mathbb D}^{[1]}_0, \\qquad AX,~AY\\in {\\mathbb D}^{[1]}_1,\\nonumber\\\\ {\\mathbb D}^{[1]}_{{\\mathbb C};2} &:& \\qquad I I,~IA\\in {\\mathbb D}^{[1]}_0, \\qquad YX,~YY \\in {\\mathbb D}^{[1]}_1,\\nonumber\\\\\n {\\mathbb D}^{[1]}_{{\\mathbb C};3} &:& \\qquad II,~IA \\in {\\mathbb D}^{[1]}_0, \\qquad ~ AI, ~AA \\in {\\mathbb D}^{[1]}_1,\n\\eea\nwhere ${\\mathbb D}^{[1]}_0$ (${\\mathbb D}^{[1]}_1$) denote the respective even and odd sectors.\\par\nThe alternative presentation $ II,~IA \\in {\\mathbb D}^{[1]}_0,$ $ YI, ~YA \\in {\\mathbb D}^{[1]}_1$ produces a superdivision algebra which is isomorphic to $ {\\mathbb D}^{[1]}_{{\\mathbb C};3} $.\\par\n\nWe have that:\\\\\n\n~\n\\\\\n$~~$ {\\it i}) ${\\mathbb D}^{[1]}_{{\\mathbb C};1}~$ corresponds to a ${\\mathbb Z}_2$-grading of the quaternions ${\\mathbb H}$, realizing a graded representation of the $Cl(0,3)$ Clifford algebra;\\\\\n$~$ {\\it ii}) ${\\mathbb D}^{[1]}_{{\\mathbb C};2}~$ corresponds to a ${\\mathbb Z}_2$-grading of the split-quaternions ${\\widetilde{\\mathbb H}}$, realizing a $4\\times4$  graded matrix representation of the $Cl(2,1)$ Clifford algebra; \\\\\n{\\it iii}) ${\\mathbb D}^{[1]}_{{\\mathbb C};3}~$ corresponds to a ${\\mathbb Z}_2$-grading of an algebra of commuting matrices.\n\n\n\\subsection{The quaternionic Z2-graded superdivision algebras}\nTwo inequivalent ${\\mathbb Z}_2$-graded superdivision algebras are obtained from the quaternionic series. On the basis of \nformulas (\\ref{mat22}, \\ref{z2sectors}, \\ref{z2even}) in Appendix ${\\bf A}$ they can be expressed as\n\\bea\\label{z2quat}\n {\\mathbb D}^{[1]}_{{\\mathbb H};1} &:& \\qquad  III, IIA, IAY, IAX\\in {\\mathbb D}^{[1]}_0, \\qquad AII, AIA, AAY, AAX\\in {\\mathbb D}^{[1]}_1,\\nonumber\\\\ {\\mathbb D}^{[1]}_{{\\mathbb H};2} &:& \\qquad \n III, IIA, IAY, IAX \\in {\\mathbb D}^{[1]}_0, \\qquad YII, YIA, YAY, YAX \\in {\\mathbb D}^{[1]}_1,\n\\eea\nwhere ${\\mathbb D}^{[1]}_0$ (${\\mathbb D}^{[1]}_1$) denote the respective even and odd sectors.\\par\n~\\par\nThe inequivalence of the two superdivision algebras given above  is spotted by taking the squares of the odd generators.\nThey produce, up to a sign, the $8\\times 8$ identity matrix. The signs are $(-+++)$  for \n$ {\\mathbb D}^{[1]}_{{\\mathbb H};1}$ and  $(+---)$  for \n$ {\\mathbb D}^{[1]}_{{\\mathbb H};2}$ .\\par\n~\\par\nIn this Section we recovered, within the alphabetic presentation, the $7$ superdivision algebras discussed in \\cite{bae} and presented, under a different name, in \\cite{frmo}. \n\n\n\\section{The  13 inequivalent  Z2xZ2-graded superdivision algebras}\nThe alphabetic construction of the ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras has been discussed in Appendix {\\bf A}. We present here the results. We obtain $13$ inequivalent cases (the defining classes of equivalence have been introduced in Section {\\bf 3}). A representative, for each class of equivalence, is given below.\\par\nThe $13$ cases are split into $13=4+5+4$ subcases; $4$ subcases are obtained from the real series, $5$ subcases from the complex series, $4$ subcases from the quaternionic series. The $13$ ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras will be named as follows:\\par\n~\\par\n${\\mathbb D}^{[2]}_{{\\mathbb R};\\ast}~$ for the real series (with $\\ast=1,2,3,4$),\\par\n ${\\mathbb D}^{[2]}_{{\\mathbb C};\\ast}~$ for the complex series (with $\\ast=1,2,3,4,5$) and \\par ${\\mathbb D}^{[2]}_{{\\mathbb H};\\ast}~$ for the quaternionic series (with $\\ast=1,2,3,4$).\\par\n~\\par\nAs mentioned in Appendix {\\bf A}, see formula (\\ref{ssubalg}), any ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebra can be characterized by its subalgebras ${\\mathbb S}_{01}, {\\mathbb S}_{10}, {\\mathbb S}_{11}$, obtained by restricting the generators to, respectively, the sectors $00 \\& 01$,  $00\\& 10$, $00\\& 11$. The subalgebras $ {\\mathbb S}_{01}, {\\mathbb S}_{10}, {\\mathbb S}_{11}$ are ${\\mathbb Z}_2$-graded superdivision algebras.  Since, as recalled in Section {\\bf 3}, see table ({\\ref{alphabetagamma}}), the sectors $01,10,11$ are on equal footing, the ${\\mathbb Z}_2$-graded subalgebra projections of a ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebra can be characterized by the triple\n\\bea\\label{sabc}\n&({\\mathbb S}_\\alpha\/{\\mathbb S}_\\beta\/{\\mathbb S}_\\gamma),&\n\\eea\nwhere the order of the subalgebras is inessential.\\par\nIn the following subsections we separately present the ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras for, respectively, the real, complex and quaternionic series.\n\n\\subsection{The real Z2xZ2 superdivision algebras}\nThe four inequivalent ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras ${\\mathbb D}^{[2]}_{{\\mathbb R};\\ast}~$ of the real series possess four generators. The matrix representatives of each class of equivalence can be expressed as in the table below, which gives the matrix generator of the corresponding graded sector:\n~\\par\n\\bea\\label{fourz2z2r}\n&\\begin{array}{|c|c|c|c|c|}\\hline &00&01&10&11\\\\ \\hline {\\mathbb D}^{[2]}_{{\\mathbb R};1}:&II&IA&AX&AY\\\\  \\hline\n{\\mathbb D}^{[2]}_{{\\mathbb R};2}:&II&IA&YX&YY\\\\  \\hline\n{\\mathbb D}^{[2]}_{{\\mathbb R};3}:&II&IA&AI&AA\\\\  \\hline{\\mathbb D}^{[2]}_{{\\mathbb R};4}:&II&IY&YI&YY\\\\  \\hline\n\\end{array}&\n\\eea\n~\\par\nSome comments are in order.\\\\\n~\\\\\n{\\it Comment} $1$ - squaring the matrices entering the $01,10,11$ sectors gives the signs\n\\bea\n& {\\mathbb D}^{[2]}_{{\\mathbb R};1}:~---; \\quad ~~{\\mathbb D}^{[2]}_{{\\mathbb R};2}:~++-;\\quad~~ {\\mathbb D}^{[2]}_{{\\mathbb R};3}:~+--;\\quad~~\n {\\mathbb D}^{[2]}_{{\\mathbb R};4}:~+++.&\n\\eea\n{\\it Comment} $2$ - \nthe projections to the ${\\mathbb Z}_2$-graded subalgebras, see formula (\\ref{sabc}), are given by\n\\bea\n&{\\mathbb D}^{[2]}_{{\\mathbb R};1}: ({\\underline 1}\/{\\underline 1}\/{\\underline 1});  \\qquad\n{\\mathbb D}^{[2]}_{{\\mathbb R};2}: ({\\underline 1}\/{\\underline 2}\/{\\underline 2});  \\qquad\n{\\mathbb D}^{[2]}_{{\\mathbb R};3}: ({\\underline 1}\/{\\underline 1}\/{\\underline 2});  \\qquad\n{\\mathbb D}^{[2]}_{{\\mathbb R};4}: ({\\underline 2}\/{\\underline 2}\/{\\underline 2}),&\n\\eea\nwhere, for simplicity, we denoted ${\\underline 1}:=\n{\\mathbb D}^{[1]}_{{\\mathbb R};1}$ and $  {\\underline 2}:=\n{\\mathbb D}^{[1]}_{{\\mathbb R};2}.$\\\\\n{\\it Comment} $3$ -  the ${\\mathbb D}^{[2]}_{{\\mathbb R};1}$ superdivision algebra is a ${\\mathbb Z}_2\\times{\\mathbb Z}_2$ gradation of the quaternions ${\\mathbb H}$, \n  the ${\\mathbb D}^{[2]}_{{\\mathbb R};2}$ superdivision algebra is a ${\\mathbb Z}_2\\times{\\mathbb Z}_2$ gradation\n of the split-quaternions ${\\widetilde {\\mathbb H}}$,  the superdivision algebras ${\\mathbb D}^{[2]}_{{\\mathbb R};3}$ and ${\\mathbb D}^{[2]}_{{\\mathbb R};4}$ are commutative.\n\n\n\\subsection{The complex Z2xZ2 superdivision algebras}\nThe  ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras ${\\mathbb D}^{[2]}_{{\\mathbb C};\\ast}~$ of the complex series possess eight generators. The multiplication tables of the admissible alphabetic presentations are grouped into $5$ classes of equivalence. The matrix representatives of each class of equivalence can be expressed as in the table below, which gives the matrix generators of the corresponding graded sector:\n~\\par\n\\bea\\label{fivez2z2c}\n&\\begin{array}{|c|c|c|c|c|}\\hline &00&01&10&11\\\\ \\hline {\\mathbb D}^{[2]}_{{\\mathbb C};1}:&III, ~IIA&IAX, ~IAY&AIX,~AIY&AAI, ~AAA\\\\  \\hline\n{\\mathbb D}^{[2]}_{{\\mathbb C};2}:&III,~IIA&AIX,~AIY&IYX,~IYY&AYI, ~AYA\\\\  \\hline\n{\\mathbb D}^{[2]}_{{\\mathbb C};3}:&III,~IIA&YIX,~YIY&IYX,~IYY&YYI, ~YYA\\\\  \n\\hline{\\mathbb D}^{[2]}_{{\\mathbb C};4}:&III,~IIA&YII,~YIA&XYI,~XYA&AYI,~AYA\\\\  \\hline\n{\\mathbb D}^{[2]}_{{\\mathbb C};5}:&III,~IIA&YII,~YIA&IYI,~IYA&YYI,~YYA\\\\  \\hline\n\\end{array}&\n\\eea\nFollowing the construction outlined in Appendix {\\bf A}, by setting the ${\\mathbb Z}_2$-graded superdivision algebras\n${\\mathbb S}_{01}$ (obtained from the $00$ and $01$ sectors) and ${\\mathbb S}_{10}$ (obtained from the $00$ and $10$ sectors), the \n${\\mathbb Z}_2$-graded superdivision algebra ${\\mathbb S}_{11}$ is determined.  It is convenient to denote here\nthe three complex ${\\mathbb Z}_2$-graded superalgebras as ${\\underline 1}:=\n{\\mathbb D}^{[1]}_{{\\mathbb C};1},~$  $  {\\underline 2}:=\n{\\mathbb D}^{[1]}_{{\\mathbb C};2}~$ and $  {\\underline 3}:=\n{\\mathbb D}^{[1]}_{{\\mathbb C};3}$.\\\\\nThe table below (where the first underlined number denotes ${\\mathbb S}_{01}$, the second number ${\\mathbb S}_{10}$ and the arrow gives the ${\\mathbb S}_{11}$ output) is found:\n\\bea\n&\n\\begin{array}{lll} {\\underline{1}}\\times {\\underline{1}}\\rightarrow {\\underline{3}},\\qquad\\quad&{\\underline{1}}\\times {\\underline{2}}\\rightarrow {\\underline{3}},\\qquad\\quad&{\\underline{1}}\\times {\\underline{3}}\\Rightarrow^{\\nearrow ~^ {\\underline{1}}_{}}_{\\searrow{ ~^{}_{\\underline{2}}}},\n\\quad\\\\\n&&\\\\\n{\\underline{2}}\\times {\\underline{1}}\\rightarrow {\\underline{3}},\\qquad&{\\underline{2}}\\times {\\underline{2}}\\rightarrow {\\underline{3}},\\quad&{\\underline{2}}\\times {\\underline{3}}\\Rightarrow^{\\nearrow ~^ {\\underline{1}}_{}}_{\\searrow{ ~^{}_{\\underline{2}}}},\n\\quad\\\\\n&&\\\\\n{\\underline{3}}\\times {\\underline{1}}\\Rightarrow^{\\nearrow ~^ {\\underline{1}}_{}}_{\\searrow{ ~^{}_{\\underline{2}}}},&{\\underline{3}}\\times {\\underline{2}}\\Rightarrow^{\\nearrow ~^ {\\underline{1}}_{}}_{\\searrow{ ~^{}_{\\underline{2}}}},&{\\underline{3}}\\times {\\underline{3}}\\Rightarrow {\\underline{3}}^{(\\ast)}.\n\\quad \n\\end{array}\n&\n\\eea\nThe presence of the split arrows indicates the cases where the ${\\mathbb S}_{11}$ output depends on the chosen sets of ${\\mathbb S}_{01}$, ${\\mathbb S}_{10}$ generators. The double arrow and the asterisk of the ${\\underline 3}\\times {\\underline 3}$ entry indicates that, depending on the chosen generators, two inequivalent ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras with the same ${\\mathbb Z}_2$-graded projections are encountered.\\par\nWith the above conventions on complex ${\\mathbb Z}_2$-graded superdivision algebras the  projections, see formula (\\ref{sabc}), to the ${\\mathbb Z}_2$-graded subalgebras are given by\n\\bea\n&\n{\\mathbb D}^{[2]}_{{\\mathbb C};1}: ({\\underline 1}\/{\\underline 1}\/{\\underline 3});\\qquad\n{\\mathbb D}^{[2]}_{{\\mathbb C};2}: ({\\underline 1}\/{\\underline 2}\/{\\underline 3});  \\qquad\n{\\mathbb D}^{[2]}_{{\\mathbb C};3}: ({\\underline 2}\/{\\underline 2}\/{\\underline 3});  \\qquad\n{\\mathbb D}^{[2]}_{{\\mathbb C};4}: ({\\underline 3}\/{\\underline 3}\/{\\underline 3});\\qquad\n{\\mathbb D}^{[2]}_{{\\mathbb C};5}: ({\\underline 3}\/{\\underline 3}\/{\\underline 3}).&\\nonumber\\\\&&\n\\eea\n~\\\\\n{\\it Remark}:\nthe inequivalent ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras ${\\mathbb D}^{[2]}_{{\\mathbb C};4},~ {\\mathbb D}^{[2]}_{{\\mathbb C};5}$ are not discriminated by their ${\\mathbb Z}_2$-graded projections. Their difference is spotted as follows:\\\\\n{\\it i}) in the ${\\mathbb D}^{[2]}_{{\\mathbb C};4}$ superdivision algebra any pair of $g, g'$ generators belonging to different $01,10,11$ graded sectors {\\it anticommute};\\\\\n{\\it ii}) in the ${\\mathbb D}^{[2]}_{{\\mathbb C};5}$ superdivision algebra any pair of $g, g'$ generators belonging to different $01,10,11$ graded sectors {\\it commute}.\n\n\\subsection{The quaternionic Z2xZ2 superdivision algebras}\nThe four inequivalent ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras ${\\mathbb D}^{[2]}_{{\\mathbb H};\\ast}~$ of the quaternionic series possess sixteen generators. The matrix representatives of each class of equivalence are given below. In all four cases the generators of the $00$ sector can be given by\n\\bea\\label{z2z2quata}\n00&:& ~ IIII,~IIIA,~IIAX,~IIAY.\n\\eea\nThe generators of the $01$, $10$, $11$  sectors are expressed as\n~\\par\n{\\small{\\bea\\label{z2z2quatb}\n&\\begin{array}{|c|c|c|c|}\\hline &01&10&11\\\\ \\hline {\\mathbb D}^{[2]}_{{\\mathbb H};1}:& IAII,~IAIA,~IAAX,~IAAY&AXII,~AXIA,~AXAX,~AXAY&AYII,~AYIA,~AYAX,~AYAY\\\\  \\hline\n{\\mathbb D}^{[2]}_{{\\mathbb H};2}:&IAII,~IAIA,~IAAX,~IAAY&AIII,~AIIA,~AIAX,~AIAY&AAII~,AAIA~,AAAX,~AAAY\\\\  \\hline\n{\\mathbb D}^{[2]}_{{\\mathbb H};3}:&IAII,~IAIA,~IAAX,~IAAY&YXII,~YXIA,~YXAX,~YXAY&YYII~,~YYIA,~YYAX,~YYAY\\\\  \\hline{\\mathbb D}^{[2]}_{{\\mathbb H};4}:&IYII,~IYIA,~IYAX,~IYAY&YIII,~YIIA,~YIAX,~YIAY&YAII,~YAIA,~YAAX,~YAAY\\\\  \\hline\n\\end{array}&\\nonumber\\\\&&\n\\eea}}\nLet us redefine ${\\underline 1}$, ${\\underline 2}$ as the quaternionic ${\\mathbb Z}_2$-graded superdivision algebras, so that,  ${\\underline 1} :=  {\\mathbb D}^{[1]}_{{\\mathbb H};1}$ and ${\\underline 2} :=  {\\mathbb D}^{[1]}_{{\\mathbb H};2}$. The quaternionic projections, see formula (\\ref{sabc}), are then given by\n\\bea\n&\n{\\mathbb D}^{[2]}_{{\\mathbb H};1}: ({\\underline 1}\/{\\underline 1}\/{\\underline 1});\\qquad\n{\\mathbb D}^{[2]}_{{\\mathbb H};2}: ({\\underline 2}\/{\\underline 2}\/{\\underline 2});  \\qquad\n{\\mathbb D}^{[2]}_{{\\mathbb H};3}: ({\\underline 1}\/{\\underline 1}\/{\\underline 2});  \\qquad\n{\\mathbb D}^{[2]}_{{\\mathbb H};4}: ({\\underline 1}\/{\\underline 2}\/{\\underline 2}).\n\\eea\n~\\par\n{\\it Comment}:  the $13$ inequivalent multiplication tables of the ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras are straightforwardly recovered from the matrix representations of the generators, given in  (\\ref{fourz2z2r}) for the real series, (\\ref{fivez2z2c}) \nfor the complex series and ({\\ref{z2z2quata},\\ref{z2z2quatb}) for the quaternionic series. To save space they will not be reported here.\\par\n\n\n\\section{Conclusions}\n\nIn this paper we showed that the $7$ inequivalent ${\\mathbb Z}_2$-graded superdivision algebras admit an alphabetic presentation; we extended this framework to individuate $13$ classes of equivalence of the ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded superdivision algebras. This result has implications both in mathematics and in physical applications.  On the mathematical side we recall the investigations of ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  structures (see, e.g., \\cite{brgr} for a state of the art account of Riemannian structures on ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  manifolds). On the physical side, the seemingly most promising application is the extension of the periodic table of topological insulators and superconductors to accommodate a ${\\mathbb Z}_2\\times {\\mathbb Z}_2$ grading. In a forthcoming paper we will  present the ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  supercommutants, that is the algebras of ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  matrices which (anti)commute with the ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras.\nThe notion of graded supercommutant is central in extending the $10$ classes of fermionic Hamiltonians discussed in \\cite{rsfl} to ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  parafermionic  Hamiltonians.  We recall that ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  parafermions are theoretically detectable, see \\cite{top1}.  The expectation is that the ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  parafermionic Hamiltonians induce new features which are not encoded in the $10$-fold way.\n\n\n~\n\n  \\renewcommand{\\theequation}{A.\\arabic{equation}}\n  \\setcounter{equation}{0} \n\\par\n~\\\\\n{\\bf{\\Large{Appendix A: alphabetic presentation of superdivision algebras}}}\n~\\par\n~\\par\nWe extend here the alphabetic presentation (described in Section {\\bf 2} for Clifford algebras) to the cases of ${\\mathbb Z}_2$ and ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras.\\par\nAny homogeneous element $g$ of a superdivision algebra is represented by an invertible real matrix which takes the form $g=M\\otimes N$, where\nthe matrix $M$ encodes the information of the grading, either ${\\mathbb Z}_2$ or ${\\mathbb Z}_2\\times{\\mathbb Z}_2$, while  the matrix $N$ encodes the information of the real, complex or quaternionic structure. \\\\\nThe matrix size for $M$ is\n\\bea\n&{\\mathbb Z}_2{\\textrm{-grading}}: ~(2\\times 2);\\qquad \\qquad\n{\\mathbb Z}_2\\times\n{\\mathbb Z}_2{\\textrm{-grading}}: ~(4\\times 4).&\n\\eea\nThe matrix size for $N$ is\n\\bea\n&{\\mathbb R}{\\textrm{-series}}: ~(1\\times 1);\\qquad \\quad\n{\\mathbb C}{\\textrm{-series}}: ~(2\\times 2);\\qquad \\quad\n{\\mathbb H}{\\textrm{-series}}: ~(4\\times 4).&\n\\eea\nConcerning the ${\\mathbb Z}_2$ grading, the even (odd) sector is denoted as $M_0$ ($M_1$); the nonvanishing elements are accommodated according to  {\\small{\\bea\\label{mat22}\n\\left(\\begin{array}{cc}\\ast&0\\\\0&\\ast\\end{array}\\right)\\in M_0,&\\qquad& \\left(\\begin{array}{cc}0&\\ast\\\\ \\ast&0\\end{array}\\right)\\in M_1.\n\\eea\n}}\nThe tensor products of the ${\\mathbb Z}_2$-graded matrices (\\ref{mat22}) produce the\n ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  matrices  $M_{ij}$ ($ij$ denotes the grading) according to\n\n{\\footnotesize{\\bea\\label{fromz2toz2z2}\n\\left(\\begin{array}{cc}\\ast&0\\\\0&\\ast\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}\\ast&0\\\\0&\\ast\\end{array}\\right)\\mapsto \\left(\\begin{array}{cccc}\\ast&0&0&0\\\\ 0&\\ast&0&0\\\\0&0&\\ast&0\\\\ 0&0&0&\\ast \\end{array}\\right)\\in M_{00},&&\n\\left(\\begin{array}{cc}\\ast&0\\\\0&\\ast\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}0&\\ast\\\\\\ast&0\\end{array}\\right)\\mapsto \\left(\\begin{array}{cccc}0&\\ast&0&0\\\\ \\ast&0&0&0\\\\0&0&0&\\ast\\\\ 0&0&\\ast&0 \\end{array}\\right)\\in M_{01},\\nonumber\\\\\n\\left(\\begin{array}{cc}0&\\ast\\\\ \\ast&0\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}\\ast&0\\\\0&\\ast\\end{array}\\right)\\mapsto \\left(\\begin{array}{cccc}0&0&\\ast&0\\\\ 0&0&0&\\ast\\\\\\ast&0&0&0\\\\ 0&\\ast&0&0 \\end{array}\\right)\\in M_{10},&&\n\\left(\\begin{array}{cc}0&\\ast\\\\\\ast&0\\end{array}\\right)\\otimes \\left(\\begin{array}{cc}0&\\ast\\\\\\ast&0\\end{array}\\right)\\mapsto \\left(\\begin{array}{cccc}0&0&0&\\ast\\\\ 0&0&\\ast&0\\\\0&\\ast&0&0\\\\ \\ast&0&0&0 \\end{array}\\right)\\in M_{11}.\\nonumber\\\\\n&&\n\\eea\n}}\nIn the ${\\mathbb Z}_2$-grading the $M_0$, $M_1$ sectors can be spanned by the matrices denoted by the letters,\n with the (\\ref{letters}) identification, given by:\n\\bea\n\\label{z2sectors}\nM_{0}: \\quad I,~X; && M_{1}: \\quad Y, ~A.\n\\eea\nIn the ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-grading the $M_{00}, M_{01}, M_{10}, M_{11}$ sectors can be spanned by the matrices denoted by the $2$-character words\n\\bea \nM_{00}: \\quad II,~IX,~XI, ~XX; && M_{01}: \\quad IA, ~IY, ~XA, ~XY;\\nonumber\\\\\nM_{10}: \\quad AI,~AX,~YI, ~YX; && M_{11}: \\quad AA, ~AY, ~YA, ~YY.\n\\eea\nIn Section {\\bf 4} we showed that each one of the $7$ inequivalent ${\\mathbb Z}_2$-graded superdivision algebras admits an alphabetic presentation in terms of equal-length words. Without loss of generality (up to similarity transformations) the even sector ${\\mathbb D}_0^{[1]}$ can be expressed as\n\\bea\\label{z2even}\n&{\\mathbb R}{\\textrm{-series}}: ~ I;\\qquad ~{\\mathbb C}{\\textrm{-series}}:  ~II,~IA;\\qquad ~{\\mathbb H}{\\textrm{-series}}:  ~III, ~IIA,~IAX,~ IAY. &\n\\eea\nThe odd sectors  ${\\mathbb D}_1^{[1]}$ are presented in table (\\ref{z2real}) (for the ${\\mathbb R}$-series), table\n(\\ref{z2complex})  (for the ${\\mathbb C}$-series), table (\\ref{z2quat}) (for the ${\\mathbb H}$-series).\\par\nThe alphabetic presentation is extended to the ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebras by taking into account that:\n\\\\\n{\\it i}) without loss of generality (up to similarity transformations) the even sector ${\\mathbb D}_{00}^{[2]}$ can be expressed as\n\\bea\n&{\\mathbb R}{\\textrm{-series}}: ~ II;\\quad ~{\\mathbb C}{\\textrm{-series}}:  ~III,~IIA;\\quad ~{\\mathbb H}{\\textrm{-series}}:  ~IIII, ~IIIA,~IIAX,~ IIAY;&\n\\eea\n{\\it ii}) each one of the three subalgebras ${\\mathbb S}_{10}, {\\mathbb S}_{01}, {\\mathbb S}_{11}\\subset {\\mathbb D}^{[2]}$, given by the direct sums\n\\bea\\label{ssubalg}\n&{\\mathbb S}_{01}:= {\\mathbb D}_{00}^{[2]}\\oplus{\\mathbb D}_{01}^{[2]},\\qquad\n{\\mathbb S}_{10}:= {\\mathbb D}_{00}^{[2]}\\oplus{\\mathbb D}_{10}^{[2]},\\qquad\n{\\mathbb S}_{11}:= {\\mathbb D}_{00}^{[2]}\\oplus{\\mathbb D}_{11}^{[2]},&\n\\eea \nis isomorphic to one (of the seven) ${\\mathbb Z}_2$-graded superdivision algebra;\\\\\n{\\it iii}) the alphabetic presentation can be assumed for ${\\mathbb S}_{01}$ and, since the second  ${\\mathbb Z}_2$ grading is independent from the first one, ${\\mathbb S}_{10}$. The closure under multiplication for any $g\\in {\\mathbb D}_{01}^{[2]},~ g'\\in\n{\\mathbb D}_{10}^{[2]}$ implies that $gg'\\in {\\mathbb D}_{11}^{[2]}$ is alphabetically presented.\n\\par\nAs discussed in Section {\\bf 5}, a ${\\mathbb Z}_2\\times{\\mathbb Z}_2$-graded  superdivision algebra can be associated with its ${\\mathbb Z}_2$-graded superdivision algebra projections ${\\mathbb S}_{01},~{\\mathbb S}_{10},~{\\mathbb S}_{11}$. \n\n\\par\n\n~\n\\par\n~\\par\n\\par\n\\par {\\Large{\\bf Acknowledgments}}\n{}~\\par{}~\\par\n\n The work was supported by CNPq (PQ grant 308095\/2017-0).\n  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nThe recent finding of fullerene-like empty cage structure\\cite{B40} of B$_{40}$ using a \ncombined experimental and computational study as well as the occurrence of quasi-planar \nstructures for many boron clusters has spurred a lot of interest in \nexperimental and theoretical studies of boron clusters to understand their growth behavior and stability. \nAlso a layer of boron, called $\\alpha$-sheet as well as quasi-planar structures of clusters are stabilized with hexagonal holes in otherwise triangular structures. These results are exciting because the possibility of the formation of carbon-like nanotubular or graphene-like planar structures based on a hexagonal lattice was ruled out due to the deficiency of electrons in boron which is known to favor three center bonding. It has been suggested that the stability of $\\alpha$-sheet as well as quasi-planar structures of clusters is due to a mixture of triangular and hexagonal network that is energetically favorable over only a triangular network which has excess of electrons. Several studies on small boron clusters having up to around 36 atoms show that \nthey have planar or quasi-planar or tubular structures as their ground state\\cite{Zhai2003a,Zhai2003b,Kiran2005,Lau2005,Huang2010,Popov2013,Oger2007} with the exception of B$_{14}$ for which a cage-like structure has been proposed\\cite{Cheng2012}. A double-ring tubular (DRT) structure has been suggested for neutral B$_{20}$\\cite{Kiran2005} and B$_{24}$\\cite{Chacko2003} while a cage structure has been suggested for B$_{38}$\\cite{B38} using \nparticle swarm algorithm combined with density functional theory (DFT) calculation. A recent study by Chen {\\it et al.} has also identified a cage structure with axially chiral feature for B$_{39}$\\cite{B39} and another recent theoretical \nstudy\\cite{b28-zhao2015} on neutral B$_{28}$ suggests a filled cage structure to be the lowest in energy while a planar isomer is nearly degenerate. Also, although for neutral B$_{40}$ a fullerene-like cage structure is the most stable one, a planar isomer is favoured for B$_{40}$ anion and therefore a quasi-planar isomer competes in energy for B$_{40}$. A similar situation may arise for other sizes as well that charged clusters have another competing structure. We ask the question if smaller cages of boron can be stabilized such as by metal (M) atom encapsulation. \n\nStudies on a large number of bulk boron compounds show that an empty center icosahedral cage of B$_{12}$ is the major building block of their structures. However, an isolated B$_{12}$ icosahedral cage is not stable due to the \navailability of 36 valence electrons, while 26 electrons are required by Wade's rules\\cite{Wade1971} to stabilize this cage. One way to stabilize the icosahedral cage is to attach ligands such as hydrogen atoms exohedrally. \nAs an example, cage-like borane structure B$_{12}$H$_{12}$$^{2-}$ is stabilized by 26 valence electrons excluding those in the B-H bonds.\\cite{Lipscomb1969} Another way to form cage structures is to dope endohedrally an M atom that may interact with boron atoms strongly and stabilize \ncage structures. Such a strategy has been successfully used to stabilize non-carbon cage structures such as those of silicon and other elements.\\cite{Kumar2001,Kumar2002,Kumar2002-2,Kumar2002-3,Kumar2003-1,Kumar2003-2,Kumar2003-3,Kumar2007} In particular, exceptional stability has been suggested for Zr@Si$_{16}$ \nfullerene and Ti@Si$_{16}$ Frank-Kasper polyhedron structures\\cite{Kumar2001} that have been subsequently realized in laboratory and even assemblies have been formed.\\cite{nakajima} The size of a B$_{12}$ icosahedron is too small to encapsulate an M atom. As boron atom is smaller \nin size compared to a silicon atom, a possibility to form boron cages may lie around the size of about 20 atoms. A recent independent study has indeed proposed stabilization \nof M doped cages for B$_{24}$ by encapsulation of Mo and W, whereas for Cr a less symmetric configuration has been reported for this size.\\cite{Lv2015} Disk-like or wheel-shaped structures have also been predicted for boron clusters \nhaving up to about 11 boron atoms by doping a transition M  \natom\\cite{Romanescu2011,Li2011,Romanescu2012,Galeev2012,Romanescu2013-1,Romanescu2013-2,Zhao2014,Zhao2015} as well as other elements.\\cite{saha2016} The stability of some disk-shaped clusters has been correlated with electronic shell closing at 12 valence electrons.\\cite{saha2016} Further, recently bowl-shaped\\cite{SDLi2006,Boyukata2011,Gu2012,Pham2015,saha2017} and drum-shaped\\cite{laisheng2015,saha2017} structures of boron have been predicted to be stabilized by M atom dopants such as M@B$_{14}$ drum with M = Cr, Fe, Co, and Ni, and also with 16 boron atoms such as Co@B$_{16}^-$. \n\nWe have studied M atom encapsulated boron clusters in the size range of 18 to 24 atoms in order to find the smallest cage of free boron clusters besides the drum structures. Note that in this size range, pure boron clusters have quasi-planar or tubular structures. As the bonding in boron is stronger compared with silicon, we considered strongly interacting transition M atoms with half-filled $d$ states such as Cr, Mo, and W to explore the possibility of stabilizing boron cages in this size range. Based on our systematic study, we predict fullerene-like cages Cr@B$_{20}$, M@B$_{22}$ and M@B$_{24}$ with M = Cr, Mo, and W in which one M atom is encapsulated in the cage. Also we find bicapped drum structures for Cr@B$_{18}$ and M@B$_{20}$ (M = Mo and W) and a drum structure for M@B$_{18}$, M = Mo and W. The stability of these structures has been studied by analysing the molecular orbitals (MOs) as well as the electronic charge density. Our results suggest that Mo and W are well suited to produce endohedrally doped novel cage structures of boron as the doping of M atom has the effect of increasing the binding energy significantly. The doping of Cr atom also leads to cage formation, but the binding energy of the doped clusters has similar values as for the pure boron clusters. We also present results for IR and Raman spectra of the neutral and cation clusters as well as for the electronic structure of anion clusters that would help to identify the atomic structure of these doped clusters when experimental data may become available.   \n\n\n\\section{COMPUTATIONAL DETAILS}\nWe used generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE)\\cite{PBE} for the exchange-correlation functional and projector augmented wave (PAW) pseudopotential plane wave method\\cite{blochl,kresse} in Vienna $\\it{ab}$ $\\it{initio}$ simulation package (VASP)\\cite{vasp} to explore several isomers for the doped boron clusters. The calculations were considered to be converged when the absolute value of the force on each ion was less than 0.005 eV\/$\\AA$ with a convergence in the total energy of 10$^{-5}$ eV. Further calculations were performed for the lowest energy isomers of the neutral clusters using Gaussian09 code\\cite{g09} and PBE0 functional. Also, we have calculated the vibrational modes of cation and neutral clusters using Gaussian09 code, and in almost all cases we obtained real frequencies suggesting the dynamical stability of the obtained atomic structures. For these calculations we used B3PW91 hybrid exchange-correlation functional as well as PBE0 functional and 6-311+G basis set\\cite{Wachters1970,Hay1977} for Cr doping and LANL2DZ basis set\\cite{Hay1985,Hay1985a} for Mo and W while 6-311+G basis set has been used for B atoms in all the cases. The atomic structures were optimized and used to calculate IR and Raman spectra for the neutral and cation clusters. Calculations using PBE0 exchange-correlation functional on the lowest energy isomers of the pure and M doped boron clusters also suggest stabilization of boron cages with M encapsulation. The bonding characteristics have been studied by performing adaptive natural density partitioning (AdNDP)\\cite{AdNDP2008} analysis at the PBE level of the theory and using the same basis sets in the Gaussian09 code and also from the Laplacian L = -(1\/4)$\\Delta^2 \\rho(\\bf{r})$, of the electronic charge density $\\rho(\\bf{r})$ using AIMALL.\\cite{AIMALL} We also calculated the electron localization function (ELF) using the charge density distribution obtained from VASP for the lowest energy isomers of the neutrals and the electron localization-delocalization index using AIMLDM script.\\cite{AIMLDM} Further calculations have been done on cation and anion clusters using PBE0 functional in Gaussian09 code. Some results are also included for the isoelectronic anions of V, Nb, and Ta doped clusters. The MOs have been analysed using the Gaussian 09 code. We used VESTA 3,\\cite{VESTA} Molekel 5.4,\\cite{Molekel}, XCrysden 1.5 \\cite{XCrysden}, AIMALL, and Gaussview\\cite{Gaussview} for visualization. In all cases of charged and neutral clusters the atomic structures were again optimized when using Gaussian09 code.\n\n\n\\section{RESULTS and DISCUSSION}\n\n\\subsection{Atomic structures and binding energies}\n\nWe studied doping of an M atom (M = Cr, Mo, and W) in several different structures of boron clusters in the size range of B$_{18}$ to B$_{24}$ in the hope of finding the smallest cage of boron. To begin with we doped an Mo atom inside a 16-atom Frank-Kasper polyhedron of boron because all the faces in this structure are triangular and boron has preference to form 3-center bonds as in an icosahedron. But upon optimization it ends up in an open structure indicating that the formation of a boron cage for this size is unlikely with endohedral doping of an Mo atom. However, recently drum-shaped clusters of 16 boron atoms have been obtained with the doping of a Co atom, while bowl-shaped clusters have been shown\\cite{saha2017} to be formed for many of the 3d, 4d, and 5d transition M atoms. Also 14-atom drum-shaped boron clusters have been shown\\cite{saha2017} to be favored by doping of some 3d transition M atoms such as Cr, Fe, Co, and Ni. These results suggest that a cage structure, besides drum-shaped clusters, is unlikely to be favorable for a 16-atom boron cluster with these M atoms. \n\nNext we focussed our attention on exploring cage formation with 18 and 20 boron atoms. We added one Mo (and also Cr and W) atom on the planar B$_{18}$ isomer and at the center of the DRT structure of B$_{20}$. Upon optimization, the planar B$_{18}$ structure transforms into a bowl-shaped isomer as shown in Figure S1 (see isomer III and other similar bowl-shaped structures IV and VIII) in Supplementary Information. This clearly showed that the doping of these M atoms tends to form a cage of boron. We rearranged the atoms and tried to construct a cage-like structure of boron but upon re-optimization it becomes again a bowl-shaped open structure indicating that more boron atoms are needed to complete a cage. Some such open as well as cage type structures are shown in Fig. S1 in Supplementary Information, but all of these lie higher in energy than the lowest energy isomer. Figure S1(I) in Supplementary Information and Fig. \\ref{fig:B18-B24}(a) show the lowest energy isomer for CrB$_{18}$ which is a bicapped 16-atom drum-shaped structure. On the other hand for MoB$_{18}$ and WB$_{18}$, a tubular drum-shaped structure shown in Fig. \\ref{fig:B18-B24}(b) and (c), respectively, and also in Fig. S1(II) in Supplementary Information, has the lowest energy, while the bicapped drum-shaped isomer lies 0.36 eV and 0.42 eV higher in energy for Mo and W, respectively. On the other hand the drum-shaped isomer of Cr@B$_{18}$ lies 1.25 eV higher in energy compared with the lowest energy isomer. In the case of MB$_{20}$ (M = Cr, Mo, and W), we attempted many structures, both open and cage type, and these are shown in Fig. S2 in Supplementary Information. We added two boron atoms in a symmetric fashion to the bowl-shaped MoB$_{18}$ cluster and optimized the atomic structure. The resulting structure was found to be lower in energy than the one obtained from the doping of an Mo atom in B$_{20}$ DRT structure. However, the optimized structure is still open, again suggesting that 20 boron atoms are not sufficient to form a cage with Mo encapsulation. Among the different structures we explored, it is found that a bicapped 18-atom drum-shaped isomer (Fig. \\ref{fig:B18-B24}(e) and (f)) has the lowest energy for Mo and W doping. However, as we shall show below, Cr is able to form a cage (Fig. \\ref{fig:B18-B24}(d)) with 20 boron atoms, and it is the smallest cage of boron that we have obtained. \n\nIt is clear from the above that there is room to accommodate a few more B atoms on the bicapped drum of Mo@B$_{20}$ to form a cage. Therefore, we added two more B atoms on the other side of the drum so that the two capping dimers form a cross leading to Mo@B$_{22}$ cluster. Also we added four boron atoms to MoB$_{20}$ open structure to form a Mo@B$_{24}$ cage structure in the hope of finding a symmetric isomer, as intuitively a 22-atom cage structure is unlikely to be very symmetric. The optimized structures show that the Mo@B$_{22}$ isomer converges with the same local structure while the optimized atomic structure of Mo@B$_{24}$ had a cage form, but it was slightly lower in symmetry. We changed the positions of two boron atoms in this Mo@B$_{24}$ structure in order to create a symmetric structure and re-optimized it. In two such steps we obtained a very symmetric cage structure of Mo@B$_{24}$ as shown in Figure \\ref{fig:B18-B24} (k). This is about 2 eV lower in energy compared with the initially optimized structure (without rearranging boron atoms). The pathway starting from B$_{18}$ to the lowest energy isomer of Mo@B$_{24}$ is shown in Figure \\ref{fig:B24-path}. The lowest energy structure for Mo@B$_{24}$ has (D$_{3h}$) symmetry and a large highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) gap (GGA) of 2.46 eV indicating its high stability. We tried several other isomers for Mo@B$_{24}$ but all of them lie higher in energy than the D$_{3h}$ cage isomer, except in one case where we considered an Mo atom doped at the center of a truncated cube and it converged to the same lowest energy D$_{3h}$ isomer. This gave us further support that our structure may be the lowest in energy. The pathway is indicated in Figure \\ref{fig:B20-22-24}. Interestingly the same isomer has been also independently obtained for Mo@B$_{24}$ by Lv et al.\\cite{Lv2015} using particle swarm optimization algorithm implemented in the CALYPSO package, which involved calculations of a few thousand structures. In these calculations hybrid PBE0 method has been used for the structure optimization with 611G* basis set in Gaussian 09 package. This gave us further confidence that our structure may indeed be the lowest energy structure for Mo@B$_{24}$. Further calculations on W doping led to the same isomer to be of the lowest energy. The optimized structures of some of the isomers we attempted for M@B$_{22}$ and M@B$_{24}$ (M = Cr, Mo, and W) are shown in Figs. S3 and S4, respectively in Supplementary Information. It can be noted that for Cr doping many isomers lie in a narrow window of energy. \n\nAfter obtaining neutral Mo@B$_{24}$ and W@B$_{24}$ doped boron clusters, both of which are almost identical, we asked ourselves the question if we can stabilize a smaller cage of boron, and in particular if a 24 boron-atom cage is the smallest cage for Mo and W. It is to be noted that when Cr is doped in the lowest energy isomer of Mo@B$_{24}$, the optimized structure is a somewhat distorted cage (Fig. \\ref{fig:B18-B24} (j)) in which the Cr atom does not interact with all the boron atoms optimally. This is because the size of a Cr atom is not optimal for the B$_{24}$ cage as it is slightly smaller than Mo and W atoms. The natural question we further asked was: what is the smallest cage stabilized by Cr. To address this question we further explored different isomers for the size of 22 boron atoms, as in the 24-atom cage we can remove two B atoms and still keep the cage structure, and also some cage structures for 20 boron atoms. Below we first discuss the formation of a B$_{20}$ cage and then the B$_{22}$ cage.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.85\\linewidth]{Fig-1.pdf}\n\\caption{Atomic structures of the lowest energy configurations of clusters having 18, 20, 22, and 24 boron atoms doped with a Cr, Mo, and W atom. In each case the symmetry is also given.}\n\\label{fig:B18-B24} \n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{Fig-2.pdf}\n\\caption{Atomic structures showing the pathway from (a) planar B$_{18}$ to (g) D$_{3h}$ Mo@B$_{24}$ cage clusters. (b) is the optimized structure when an Mo atom is placed above the planar isomer (a). (c) is slightly modified structure obtained from (b) by adjusting some atoms in an effort to make a cage. In going from (c) to (d) two boron atoms are added and then four boron atoms are added in the configuration (d). The resulting structure (e) is not symmetric. (f) and (g) are obtained by adjusting B atoms in (e) in order to close the cage and make the cluster symmetric.}\n\\label{fig:B24-path} \n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{Fig-3.pdf}\n\\caption{Optimized atomic structures for (a) D$_3$ Cr@B$_{20}$, (b) D$_2$ Mo@B$_{22}$, and (c) D$_{3h}$ Mo@B$_{24}$ clusters from their respective starting atomic configurations, namely 18-atom three-ring tubular structure which was capped with two B atoms, a 20-atom dodecahedron which was capped with two B atoms, and 24-atom truncated cube, respectively. In each case Mo atom was placed inside the cage.}  \n\\label{fig:B20-22-24} \n\\end{figure}\n\nIn order to study the doping of a Cr, Mo, and W atom in B$_{20}$ cluster, we tried several different initial structures including C$_{20}$ type B$_{20}$ fullerene with all pentagonal faces, quasi-planar, and tubular structures. All of them were optimized and also atoms were rearranged to possibly find symmetric lowest energy structures for all the three M atom dopants. The optimized structures are shown in Fig. S2 in Supplementary Information. First note that for Cr, the CrB$_{18}$ structure remains open and forms a bicapped-drum structure as discussed above, suggesting that eighteen atoms of boron are not sufficient to form a cage. The lowest energy isomers for MB$_{18}$, (M = Cr, Mo, and W) show (Figure \\ref{fig:B18-B24}(a)-(c)) that the structure of CrB$_{18}$ is different from the one for the MoB$_{18}$ and WB$_{18}$ clusters. CrB$_{18}$ structure has two boron atoms capped on to a B$_{16}$ DRT structure forming a bicapped-drum structure with $C_{2v}$ symmetry. This structure has two pentagons arranged in base sharing fashion. For Cr@B$_{20}$, we obtained a $D_3$ symmetric cage structure as shown in Figure \\ref{fig:B18-B24}(d) to be of the lowest energy. This smallest cage in our calculations has three empty boron hexagons that are inter-linked through three 2-atom chains to two capped boron hexagons. However, for MoB$_{20}$ and WB$_{20}$, a bicapped drum structure with two boron atoms capping an 18-atom boron DRT with $C_s$ symmetry and M atom at the centre has the lowest energy. This structure has one pentagon and one hexagon arranged in base sharing fashion. For the CrB$_{20}$ case, the bicapped drum isomer with $C_s$ symmetry is 0.32 eV higher in energy than the cage structure. Also the $D_3$ fullerene structure for MoB$_{20}$ and WB$_{20}$ cases is, respectively, 1.79 eV and 2.06 eV higher in energy than the lowest energy bicapped drum ($C_s$) structure. We also obtained an unsymmetric cage ($C_1$) structure for Mo@B$_{20}$ and W@B$_{20}$ (see isomer X in Fig. S2 in Supplementary Information) which is 0.88 eV and 1.02 eV, respectively, higher in energy than the lowest energy structure. Several other isomers that we tried are shown in Fig. S2 in Supplementary Information using PBE in VASP. These results suggest that for Mo and W, the cage need to have more than 20 boron atoms.\n\nIn order to explore a cage of B$_{22}$, we tried several configurations of neutral Mo@B$_{22}$. The lowest energy configuration has $D_2$ symmetry as shown in Figure \\ref{fig:B18-B24}(h). This is obtained by adding two boron atoms in the $C_1$ isomer of Mo@B$_{20}$ with slight rearrangement. This structure has four heptagons that are interlinked through two 2-atom chains and three boron atoms. Another calculation starting from C$_{20}$ fullerene for B$_{20}$ and capping two opposite pentagons with two boron atoms converged to the same $D_2$ structure after optimization for Mo@B$_{22}$. This is shown in Fig. \\ref{fig:B20-22-24}(b). We also tried several isomers for Cr and W doping and obtained the same $D_2$ structure for W@B$_{22}$ to be of the lowest energy as shown in Figs. \\ref{fig:B18-B24}(i), but for Cr@B$_{22}$, while in VASP calculation, the same isomer is of the lowest energy, a slightly different isomer shown in Fig.\\ref{fig:B18-B24}(g) is obtained from Gaussian calculation as the lowest energy one. This structure also shows that less than 22 boron atoms are better to have an optimal cage with Cr doping. Interestingly when this isomer is reoptimized in VASP, it reverts to the same structure as before in VASP. Several isomers including the lowest energy isomers for Cr, Mo, and W doped B$_{22}$ are shown in Fig. S3 in Supplementary Information. As can be seen, there are a few isomers for Cr doping that lie close in energy within about 0.30 eV of each other. The isomer in which the M atom interacts with a quasi-planar isomer of B$_{22}$ (Fig. S3 XVI in Supplementary Information) lies more than 2 eV higher in energy than the lowest energy isomer.\n\nFor the case of the B$_{24}$ cage, we further explored different structures for M@B$_{24}$ with M = Cr, Mo, and W. These are shown in Fig. S4 in Supplementary Information. The lowest energy configuration for Cr@B$_{24}$ has C$_1$ symmetry as shown in Fig. \\ref{fig:B18-B24}(j). The isomer with D$_{3h}$ configuration for Cr lies 0.25 eV higher in energy. Although our result of the lowest energy structure of Cr@B$_{24}$ is in agreement with that reported by Lv {\\it et al.}\\cite{Lv2015}, they reported this structure to have quintet spin-multiplicity. But in our calculations a singlet configuration has the lowest energy. The triplet and quintet configurations are, respectively, 0.97 eV and 2.02 eV higher in energy. As stated earlier, the lowest energy isomers for Mo@B$_{24}$ and W@B$_{24}$ are similar and are shown in Figs. \\ref{fig:B18-B24}(k) and (l), respectively.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\linewidth]{Fig-4.pdf}\n\\caption{Calculated binding energies for the lowest energy isomers.}\n\\label{fig:BE} \n\\end{figure}\n\n\nIt is to be noted that neutral pure boron clusters, B$_n$ with $n$ = 18-24, prefer quasi-planar or tubular structures in their ground states and these become drum-shaped or form cages after doping of a transition M atom. We calculated the energy of the drums\/cages by removing the M atom and keeping the atomic positions of the boron atoms the same as in the optimized doped cluster. \nIt is seen that the empty cage structures of B$_n$ ($n$ = 20, 22, and 24) lie 1.50-3.50 eV higher in energy than their respective ground state isomers within PBE. Therefore, the interaction of the M atom due to endohedral doping lowers the energy of the cage significantly. The doping energy has been calculated from $\\Delta$$E$ = $E(M atom) + E(B_n cage) - E(M@B_n)$ and the values are given in Table 1. In general large values ($>$ 8 eV) of $\\Delta$$E$ are obtained for W doping and for Mo in B$_{22}$ cage. \nThis also leads to significant enhancement in the binding energy of the doped boron clusters compared with that of the undoped clusters. The binding energy is defined as the energy per atom of the cluster with respect to the energy of the free atoms, [$nE(B) + E(M) - E(M@B_n)]\/(n+1)$, and it is shown in Fig. \\ref{fig:BE} using PBE in VASP and PBE0 in Gaussian 09 program for the lowest energy isomers of all the sizes and with different dopants. Here $E(B)$, $E(M)$, and $E(M@B_n)$ are the energies of B atom, M atom, and the doped cluster, respectively. The overall trend of the binding energy in both the methods is similar, but the values are slightly lower when PBE0 is used. These results also show that the binding energy increases in going from Cr doping to Mo and then to W, indicating that W doping is the strongest among the three dopants. There is a slight increase in the binding energy in going from 18-atom to 20-atom boron clusters for all the dopants. For Cr doping the binding energy remains nearly the same in going from Cr@B$_{20}$ to Cr@B$_{22}$. However, it slightly increases to 5.59 eV for Cr@B$_{24}$ cluster within PBE. In the case of Mo and W doping the trend shows that the increase in the binding energy in going from Mo@B$_{20}$ to Mo@B$_{22}$ is more than the increase in going from Mo@B$_{22}$ to Mo@B$_{24}$. Similarly, in the case of W doping the binding energy increases in going from W@B$_{20}$ to W@B$_{22}$, but it remains nearly the same for W@B$_{22}$ and W@B$_{24}$ within PBE. Our results suggest that both 20 and 22 boron atom clusters doped with these M atoms are likely to be abundant, as the second derivative of energy ($\\Delta$$E$ = 2$E$(M@B$_n$) - $E$(M@B$_{n-2}$) - $E$(M@B$_{n+2}$))\/4 is negative for both these sizes. We have also plotted the values of the binding energy for the pure boron clusters with 18, 20, 22, and 24 atoms. These results show that the doping of Mo and W atoms leads to a significant increase in the binding energy of the boron clusters and therefore, Mo and W are favorable to form endohedral cage structures of boron. However, in the case of Cr doping, the binding energy is either nearly the same or marginally higher (in the case of CrB$_{18}$) or lower (Cr@B$_{24}$) than the value for the corresponding pure boron cluster within PBE in VASP (see Table 1). But using PBE0 in Gaussian 09 program the binding energy of CrB$_{n}$ clusters is slightly lower than the values for the pure boron clusters except for the Cr@B$_{18}$ case as shown in Fig. \\ref{fig:BE}. Also we find that within PBE0, there is a large HOMO-LUMO gap for Cr@B$_{18}$. Therefore, we conclude that the stabilization of boron cages with Cr doping is less than those of Mo and W doped clusters, and the magnetic moments on the M atom is quenched. Using PBE0 the second derivative of energy ($\\Delta E$) for Cr@B$_{20}$ is -0.165 eV, while for MoB$_{20}$ and WB$_{20}$ doping it is -0.018 eV and -0.072 eV, respectively. \nSo M@B$_{20}$ is weakly magic for Mo and intermediate for W doping. Also for M@B$_{22}$, it is -0.015 eV, -0.043 eV, -0.050 eV, respectively for M = Cr, Mo, and W. Accordingly Mo@B$_{22}$ and W@B$_{22}$ are likely to be more abundant than Cr@B$_{22}$. Therefore, MoB$_{22}$ is magic within PBE0 while W@B$_{20}$ is likely to be more abundant than W@B$_{22}$. In the following we discuss Cr as well as Mo and W doping to understand the cage stability and bonding characteristics in the doped clusters.\n\n\\begin{table}\n\\begin{center}\n\\caption{Calculated embedding energy (E$_n$(M)) of the M atom in n-atom boron cage, BE, HOMO-LUMO gap (E$_g$), vertical electron affinity (VEA) and vertical ionization potential (VIP) for the lowest energy isomers of M@B$_n$ (n = 18, 20, 22, and 24; M = Cr, Mo, W) clusters. VEA and VIP are calculated using Gaussian 09 program. For comparison, the binding energy and HOMO-LUMO gap calculated within PBE0 are also given. Results have also been given for the lowest energy isomers of pure B$_n$ clusters ($n$ = 18, 20, 22, and 24).}\n\\begin{tabular}{|c|ccc|ccccc|}\n\\hline\n&\\multicolumn{3}{c|}{PBE} &\\multicolumn{5}{c|}{PBE0} \\\\\n\\hline\nCluster & E$_n$(M) &BE per atom&E$_g$ &  E$_n$(M) & BE per atom&E$_g$ &VEA&VIP\\\\\n        & (eV)&(eV)&(eV)& (eV)&(eV)& (eV)&(eV)&(eV) \\\\\n\\hline\nB$_{18}$& - & 5.48 & 0.47 & -&4.93&1.39&-&-\\\\\nB$_{20}$& - & 5.56 & 1.36 & -&5.05&2.72&-&-\\\\\nB$_{22}$& - & 5.57 & 0.29 & -&5.06&1.35&-&-\\\\\nB$_{24}$& - & 5.63 & 1.30 & -&5.10&2.18&-&-\\\\\n\nCr@B$_{18}$& 6.67 & 5.55 & 1.24 & 5.65 & 4.97 &2.73  &2.72 &7.63 \\\\\nMo@B$_{18}$& 8.28 & 5.63 & 0.40 & 7.53& 5.07 & 1.71 &3.33  &7.37\\\\\nW@B$_{18}$& 9.61 & 5.70 & 0.47  & 9.51& 5.17 & 1.71 & 3.37 &7.44\\\\\nCr@B$_{20}$& 5.62 & 5.57 & 2.14 &  4.28& 5.02 & 3.85 &1.88 & 7.94\\\\\nMo@B$_{20}$& 7.76 & 5.67 &1.76 & 6.54& 5.12 & 3.21&2.06 &6.99\\\\\nW@B$_{20}$& 9.22 & 5.74 &1.79  & 8.72& 5.23 & 3.31 &2.00 &7.06 \\\\\nCr@B$_{22}$& 5.29 & 5.57 & 2.26 &  3.99& 5.03 & 3.78 &2.05 & 7.05\\\\\nMo@B$_{22}$& 8.25 & 5.70 & 2.51 & 7.59& 5.17 & 3.87&1.97&8.01\\\\\nW@B$_{22}$& 9.67 & 5.76 & 2.50 &  9.76& 5.26 & 3.85&1.99&8.03\\\\\nCr@B$_{24}$& 4.59 & 5.59 & 0.89 &  3.32& 5.03 & 2.35 &3.44&7.11\\\\\nMo@B$_{24}$& 7.33 & 5.71 & 2.46 &  7.42& 5.20 & 4.40 &1.59&7.86\\\\\nW@B$_{24}$& 8.69 & 5.76 & 2.45 &  9.55& 5.28 & 4.37&1.56&7.89\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nFrom Table 1 the calculated HOMO-LUMO gaps (E$_g$) within PBE are 1.24 eV, 0.40 eV, and 0.47 eV for MB$_{18}$, M = Cr, Mo, and W, respectively. Note that GGA underestimates E$_g$ and the values obtained using PBE0 are much larger. \nThe values of E$_g$ for these cases are, respectively, 2.72 eV, 1.71 eV and 1.71 eV when PBE0 functional is used. Here note the large value for the case of Cr doping. For MB$_{20}$, the HOMO-LUMO gaps increase to 2.14 eV, 1.76 eV, and 1.79 eV, respectively with M = Cr, Mo, and W, and further to 2.26 eV, 2.51 eV, and 2.50 eV for MB$_{22}$ within PBE. These large values of E$_g$ suggest the stability of the cage structures. However, for Cr@B$_{24}$ the HOMO-LUMO gap decreases to 0.89 eV as the cage becomes less symmetric and Cr is not bonded well with all the B atoms. Mo@B$_{24}$ and W@B$_{24}$ clusters also have large E$_g$ (2.46 eV and 2.45 eV, respectively), but the values are slightly lower than for Mo@B$_{22}$ and W@B$_{22}$ cages. The calculated E$_g$ for the M doped B$_{22}$ and B$_{24}$ cage structures using PBE0 are again much larger than the values obtained within PBE, and these are given in Table 1. The largest value (4.40 eV) of E$_g$ within PBE0 is obtained for Mo@B$_{24}$ and a quite similar value for W@B$_{24}$. The largest value (5.28 eV\/atom) of the binding energy within PBE0 is obtained for W@B$_{24}$ and a slightly smaller value (5.26 eV\/atom) for W@B$_{22}$. This is in contrast to the values of 5.06 eV\/atom and 5.10 eV\/atom for the binding energies of elemental boron clusters B$_{22}$ and B$_{24}$, respectively, within PBE0. Therefore, M doping enhances the stability of boron clusters and leads to the formation of the cage structures of boron for smaller sizes compared with pure boron clusters.   \n\n\\subsection{Bonding characteristics}\n\nIn order to understand the bonding character in these structures, we noted that the shortest B-B bond lengths for Cr@B$_{18}$, Mo@B$_{18}$, and W@B$_{18}$ are 1.61 {\\AA}, 1.58 {\\AA}, and 1.58 {\\AA} respectively, whereas B-Cr, B-Mo, and B-W bond lengths are in the range of 2.12-2.36 {\\AA}, 2.32-2.63 {\\AA}, and 2.33-2.60 {\\AA}, respectively using PBE. The shortest B-B bonds are the two-center (2c) sigma bonds. As the cluster size increases, we find that for Cr@B$_{20}$ the B-B and B-Cr bond distances lie in the range of 1.60-1.83 and 2.08-2.30 {\\AA}, respectively while for Mo@B$_{20}$ and W@B$_{20}$ the shortest B-B bond length is 1.57 {\\AA}. The B-Mo and B-W bond lengths are in the range of 2.13-2.57 {\\AA} and 2.14-2.58 {\\AA}, respectively. These Mo-B and W-B bond lengths are shorter than for Mo@B$_{18}$ and W@B$_{18}$, respectively. The shorter bonds are formed by the capping boron atoms with the M atom. For the M doped B$_{22}$ cage, some of the B-B bonds are shorter than those in the M doped B$_{20}$ cage. There are two shortest B-B bonds for the B$_{22}$ cage with bond length 1.56 {\\AA} for Cr encapsulation, and 1.57 {\\AA} for Mo and W encapsulation, but the M-boron bond lengths are slightly elongated and lie in the range of 2.31-2.54 {\\AA} for Cr encapsulation and 2.38-2.59 {\\AA} for Mo and W encapsulation as compared to those in the respective doped B$_{20}$ cages. In the B$_{24}$ cage the shortest B-B bond length increases to 1.64 {\\AA} for Mo and W encapsulation. Also some of the M-boron bond lengths are longer and lie in the range of 2.43-2.55 {\\AA} for Mo encapsulation and 2.44-2.55 {\\AA} for W encapsulation. For Cr encapsulation in B$_{24}$, the B-B bond lengths are in the range of 1.54-1.84 {\\AA} while B-Cr bond lengths have wide variation and are in the range of 2.18-2.66 {\\AA}. This is becuase the Cr atom is not at the center of the cage. \n\nWe further calculated the total charge density isosurfaces, ELF, and molecular orbitals for the doped clusters. We also performed AdNDP analysis on some of the clusters. Details of these results are presented in Supplementary Information and in Figs. \\ref{fig:MB20-22-24-elf} and \\ref{fig:B20Cr-bonding}. Earlier studies on pure boron clusters have suggested the existence of multi-center bonds namely 2c, 3c, 4c, 6c, and 7c bonds using AdNDP analysis\\cite{AdNDP2008}. However, in our experience the AdNDP analysis is not straightforward as there is not a unique way to identify multi-center bonds. We were greatly assisted by the analysis of the charge density, molecular orbitals, and ELF. In an earlier study\\cite{B84} on B$_{84}$ cluster, the total charge density and ELF were used as tools to understand 2c and 3c bonds in boron clusters along with the AdNDP analysis. This was helpful in further calculating the multi-center bonds with AdNDP. We followed a similar strategy for some of the M doped boron clusters studied here. Further, we have calculated Laplacian of the electron density and bond- as well as ring-critical points to analyse the bonding character.  \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{Fig-5.pdf}\n\\caption{Isosurfaces of ELF for (a) Cr@B$_{20}$, (b) Mo@B$_{22}$, and (c) Mo@B$_{24}$. Two orientations are shown in each case. In (a) the lobes shown by purple color have strong localization of charge with high value of ELF (0.94) representing 2c-2e bonds. As the value of the iso-surface is decreased (0.84), these lobes become bigger and new lobes appear. These other lobes shown with yellowish color are 3c or higher center $\\sigma$ bonds. In (b) there are 2 purple color lobes with ELF = 0.96 (2c) and sixteen yellowish color lobes with ELF = 0.88 (3c). In (c) there are six 2c (purple color) and eighteen 3c (yellowish) bonds.}\n\\label{fig:MB20-22-24-elf} \n\\end{figure}\n\n\nFigure \\ref{fig:MB20-22-24-elf} shows ELF for the smallest cage Cr@B$_{20}$ and also for Mo@B$_{22}$ and Mo@B$_{24}$ clusters. For Cr@B$_{20}$ the bond length analysis showed that there are six short B-B bonds with 1.60 {\\AA} length. These bonds are along the three 2-atom chains joining the three empty heptagons and can be detected in the charge density and ELF plots at a high value of the isosurface (see Fig. S5 in Supplementary Information). A high value of ELF shows strong localization of charge as in covalent bonds. These six lobes are indicated by purple color in Fig. \\ref{fig:MB20-22-24-elf}(a) at the ELF value of 0.94. We identify these bonds as six 2c-2e bonds. Further decreasing the ELF value to 0.84, these six lobes become bigger (shown by yellowish color around the purple color lobes) and twelve more lobes can be seen. These twelve additional lobes (yellowish in color in Fig. \\ref{fig:MB20-22-24-elf}(a)) represent twelve 3c $\\sigma$ bonds. This identification was also supported by the AdNDP analysis as we shall discuss below. Figs. \\ref{fig:MB20-22-24-elf}(b) and (c) show ELF plots for Mo@B$_{22}$ and Mo@B$_{24}$, respectively. For Mo@B$_{22}$, the two lobes shown by purple color are obtained at the ELF value of 0.96 and they represent two 2c bonds with the bond length of 1.57 {\\AA}. There are sixteen more lobes shown by yellowish color at the ELF value of 0.88. Similar to the case of the B$_{20}$ cage, they represent sixteen 3c bonds. For Mo@B$_{24}$ the analysis of the charge density and ELF shows that this cluster has six 2c $\\sigma$-bonds (purple color lobes in ELF) and eighteen 3c $\\sigma$-bonds. Further analysis using AdNDP shows six 5c $\\sigma$-bonds (see Fig. \\ref{fig:MB20-22-24-elf}(c)). The remaining eighteen valence electrons are involved in $\\pi$-bonding as well as bonding with the M atom, and can be detected in the AdNDP analysis\\cite{Lv2015}. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.4\\linewidth]{Fig-6.pdf}\n\\caption{Different multi-center bonds in the lowest energy isomer for Cr@B$_{20}$ as obtained from the AdNDP analysis.}\n\\label{fig:B20Cr-bonding} \n\\end{figure*}\n\n\nWe consider Cr@B$_{20}$ cage cluster as the smallest cage, and its AdNDP analysis shows (Fig. \\ref{fig:B20Cr-bonding} (a)) that there are six 2c-2e bonds, as also concluded from ELF. Further, there are twelve 3c bonds, six are along the three 2-atom chains, and the remaining six are placed alternatively on the three edges of each capped hexagon as shown in Figure \\ref{fig:B20Cr-bonding}(b). These bonds were also detected in the total charge density and ELF analysis as discussed above. In addition to these eighteen $\\sigma$-bonds there are four 5c-2e $\\sigma$-bonds and two 6c-2e $\\sigma$-bonds as shown in Figure \\ref{fig:B20Cr-bonding}(c) and (d), respectively. These bonds connect the atoms on the hexagons to the atoms on the three 2-atoms chains. All these 24 $\\sigma$-bonds involve bonding among boron atoms. There are six 5c-2e $\\pi$-bonds, each connecting three B atoms on one capped hexagon to one B atom on the edge through the Cr atom. There are also three 6c-2e $\\pi$-bonds along the three 2-atom chains, each involving six B atoms on the chain and the Cr atom. It is interesting to see that there are altogether nine $\\pi$-bonds which involve bonding of the boron atoms on the cage and the Cr atom. This indicates that the stability of the cage due to the Cr atom doping is governed by the completion of an electronic shell with 18 valence electrons, as we shall further show in the following from the analysis of the MOs of Cr@B$_{20}$. \n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.6\\linewidth]{Fig-7.pdf}\n\\caption{The $\\pi$-bonded MOs for the bare B$_{20}$ cage and Cr@B$_{20}$ calculated with Gaussian 09 program.}\n\\label{fig:B20Cr-MO} \n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.6\\linewidth]{Fig-8.pdf}\n\\caption{Some of the $\\pi$-bonded MOs for the bare B$_{22}$ and Mo@B$_{22}$.}\n\\label{fig:B22-MO} \n\\end{figure*}\n\nIn order to understand the electronic origin of the stability of these cages, we calculated the MOs for the M doped clusters as well as the corresponding boron cage by removing the M atom and keeping the atomic positions fixed. For the Cr@B$_{20}$ case, six electrons of the Cr atom and twelve $\\pi$-bonded electrons from the B$_{20}$ cage contribute to the stability with electronic shell closing at eighteen $\\pi$ bonded valence electrons. Figure 7 shows that the bare B$_{20}$ cage has six occupied $\\pi$-bonded MOs (doubly degeberate HOMO and HOMO-4, HOMO-5, and HOMO-9). The two HOMO levels have {\\it d} angular momentum character. There are three unoccupied $\\pi$-bonded MOs (LUMO and LUMO+1) of {\\it d} angular momentum character. For Cr@B$_{20}$ these five MOs of {\\it d} angular momentum character hybridize with the {\\it d} orbitals of Cr atom forming bonding and anti-bonding MOs. The six electrons (3d$65$ 4s$^1$) of the Cr atom are accommodated in the bonding MOs and there is a large HOMO-LUMO gap which results in the stability of the endohedral Cr@B$_{20}$ cage cluster. \n\nFor the Mo@B$_{22}$ cage, the $\\pi$-bonded MOs for the bare B$_{22}$ cage and also the MOs for Mo@B$_{22}$ along with the corresponding eigenvalues are shown in Fig. \\ref{fig:B22-MO} as obtained from the Gaussian calculation with the PBE0 functional. It is seen that the stability of the Mo encapsulated B$_{22}$ cage is governed by the nine $\\pi$-bonded MOs with electronic shell closing at 18 valence electrons. For the bare B$_{22}$ cage there are six occupied $\\pi$-bonded MOs and three empty $\\pi$-bonded MOs (not shown) of $d$ angular momentum character just above the HOMO. The five $\\pi$-bonded MOs with {\\it d} angular momentum character hybridize strongly with the {\\it d} orbitals of the Mo atom forming five bonding MOs and five anti-bonding MOs. The bonding MOs are fully occupied and similar to the case of 18 valence electron stability of the B$_{20}$ cage, the stability of the Mo@B$_{22}$ cluster also arises from 18 valence electrons. The ordering of the MOs can be seen in the figure. Note that the bare B$_{22}$ cage has very small HOMO-LUMO gap but it is very much increased by Mo encapsulation. Further, the stability of the Mo@B$_{24}$ cluster also arises from the occupation of nine $\\pi$ bonded MOs corresponding to 18 valence electrons. A similar result will hold for the case of W doping. We have also performed analysis of the MOs of cr@B$_{18}$ bicapped drum structure, and in this case also the stability is associated with 18 $\\pi$ bonded valence electrons. On the other hand the stability of the disk-shaped M@B$_{20}$, for M = Mo and W arises from 20 $\\pi$ bonded valence electrons. \n\n\nWe have further performed analysis of the electronic charge density $\\rho(\\bf{r})$ of Cr@B$_{22}$ as representative of the 22-atom boron cage by calculating contours of the Laplacian of $\\rho(\\bf{r})$ as well as Laplacian at bond critical points (BCPs) and ring critical points (RCPs). The method of calculation has been discussed earlier.\\cite{saha2016} Figure \\ref{fig:laplacian} (a) shows the atomic structure with BCPs (green dots) and RCPs (red dots) while (b) shows a symmetrical view of Cr@B$_{22}$ along with the electrostatic potential mapped onto a $\\rho(\\bf{r}) = 0.1 e\/Bohr^3$ electron density isosurface. Blue regions indicate negative electrostatic potential associated with the boron atoms. Of the 22 boron atoms, there are four B atoms (B1, B3, B17, and B19) that have coordination 2 in the middle of the two boron chains, while 18 boron atoms are in two semi-circular bands consisting of BBB triangles and\/or BBBB quadrilaterals. The two boron chains appear at the left and right ends of the figure, while one of the semi-circular bands appears in the foreground running diagonally from top left to bottom right (the other is partially occluded at the back, running from top right to bottom left). The two bands and the two chains loop around and encapsulate the M atom in a tetrahedral coordination. Such bands are a recurring motif in many kinds of boron clusters, including drums\\cite{saha2016,saha2017} and quasiplanar structures.\\cite{B84} The two bands join each other and the two chains at four tetra-coordinate boron atoms. The critical point analysis requires that the Poincare-Hopf relationship: $NumNACP + NumNNACP - NumBCP + NumRCP - NumCCP = 1$ be satisfied, where $NumNACP$ is the number of nuclei (here 23), $NumNNACP$ is the number of non-nuclear attractors (here 0) critical points, $NumBCP$ is the number of BCPs (here 40), $NumRCP$ is the number of RCPs (here 20), and $NumCCP$ is the number of cage critical points (here 2). In Figs. \\ref{fig:laplacian} (c)-(e) red dots indicate the locations of the RCPs of the BBBB quadrilaterals. Of the 36 B-B BCPs (green dots), there are 6 that represent covalent bonds in the two boron chains. Figure \\ref{fig:laplacian}(d) shows a contour plot of the Laplacian distribution in a plane passing approximately through the rim of one of these bands (Cr@B$_{22}$ with B10-B9-B4-B8-B7-B22). 12 BCPs correspond to bonds at the bases of BBB triangles in the semi-circular bands, and 10 corresponding to bent bonds spanning the width of one or the other of the two bands. Figure \\ref{fig:laplacian} (e) (Cr@B$_{22}$ with B2-B1-B3-B10-Cr23) shows a contour plot of the Laplacian distribution in a plane passing approximately through one of the boron chains and the M atom. 8 BCPs represent covalent bonds along rims of the semi-circular bands. The properties of these bonds are shown in Table II and Fig. \\ref{fig:laplacian1}. Large values of the charge density and large positive Laplacian with low values of bond ellipticity correspond to covalent bonds while high bond ellipticities, low positive values of the Laplacian L($\\bf r$) and delocalization index (off-diagonal localization delocalization matrix (LDM) elements) are characteristic of multi-center bonding in the BBB triangles. On the other hand negative values of Laplacian correspond to the M-Cr bonds. Silimar results have been obtained for Cr@B$_{20}$ cage and shown in Fig. \\ref{fig:laplacian1}(c).\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.6\\linewidth,]{Fig-9.pdf}\n\\caption{(a) BCPs (green dots) and RCPs of BBBB quadrilaterals (red dots) of Cr@B$_{22}$ cluster and (b) electrostatic potential mapped onto a $\\rho$(r) = 0.1 e\/Bohr$^3$ electron density isosurface. Blue regions indicate negative electrostatic potentials associated with the boron atoms. (c) - (e) Contour plots of the Laplacian in different planes passing through atoms (c) B7, B8, B15, B16, B21, B22, and Cr, (d) B4, B7, B8, B9, B10, B22, and (e) B1, B2, B3, B10, and Cr. The contour values are 0.0, 0.001, 0.002, 0.004, 0.008, 0.02, 0.04, 0.08, 0.2, 0.4, 0.8, 2.0, 4.0, 8.0, 20.0, 40.0, 80.0, 200.0, 400.0, 800.0, -0.001, -0.002, -0.004, -0.008, -0.02, -0.04, -0.08, -0.2, -0.4, -0.8, -2.0, -4.0, -8.0, -20.0, -40.0, -80.0, -200.0, -400.0, -800.0. Pink (grey) balls show B (M) atoms.}\n\\label{fig:laplacian} \n\\end{figure*}\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Different types of bonds, the number of bonds of each type, charge density ($\\rho$) in units of e\/Bohr$^3$ at the BCP, bond ellipticity, Laplacian (L) in units of e\/Bohr$^5$ at the BCPs, and the delocalization index. Small value of ellipticity and large positive value of Laplacian indicates strong 2c covalent bonds while large values of ellipticity  and small values of Laplacian indicates delocalization of charge (multi-center bonding). The numbering of boron atoms is given in Fig. (\\ref{fig:laplacian}).}\n\n\\begin{adjustbox}{width=\\textwidth}\n\\begin{tabular}{|l|ccccc|}\n\\hline\n\\small{Bond Type}&\\small{No. of Bonds}& \\small{$\\rho$} &\\small{Bond Ellipticity} & \\small{L}\t&\\small{Delocalization Index}\\\\\n\\hline\n\\small{B1-B3, B17-B19                 } &2      &0.174 &0.02 &0.104 &1.183\\\\\n\\small{B1-B2, B3-B10, B12-B19, B17-B20} &4      &0.162 &0.15 &0.085 &1.023\\\\\n\\small{B4-B8, B5-B16, B11-B22, B14-B21} &4     \t&0.160 &0.12 &0.083 &1.009 \\\\\n\\small{B4-B9, B5-B6, B11-B18, B13-B14 } &4      &0.151 &0.22 &0.069 &0.959 \\\\\n\\small{B7-B8, B7-B22, B15-B16, B15-B21} &4\t&0.145 &0.47 &0.061 &0.886\\\\\n\\small{B2-B6, B9-B10, B12-B13, B18-B20} &4      &0.128 &1.80 &0.033 &0.686\\\\\n\\small{B2-B22, B8-B12, B10-B21, B16-B20}&4\t&0.125 &2.87 &0.028 &0.555\\\\\n\\small{B2-B7, B7-B12, B10-B15, B15-B20} &4\t&0.124 &1.66 &0.030 &0.747\\\\\n\\small{B9-B21, B6-22, B16-B18, B8-B13 } &4      &0.118 &5.10-5.20 &0.018 &0.555\\\\\n\\small{B5-B11, B4-B14                 } &2      &0.115 &1.295     &0.019 &0.572\\\\\n\\hline\n\\end{tabular}\n\\end{adjustbox}\n\\end{center}\n\\end{table}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.6\\linewidth,]{Fig-10.pdf}\n\\caption{(a) Laplacian (in units of e\/Bohr$^5$) vs electron density (e\/Bohr$^3$), (b) bond ellipticity as a function of the electron density for Cr@B$_{22}$ cluster and (c) Laplacian vs electron density for Cr@B$_{20}$ cluster. }\n\\label{fig:laplacian1} \n\\end{figure*}\n\n\nWe also performed calculations on cationic and anionic clusters. For these calculations we have considered the lowest energy and some low-lying isomers of neutral clusters. The cationic and anionic clusters are calculated using Gaussian09 program. Note that for the neutral and charged clusters the calculations using Gaussian09 program give almost the same energy order of the low-lying isomers as with the VASP calculations, but we find that in some cases the lowest energy neutral isomer does not have the lowest energy when charged. As an example the cation of W@B$_{22}$ has a double bicapped drum structure, while the anion of the lowest energy neutral isomer continues to have the lowest energy. Also a cage structure of anions of M@B$_{20}$ (M = Mo and W) has the lowest energy. More details are given in Supplementary Information. We also performed calculations on isoelectronic neutral and anionic clusters of B$_{18}$, B$_{20}$, and B$_{22}$ with the doping of V, Nb, and Ta atoms using Gaussian 09 code with PBE0 functional. Broadly the trends are similar as obtained for the cases of Cr, Mo, and W doping. For VB$_{18}$ anion we obtained a bicapped drum to be most favorable as for isoelectronic Cr doping, but neutral cluster favors a drum structure over the bicapped drum isomer by 0.27 eV. On the other hand for Nb and Ta doping, the drum isomer is lowest in energy for both neutral and anion similar to the case of isoelectronic Mo and W, respectively. In the case of VB$_{20}$ anion and neutral, a bicapped drum is 0.375 eV and 1.281 eV, respectively, lower in energy than a cage structure which is the most favorable for the neutral Cr case. However, for Nb and Ta doped B$_{20}$ anions a bicapped drum is 3.084 eV and 3.065 eV, respectively, lower in energy than a cage isomer similar to the case of Mo and W doping. Also in both cases the neutral NbB$_{22}$ and TaB$_{22}$ clusters also favor the bicapped isomer over the cage isomer by 2.181 and 1.395 eV, respectively. Interestingly for the B$_{24}$ cage all the three metal atoms V, Nb, and Ta stabilize it without much distortion for both neutral and anions and the electric dipole moment in all the cases is very close to zero. This is in contrast to Cr in which case the M atom drifts away from the center and does not interact with all the B atoms properly. \n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1.0\\linewidth,]{Fig-11.pdf}\n\\caption{Calculated density of states (DOS) for neutral Cr, Mo, and W encapsulated B$_{20}$-B$_{24}$ clusters. The vertical line shows the HOMO.}\n\\label{fig:DOS-neutral} \n\\end{figure*}\n\nThe density of states (DOS) for the neutral and anionic clusters provide further information on the nature of the electronic states. The DOS for the anionic clusters (shown in the Supplementary Information) will be useful for comparison with the results of photo-electron spectroscopy experiments that may become available in the future. The DOSs for the neutral clusters with M = Cr, Mo, and W are shown in Figure \\ref{fig:DOS-neutral}. Our results show that all the cages have a large HOMO-LUMO gap as obtained from the PBE0 calculations, indicating their very good chemical stability. As seen from the MOs, all the energy levels near the HOMO (shown by the vertical line) arise from hybridization of the boron cage orbitals with the {\\it d} orbitals of the M atom. For Cr@B$_{20}$ very localized peaks can be seen below the HOMO, indicating the degeneracy arising from the symmetry of the structure. In going from Cr@B$_{20}$ to Cr@B$_{22}$ there is a spread in the distribution of the electronic energy levels. For Mo@B$_{20}$ and W@B$_{20}$, a capped drum-like structure is lower in energy and the electronic states show more localized peaks compared to the Mo@B$_{18}$ and W@B$_{18}$ cases which lack symmetry. For M@B$_{22}$ and M@B$_{24}$ some sharp peaks can be noticed. The overall nature of the DOS for Mo and W encapsulation remains similar. \n\n\n\\subsection{Vibrational spectra}\n\nWe have calculated the vibrational modes for the lowest energy neutral isomers and their cations for all the cases using PBE0 functional in Gaussian09 program. Calculations were also done using B3PW91 hybrid functional in Gaussian09 code for some cases. The general features of the spectra using the two functionals are similar, as can be seen from the data in Tables S1-S4 in Supplementary Information, though there are some deviations in intensities and frequencies. Figure \\ref{fig:IR+Raman-neutral} shows the calculated IR and Raman spectra for Cr, Mo, and W doped neutral clusters obtained using the PBE0 functional. In almost all the cases we find no imaginary frequency, indicating that the cage structures are dynamically stable. The IR and Raman spectra for the cationic clusters are given in Fig. S9 in Supplementary Information using the PBE0 functional. Tables S1 and S2 in Supplementary Information give the IR intensities and Raman activities for the neutral and cationic clusters, respectively, using B3PW91 functional. We have also given the bond distances in the neutral case. The results using PBE0 functional are given in Tables S3 and S4. Considering the IR active modes for the Cr@$B_{18}$ cluster, we find a strong peak at 424 cm$^{-1}$ corresponding to the breathing mode. In this case the Cr atom vibrates along the axis of the drum, which involves breathing of the two B$_8$ rings. The most dominant modes occur at 473 cm$^{-1}$ and 482 cm$^{-1}$ corresponding to the vibration of the Cr atom in two perpendicular directions in the plane of the B$_{16}$ disk. In one case the Cr atom vibrates parallel to the B-B bond of the two capped boron atoms. \n\nFor the drum-shaped structure of Mo@B$_{18}$ there are strong bending\/breathing modes in the range of 291-320 cm$^{-1}$ in the IR spectrum involving M atom and the boron rings. The strong peak at 466 cm$^{-1}$ corresponds to the stretching mode of the boron ring. Another stretching mode occurs with high intensity at 1317 cm$^{-1}$. In the case of W@B$_{18}$, a similar behavior of the modes has been obtained. The strong modes at 250 cm$^{-1}$ and 269 cm$^{-1}$ for W@B$_{18}$ involve swinging of the W atom in two perpendicular directions in the plane of the boron disk, and therefore the frequencies are reduced compared to the Mo case. The mode at 456 cm$^{-1}$ involves bending of the disk without the movement of the W atom, and so it is less affected compared with the Mo case. On the other hand the frequency of the stretching mode at 1332 cm$^{-1}$ has increased compared to the Mo case, which suggests stronger bonding between the boron atoms. The Raman spectrum of Cr@B$_{18}$ has a very high intensity peak at 707 cm$^{-1}$ corresponding to the symmetric breathing mode of the capped drum boron structure. For Mo@B$_{18}$ the high intensity Raman active peak at 676 cm$^{-1}$ and for W@B$_{18}$ the high intensity peaks at 616 cm$^{-1}$ and 665 cm$^{-1}$ corrpdfond to the breathing modes of the B$_{18}$ ring. These vibrations involve purely boron atoms and not the M atom. \n \nFor the IR spectrum of Mo@B$_{20}$ the strong modes occur at 353 cm$^{-1}$ and 357 cm$^{-1}$ and correspond to the swing of the Mo atom in the direction perpendicular to the capping boron dimer and parallel to the dimer, respectively. There is also associated bending of the boron rings. The mode at 371 cm$^{-1}$ is also a swing mode of the Mo atom along the boron dimer. For W@B$_{20}$ the IR spectrum is similar with two strong modes with frequencies 294 cm$^{-1}$ and 303 cm$^{-1}$, with the swing of W atom in the direction perpendicular and along the capping boron dimer, respectively. There is an imaginary frequency for MoB$_{20}$ at -122 cm$^{-1}$ and for W@B$_{20}$ at -100 cm$^{-1}$ but in both the cases the intensity is very small. The Raman spectrum of  Mo@B$_{20}$ has two strong modes at 633 cm$^{-1}$ and 762 cm$^{-1}$ while for W@B$_{20}$ there is one strong mode at 636 cm$^{-1}$. These correspond to the breathing mode of the outer capped drum structure.\n\nFor Cr@B$_{20}$, the IR spectrum has several high intensity peaks with the most dominant ones at 454 cm$^{-1}$, 455 cm$^{-1}$, 482 cm$^{-1}$, 484 cm$^{-1}$, and 508 cm$^{-1}$ corresponding to the swing modes of the Cr atom while the peaks at 577 cm$^{-1}$ and 662 cm$^{-1}$ correspond to bending modes, and the peak at 807 cm$^{-1}$ represents stretching and breathing mode. In general the IR active peaks involve stretching and bending vibrations of different B-B and B-Cr bonds. However, the Raman spectrum has only one intense peak at 742 cm$^{-1}$ which corresponds to the breathing mode of the cage. For Cr@B$_{20}$ the Raman active peaks come from vibrations involving only boron atoms. For Cr@B$_{22}$, the IR spectrum has dominant peaks at 383 cm$^{-1}$ and 391 cm$^{-1}$, whereas the Raman spectrum has dominant peaks at 608 cm$^{-1}$, 641 cm$^{-1}$, and 702 cm$^{-1}$. Similar spectra are obtained for Mo and W doped B$_{22}$ cages. For Mo@B$_{22}$, the IR modes at 313 cm$^{-1}$ and 362 cm$^{-1}$ are swing modes of the Mo atom with associated motion of the B atoms. The Raman spectrum of Mo@B$_{22}$ has two major peaks at 631 cm$^{-1}$ and 675 cm$^{-1}$ corresponding to breathing mode of the boron cage. For W@B$_{22}$ the dominant IR modes occur at frequencies 259, 275, and 309 cm$^{-1}$, whereas the Raman activity has strong peaks at 630 cm$^{-1}$ and 675 cm$^{-1}$ corresponding to the breathing modes of the cage. In general the dominant IR peaks shift towards lower frequencies compared to Cr encapsulation because of the higher mass of Mo compared to Cr atom. The peaks shift further towards lower frequencies for W encapsulation with IR active modes at 259 cm$^{-1}$ and 275 cm$^{-1}$, compared to 313 and 362 for MO@B$_{22}$ and 383 as well as 391 for Cr@B$_{22}$. \n \n\\begin{figure*}\n\\centering\n\\includegraphics[width=1.0\\linewidth]{Fig-12.pdf}\n\\caption{Calculated IR and Raman spectra for neutral clusters. }\n\\label{fig:IR+Raman-neutral} \n\\end{figure*}\n\nA similar behavior has been obtained for Mo and W encapsulated B$_{24}$ cages. For Mo@B$_{24}$, the IR active dominant peaks are at 325 cm$^{-1}$ and 327 cm$^{-1}$, whereas the Raman spectrum has a strong peak at 632 cm$^{-1}$ corresponding to breathing of the cage. For W@B$_{24}$ the strong IR modes are at frequencies 258 cm$^{-1}$, 263 cm$^{-1}$, 265 cm$^{-1}$, and 1056 cm$^{-1}$, whereas there is one strong Raman peak at 636 cm$^{-1}$ similar to the Mo@B$_{24}$ cage. In general the highest Raman intensity peak corresponds to the breathing mode, whereas the IR active and other Raman active peaks involve different stretching and bending modes. The IR and Raman spectra have more localized and high intensity peaks for the symmetric cases compared to the broader spectrum for clusters with low symmetry.\n\nWe also calculated the vibrational spectra for the cationic clusters (see Supplementary Information); Figure S9 in Supplementary Information shows the IR and Raman spectra. For Cr doped cases the IR and Raman spectra differ more significantly for neutral and cationic clusters as compared to those with Mo and W doping. For Cr@B$_{18}^+$ there is one dominant IR peak at 479 cm$^{-1}$ corresponding to swinging mode (perpendicular to the capping boron dimer) of the Cr atom, while the peak at 544 cm $^{-1}$ corresponds to swinging along the boron dimer and the peak at 638 cm$^{-1}$ corresponds to the motion of the Cr atom along z direction. There is only one strong Raman mode at 717 cm$^{-1}$ corresponding to breathing of the boron cage. In the case of Mo@B$_{18}^+$ the IR spectrum has dominant peaks at 312 cm$^{-1}$, 313 cm$^{-1}$ (swing of the M atom), 481 cm$^{-1}$ (swing of the cage), and 730 cm$^{-1}$ (stretching and bending of the B-B bonds), while the Raman spectrum has a strong peak at 676 cm$^{-1}$ corresponding to the breathing mode of the boron cage. In the case of W@B$_{18}$ cation, the strong IR modes are at 268 cm$^{-1}$ and 269 cm$^{-1}$ corresponding to swinging of the W atom, while the mode at 462 cm$^{-1}$ corresponds to the swing of the cage. The Raman spectrum has one strong peak at 667 cm$^{-1}$ corresponding to the breathing mode.  \n\nFor Cr@B$_{20}$$^+$ the dominant IR modes are at 402 cm$^{-1}$ corresponding to the swing of the M atom, 748 cm$^{-1}$ (bending of the boron network), 802 cm$^{-1}$ (swing of the cage), 960 cm$^{-1}$ (stretching of the B-B bonds), and 1208 cm$^{-1}$ (stretching of the B-B bonds). The Raman spectrum has strong peaks at 164 cm$^{-1}$ (swinging of the M atom) which is also IR active, and scissor mode at 960 cm$^{-1}$, and stretching\/breathing mode at 968 cm$^{-1}$. For Mo@B$_{20}^+$ the dominant IR modes are at 338 cm$^{-1}$, 354 cm$^{-1}$, and 359 cm$^{-1}$ (all three swing modes of the Mo atom), while the Raman spectrum has a strong peak at 650 cm$^{-1}$ corresponding to the breathing of the cage. In the case of W@B$_{20}^+$, the dominant IR modes are at 292 cm$^{-1}$ and 304 cm$^{-1}$ (corresponding to swinging of the W atom in two perpendicular directions) while the dominant Raman peak is  at 651 cm$^{-1}$ corresponding to the breathing mode. \n\nIn the case of Cr@B$_{22}^+$ the main IR peaks occur at 81 cm$^{-1}$, 321 cm$^{-1}$, and 382 cm$^{-1}$, all corresponding to the swinging of the Cr atom, while the Raman spectrum has a strong peak at 693 cm$^{-1}$ corresponding to the breathing of the cage. There is a imaginary frequency at 187 cm$^{-1}$ but the intensity is very small. For Mo@B$_{22}^+$, the main IR peaks are at 326 cm$^{-1}$ and 369 cm$^{-1}$ corresponding to swinging modes of the Mo atom, while the Raman spectrum has two dominant peaks at 615 cm$^{-1}$ and 677 cm$^{-1}$ corresponding to breathing modes. There is an imaginary frequency at 91 cm$^{-1}$ but again the intensity is low. For W@B$_{22}$ the main IR peaks are at 258 cm$^{-1}$, 297 cm$^{-1}$, and 302 cm$^{-1}$ corresponding to the swinging modes of the W atom, bending mode at 328 cm$^{-1}$, and stretching mode at 1150 cm$^{-1}$. The Raman spectrum has a dominant peak at 678 cm$^{-1}$ corresponding to a breathing mode. For Mo@B$_{24}$ the dominant IR peaks are at 314 cm$^{-1}$, 367 cm$^{-1}$, and 376 cm$^{-1}$ corresponding to swinging modes of the Mo atom, while the dominant Raman peak is at 636 cm$^{-1}$ corresponding to the breathing mode. For W@B$_{24}^+$, the IR modes are strong at 253 cm$^{-1}$, 267 cm$^{-1}$, and 297 cm$^{-1}$ corresponding to swinging modes, while the peak at 1035 cm$^{-1}$ is a stretching mode. The Raman spectrum has a strong peak at 638 cm$^{-1}$ corresponding to the breathing mode. In general for Mo@B$_{22}$$^+$, Mo@B$_{24}$$^+$, W@B$_{22}$$^+$ and W@B$_{24}$$^+$ the positions of the dominant peaks occur at similar frequencies as for the corresponding neutral cases.\n\n\n\\section{CONCLUSIONS}\nIn summary, we have performed a systematic study on M = Cr, Mo, and W doped boron clusters in the size range of 18 to 24 boron atoms. Our results show that by M encapsulation, it is possible to have fullerene-like cage structures of boron with about 20 atoms in contrast to dominantly planar or quasi-planar or tubular structures of pure boron clusters for less than 40 atoms. Our results suggest that doping of Cr is suitable to produce symmetric small cage clusters of boron with B$_{20}$, whereas B$_{22}$ is the smallest symmetric cage for Mo and W encapsulation, which is magic and is likely to be produced in high abundance in experiments. A symmetric B$_{24}$ cage is also formed with Mo and W encapsulation, as also predicted recently, but the variation in the BE of the clusters suggests that the B$_{22}$ cage has the optimal size for Mo and W encapsulation. There is a large gain in energy when a Mo or W atom is encapsulated in the cage, which also supports the strong stability of the doped clusters. We performed an analysis of the bonding nature in these clusters and found that the cage structures are stabilized by strong interaction between the M atom $d$ orbitals and the $\\pi$-bonded MOs of the bare boron cage. From this analysis we find that the stability of the Cr@B$_{18}$, Cr@B$_{20}$, M@B$_{22}$, M = Cr, Mo, and W, and M@B$_{24}$ clusters is associated with 18 $\\pi$ bonded valence electrons while for M@B$_{18}$ (M = Mo and W) disk-shaped tubular clusters, the stability is associated with 20 $\\pi$ bonded valence electrons. In most cases the IR and Raman spectra for the neutral and cationic clusters show that the cages are dynamically stable. These results as well as those of the electronic levels of the anionic clusters will help to compare our predictions with experiments. We have also studied isoelectronic anion and neutral B$_{18}$, B$_{20}$, B$_{22}$, and B$_{24}$ clusters doped with V, Nb, and Ta. In general the structural trends are similar as obtained for Cr, Mo, and W but for B$_{24}$ we obtained a symmetric cage in all cases. We hope that our results will stimulate experimental work on these M doped disk-shaped and fullerene-like structures of boron.\n\n\\section{ACKNOWLEDGMENTS}\n\nWe gratefully acknowledge the use of the high performance computing facility Magus of the Shiv Nadar University where a part of the calculations have been performed. ABR and VK thankfully acknowledge financial support from International Technology Center - Pacific. ABR acknowledges international travel support (ITS), from SERB, Govt. of India. We thank Prof. Cherif Matta for providing access to AIMLDM software.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSpielberg \\cite{SCP} has recently studied a large family of $C^*$-algebras which includes the $C^*$-algebras of higher-rank graphs \\cite{KP, RSY2} and the boundary quotients of quasi-lattice ordered groups \\cite{N, CL2}. He has also shown that the Baumslag-Solitar groups are quasi-lattice ordered with boundary quotients that are typically Kirchberg algebras, and has computed the $K$-theory of these boundary quotients \\cite{SBS}.\n\nA quasi-lattice ordered group also has a (much larger) Toeplitz algebra, and the Toeplitz algebras of groups similar to Baumslag-Solitar groups have recently  been shown to exhibit interesting phase transitions. Indeed, there is nontrivial overlap\\footnote{More precisely, the groups $\\operatorname{BS}(1,d)$ are discussed in \\cite[\\S9]{LRR}. There is similar overlap between the structural results and $K$-theory computations in \\cite{EaHR} and \\cite{SBS}.} between the Toeplitz algebras studied in \\cite{LR, LRR} and the Toeplitz algebras of the Baumslag-Solitar groups studied in \\cite{SBS} (see \\cite[\\S9]{LRR}). So one naturally wonders whether there are interesting phase transitions on the Toeplitz algebras of Baumslag-Solitar groups. Here we confirm that this is indeed the case.\n\nSuppose that $c$ and $d$ are nonzero integers. The Baumslag-Solitar group $G=\\operatorname{BS}(c,d)$ is the group generated by two elements $a,b$ subject only to the relation $ab^c=b^da$. When $c$ and $d$ are positive, we consider  the subsemigroup $P$ of $G$ generated by $a$ and $b$. This semigroup defines a partial order on $G$: $g\\leq h$ means that $g^{-1}h\\in P$. Spielberg proved in \\cite[Theorem~2.11]{SBS} that the pair $(G,P)$ is quasi-lattice ordered in the sense of Nica \\cite{N}. The Toeplitz algebra is the $C^*$-subalgebra $\\mathcal{T}(P)$ of $B(\\ell^2(P))$ generated by a left-regular representation of $P$ by isometries $\\{T_x:x\\in P\\}$ (but see \\S\\ref{back} for further discussion of our conventions). This algebra carries a natural gauge action of the circle, which we can lift to an action $\\alpha$ of $\\mathbb{R}$. We are interested in the KMS states of the dynamical system $(\\mathcal{T}(P),\\alpha)$.\n\nWe show that for inverse temperatures $\\beta$ larger than $\\ln d$, there is a large simplex of KMS$_\\beta$ states parametrised by the probability measures on the circle.  When $d$ does not divide $c$, there is a phase transition at the \\emph{critical inverse temperature} $\\ln d$ in which this simplex collapses to a single point, and the KMS$_{\\ln d}$ state factors through the boundary quotient of \\cite{CL2}. The condition ``$d$ does not divide $c$'' has previously occurred in Spielberg's analysis of the groupoid model for the boundary quotient, where it is shown to be necessary and sufficient for the groupoid to be topologically principal (which he calls ``essentially free'') \\cite[Theorem~4.9]{SBS}. \n\nWe begin with a section on background material: we discuss our conventions concerning quasi-lattice ordered groups and their Toeplitz algebras, and the normal form for elements of Baumslag-Solitar groups which we will use throughout. The normal form identifies a family of words in $P$ that play a vital role in computations in $G$ and $P$. We call these words ``stems'', and in \\S\\ref{sect:stems} we establish  some properties of the map which sends an arbitary element of $P$ to its stem. In \\S\\ref{sect:pres} we give a presentation of our Toeplitz algebra which will allow us to build Hilbert-space representations. Then in \\S\\ref{sect:char}, we turn to KMS states. The Toeplitz algebra $\\mathcal{T}(P)=C^*(\\{T_x:x\\in P\\})$ is spanned by the elements $T_xT_y^*$, and the KMS states are the states that satisfy a commutation relation involving products of two spanning elements. In Proposition~\\ref{charKMS} we give a characterisation of KMS states in terms of their values on individual spanning elements. This implies, for example, \nthat all KMS states at real inverse temperatures factor through the quotient in which the generator $T_b$ is unitary. \n\nOur main theorem about the KMS$_\\beta$ states for $\\beta>\\ln d$ is Theorem~\\ref{KMSToe}, and the rest of \\S\\ref{sect:large} is devoted to its proof. The strategy is a refinement of the one developed in \\cite{LR} and  \\cite{LRR}. To build KMS states, we exploit that all KMS states think $T_b$ is unitary: we take a carefully chosen unitary representation $W$ of the subgroup generated by $b$, and induce it to a large unitary representation $\\operatorname{Ind} W$ of $G$. The KMS states come from the isometric representation obtained by restricting $(\\operatorname{Ind} W)|_P$ to a suitable invariant (but not reducing!) subspace. Our results at the critical inverse temperature are in Proposition~\\ref{KMScrit}, and we show by example that they are sharp: when $d$ divides $c$, there is more than one KMS$_{\\ln d}$ state.  \n\nOur last main result is Theorem~\\ref{thmground}, where we identify the ground and KMS$_\\infty$ states of our system. This seems to be harder than in previous computations of KMS structure: ground states need not factor through the same quotient of $\\mathcal{T}(P)$, and hence we cannot use induced representations. But by mimicking what happens in \\S\\ref{sect:large}, we can build suitable isometric representations with our bare hands. We close with an appendix in which we prove that the quasi-lattice ordered group $(G,P)$ is amenable in the sense of \\cite{N,LRold}. This result is not strictly needed in the rest of the paper, but it does simplify things notationally because it implies that the Toeplitz algebra is universal for Nica-covariant representations of $P$ (see Corollary~\\ref{same}). \n\n\n\\section{Background}\\label{back}\n\n\\subsection{Quasi-lattice ordered groups}\nSuppose that $G$ is a group and $P$ is a subsemigroup such that $P\\cap P^{-1}=\\{e\\}$.\nThen there is a partial order on $G$ such that \n\\[\ng\\leq h\\Longleftrightarrow h\\in gP\\Longleftrightarrow g^{-1}h\\in P.\n\\]\nThis partial order is left-invariant, in the sense that $g\\leq h\\Longrightarrow kg\\leq kh$. \n\nAccording to Nica \\cite{N}, the pair $(G,P)$ is a \\emph{quasi-lattice ordered group} if every pair $g,h$ in $G$ with a common upper bound in $P$ has a least upper bound $g\\vee h$ in $P$. Subsequently, Crisp and Laca showed that it suffices to check that every element $g\\in G$ with an upper bound in $P$ has a least upper bound in $P$ \\cite[Lemma~7]{CL1} (and that useful lemma contains several other equivalent reformulations of the definition). We write $g\\vee h<\\infty$ if $g$ and $h$ have an upper bound in $P$, and $g\\vee h=\\infty$ otherwise.\n\nSuppose that $(G,P)$ is quasi-lattice ordered. We consider the Hilbert space $\\ell^2(P)$ with the orthonormal basis $\\{e_x:x\\in P\\}$ of point masses. For each $x\\in P$, there is an isometry $T_x$ on $\\ell^2(P)$ such that $T_xe_y=e_{xy}$ for $y\\in P$. We have $T_e=1$ (the identity operator), and $T_xT_y=T_{xy}$. In other words, $T$ is a homomorphism of the monoid $P$ into the monoid of isometries on $\\ell^2(P)$, and we say that $T$ is an \\emph{isometric representation} of $P$. Nica observed that the representation $T$ has the extra property \n\\begin{equation}\\label{Nicacov}\nT_xT_x^*T_yT_y^*=\\begin{cases}T_{x \\vee y} T_{x \\vee y}^*&\\text{if $x\\vee y<\\infty$}\\\\\n0&\\text{if $x\\vee y=\\infty$.}\n\\end{cases}\n\\end{equation}\nNow we say that an isometric representation satisfying \\eqref{Nicacov} is  \\emph{Nica covariant}. Nica covariance is equivalent to\n\\begin{equation}\\label{altNica}\nT_x^*T_y =\\begin{cases}\nT_{x^{-1}(x \\vee y)}T_{y^{-1}(x \\vee y)}^*&\\text{if $x\\vee y<\\infty$}\\\\\n0&\\text{if $x\\vee y=\\infty$.}\n\\end{cases}\n\\end{equation}\n\n\nA quasi-lattice ordered group $(G,P)$ has two $C^*$-algebras: the \\emph{Toeplitz algebra} $\\mathcal{T}(P)$ is the $C^*$-subalgebra of $B(\\ell^2(P))$ generated by the operators $\\{T_x:x\\in P\\}$, and the \\emph{universal $C^*$-algebra} $C^*(G,P)$ is generated by a universal Nica-covariant representation $i:P\\to C^*(G,P)$. Nica covariance implies that every word in the $i(x)$ and their adjoints reduces to one of the form $i(x)i(y)^*$, and hence\n\\[\nC^*(G,P)=\\overline{\\operatorname{span}}\\{i(x)i(y)^*:x,y\\in P\\}.\n\\]\nThe Toeplitz representation $T:P\\to \\mathcal{T}(P)$ induces a surjection $\\pi_T:C^*(G,P)\\to \\mathcal{T}(P)$, and a major issue considered in \\cite[\\S4]{N} is when $\\pi_T$ is an isomorphism. \n\nBecause $x\\vee y=y\\vee x$, Nica covariance implies that the range projections $i(x)i(x)^*$ commute with each other, and then \n$D:=\\overline{\\operatorname{span}} \\{i(x)i(x)^*:x\\in P\\}$ is a commutative $C^*$-subalgebra. There is a positive norm-decreasing linear map $E:C^*(G,P)\\to D$ such that $E(i(x)i(y)^*)=\\delta_{x,y}i(x)i(x)^*$ (see \\cite[\\S4.2]{N} or \\cite[Proposition~3.1]{LRold}), and we say that $(G,P)$ is \\emph{amenable} if $E$ is faithful. Nica proved that if $(G,P)$ is amenable, then the Toeplitz representation $\\pi_T$ is injective (see  \\cite[\\S4.2]{N} or \\cite[Corollary~3.9]{LRold}). This implies that the Toeplitz algebra has the universal property of $(C^*(G,P),i)$, and justifies the following:\n\n\\begin{con}\nAll the quasi-lattice ordered groups $(G,P)$ in this paper are \namenable (see Theorem~\\ref{BSamenable}). \nSo it makes no difference whether we use  $C^*(G,P)$ or $\\mathcal{T}(P)$. We choose to write $C^*(G,P)$ for \nthe algebra because we want to emphasise the universal property, but write $T$ for the universal Nica-covariant representation of $P$ in $C^*(G,P)$. We write $\\mathbb{N}=\\{0,1,2,\\dots\\}$. \n\\end{con}\n \n\n\n\\subsection{Baumslag-Solitar groups} We fix positive integers $c$ and $d$. Then the \\emph{Baumslag-Solitar group} is the group\n\\[\nG := \\langle a, b : ab^c = b^da \\rangle;\n\\] \nif we want to emphasise the dependence on the numbers $c,d$, we write $G=\\operatorname{BS}(c,d)$. We consider the submonoid $P$ of $G$ generated  by\nby $a$ and $b$. Spielberg proved in \\cite[Theorem~2.11]{SBS} that (provided $c$ and $d$ are positive) $(G,P)$ is quasi-lattice ordered. For the rest of this paper, $(G,P)$ denotes one of these groups.\n\n \n\nFollowing \\cite{SBS}, we write $\\theta$ for the homomorphism $\\theta:G \\to \\mathbb{Z}$ \nsuch that $\\theta(a)=1$ and $\\theta(b) = 0$, and call $\\theta(g)$ the \\emph{height} of $g$. Baumslag-Solitar \ngroups are examples of Higman-Neumann-Neumann extensions, and each element has a unique normal form \n\\[\ng=b^{s_0}a^{\\varepsilon_1}b^{s_1}\\cdots b^{s_{k-1}}a^{\\varepsilon_k}b^{s_k}\n\\]\nin which each $\\varepsilon_i$ is $\\pm 1$, $0\\leq s_{i-1}<d$ when $\\varepsilon_i=1$, and $0\\leq s_{i-1}<c$ when $\\varepsilon_i=-1$ (see, for example, Theorem~2.1 on \\cite[page~182]{LS}).  For elements of $P$, we have $\\varepsilon_i =1$ for all $k$ and $0\\leq s_i<d$ for all $i<k$; there is no restriction on $b^{s_k}$ except that $s_k\\geq 0$, and we have $k=\\theta(x)$. \n\n\\section{Stems and their properties}\\label{sect:stems}\n\n\n\n\nEach $x\\in P$ has a unique normal form $x=b^{s_0}a b^{s_1}a\\cdots b^{s_{k-1}}ab^{s_k}$ with $0\\leq s_i<d$ for $i<k$ and $k=\\theta(x)$. \nWe then write \n\\[\n\\operatorname{stem}(x):=b^{s_0}a b^{s_1}a\\cdots b^{s_{k-1}}a\n\\]\nfor the \\emph{stem} of $x$. We write $\\Sigma_k$ for the set of possible stems with height $k$, including $\\Sigma_0:=\\{e\\}$; note that each $\\Sigma_k$ is finite with cardinality $d^k$. \n\n\nOur constructions of KMS states will involve the Hilbert space $\\bigoplus_{k\\geq 0}\\ell^2(\\Sigma_k)$, and hence properties of stems will be important throughout the paper. In this section, we describe some of these properties.\n\n\\begin{lemma}\\label{forlisa}\nSuppose that $x,y\\in P$. Then $\\operatorname{stem}(x\\operatorname{stem}(y))=\\operatorname{stem}(xy)$. If $s$ and $t$ satisfy $y=\\operatorname{stem}(y)b^t$ and $x\\operatorname{stem}(y)=\\operatorname{stem}(x\\operatorname{stem}(y))b^s$, then $xy=\\operatorname{stem}(xy)b^{s+t}$.\n\\end{lemma}\n\n\\begin{proof}\nThere exists $t$ such that $y=\\operatorname{stem}(y)b^t$, and then \n\\[\nxy=x\\operatorname{stem}(y)b^t=(\\operatorname{stem}(x\\operatorname{stem}(y))b^s)b^t\\quad\\text{for some $s\\in \\mathbb{N}$.}\n\\]\nOn the other hand, we have $xy=\\operatorname{stem}(xy)b^r$ for some $r\\in \\mathbb{N}$. Since both $\\operatorname{stem}(xy)b^r$ and $\\operatorname{stem}(x\\operatorname{stem}(y))b^{s+t}$ are normal forms for $xy$, the uniqueness of normal forms implies that $\\operatorname{stem}(x\\operatorname{stem}(y))=\\operatorname{stem}(xy)$ and that $r=s+t$. \n\\end{proof}\n\n\n\\begin{lemma}\\label{maponstems}\n\\begin{enumerate}\n\\item\\label{hkm} For all $k\\in \\mathbb{N}$ and $m\\geq 0$, the map $h_{k,m}:x\\mapsto \\operatorname{stem}(b^mx)$ is a bijection of $\\Sigma_k$ onto $\\Sigma_k$.\n\n\\item\\label{gk} For all $k\\in \\mathbb{N}$, the map $x\\mapsto \\operatorname{stem}(b^cax)$ is an injection of $\\Sigma_k$ into $\\Sigma_{k+1}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}[Proof of part~\\eqref{hkm}]\n We prove by induction on $k$ that $h_{k,m}$ is a bijection for all $m\\in \\mathbb{N}$. The result is trivial if $k=0$. \nFor $k=1$, $h_{1,m}(b^ia)=b^ja$ where $m+i=nd+j$ and $0\\leq j<d$; since addition by $m$ modulo $d$ is a bijection on $\\{0,1,\\cdots, d-1\\}$, $h_{1,m}$ is a bijection. Suppose that $h_{k,m}$ is a bijection for all $m$. Because $\\Sigma_{k+1}$ is finite, \nit suffices to see that $h_{k+1,m}$ is one-to-one. So suppose that $h_{k+1,m}(x)=h_{k+1,m}(x')$. We can write \n$x=b^iay$ for some $y\\in \\Sigma_k$.  We now define $n,j$ by  $m+i=nd+j$ and $0\\leq j<d$, and then \n\\[\nh_{k+1,m}(b^iay)=\\operatorname{stem}(b^{m+i}ay)=b^ja\\operatorname{stem}(b^{nc}y)=b^ja h_{k,nc}(y).\n\\]\nDoing the same for $x'=b^{i'}ay'$ shows that\n\\[\nb^ja h_{k,nc}(y)=b^{j'}ah_{k,n'c}(y')\\text{ where $m+i=nd+j$ and $m+i'=n'd+j'$}.\n\\]\nNow uniqueness of the normal form forces $j=j'$ and $h_{k,nc}(y)=h_{k,n'c}(y')$. Since $j=j'$ and \n$|i-i'|<d$, we must have $i=i'$ and $n=n'$ too. Now the injectivity of $h_{k,nc}$ implies that $y=y'$ and $x=b^jay=x'$.\n\\end{proof}\n\nFor the proof of part~\\eqref{gk}, we separate out a calculation. Notice that it applies with $m=c$, \nand then gives \\eqref{gk} for $k=1$. (Since $\\Sigma_0=\\{e\\}$, \\eqref{gk} is trivially true for $k=0$.)\n\n\\begin{lemma}\\label{sublem}\nSuppose that $m\\in \\mathbb{N}$, $b^ia, b^ja\\in \\Sigma_1$, and $\\operatorname{stem}(b^mab^ia)=\\operatorname{stem} (b^mab^ja)$. Then $i=j$.\n\\end{lemma}\n\n\\begin{proof}\nWrite $m=nd+m'$ with $0\\leq m'<d$. Then \n\\[\nb^mab^ia=b^{m'}ab^{nc+i}a\\ \\text{ and }\\ b^mab^ja=b^{m'}ab^{nc+j}a.\n\\]\nNow we write $nc +i=n_id+i'$ and $nc+j=n_jd+j'$, and the hypothesis gives\n\\[\nb^{m'}ab^{i'}a=\\operatorname{stem}(b^mab^ia)=\\operatorname{stem} (b^mab^ja)=b^{m'}ab^{j'}a.\n\\]\nThus the uniqueness of the normal form implies that $i'=j'$, and we have\n\\[\ni=n_id+i'-nc=n_jd+j'-nc +(n_i-n_j)d=j+(n_i-n_j)d.\n\\]\nNow $0\\leq i,j<d$ implies that $n_i-n_j=0$, and hence $i=j$.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{maponstems}\\,\\eqref{gk}]\nSuppose that $x,y\\in \\Sigma_k$ and $\\operatorname{stem}(b^cax)=\\operatorname{stem}(b^cay)$. We write $x$ and $y$ in normal form as\n\\[\nx=b^{x_0}ab^{x_1}a\\cdots b^{x_{k-1}}a\\ \\text{ and }\\ y=b^{y_0}ab^{y_1}a\\cdots b^{y_{k-1}}a,\n\\]\nand prove by induction on $n$ that $x_i=y_i$ for $0\\leq i\\leq n<k$. Since the stem of $b^cax$ \nbegins with the stem of $b^cab^{x_0}a$, and similarly for $b^cay$, Lemma~\\ref{sublem} \nimplies that $x_0=y_0$. Suppose that we have $x_i=y_i$ for $i\\leq n<k-1$. Next we put into normal form\n\\[\nb^cab^{x_0}ab^{x_1}a\\cdots b^{x_n}a=b^{s_0}ab^{s_1}a\\cdots b^{s_n}ab^m;\n\\]\nby the inductive hypothesis, we have\n\\begin{align*}\nb^cax&=b^{s_0}ab^{s_1}a\\cdots b^{s_n}ab^mb^{x_{n+1}}\\cdots b^{x_{k-1}}a,\\text{ and }\\\\ \nb^cay&=b^{s_0}ab^{s_1}a\\cdots b^{s_n}ab^mb^{y_{n+1}}\\cdots b^{y_{k-1}}a.\n\\end{align*}\nThus\n\\begin{align*}\nb^{s_0}ab^{s_1}&a\\cdots b^{s_n}a\\operatorname{stem}(b^mab^{x_{n+1}}a\\cdots b^{x_{k-1}}a)=\\operatorname{stem}(b^cax)\\\\\n&=\\operatorname{stem}(b^cay)=b^{s_0}ab^{s_1}a\\cdots b^{s_n}a\\operatorname{stem}(b^mab^{y_{n+1}}a\\cdots b^{y_{k-1}}a),\n\\end{align*}\nand we deduce that \n\\[\n\\operatorname{stem}(b^mab^{x_{n+1}}a\\cdots b^{x_{k-1}}a)=\\operatorname{stem}(b^mab^{y_{n+1}}a\\cdots b^{y_{k-1}}a).\n\\]\nIn particular, we have $\\operatorname{stem}(b^mab^{x_{n+1}}a)=\\operatorname{stem}(b^mab^{y_{n+1}}a)$, \nand Lemma~\\ref{sublem} implies that $x_{n+1}=y_{n+1}$.\n\\end{proof}\n\n\\begin{lemma}\n \\label{lem:iains}Let $x,y \\in P$ such that $x \\vee y < \\infty$.  \n\\begin{enumerate} \n \\item\\label{it:iains1} If $\\theta(y) > \\theta(x)$ then there exists $t \\in \\mathbb{N}$ such that \n$x \\vee y =yb^t$.\n\\item\\label{it:iains2} If $\\theta(x) = \\theta(y)$ then there exists $t \\in \\mathbb{N}$ such that either \n\\[x \\vee y = x = yb^t\\text{ or } x \\vee y =y=xb^t.\\] \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFor (\\ref{it:iains1}), suppose $x = \\operatorname{stem}(x)b^s$ for some $s \\in \\mathbb{N}$.  Then because $x\\vee y < \\infty$ and $\\theta(y) > \\theta(x)$,  \n$\\operatorname{stem}(y)=\\operatorname{stem}(x)\\sigma$ for some stem $\\sigma$. Then \n$y = \\operatorname{stem}(x)\\sigma b^n$ for some $n \\in \\mathbb{N}$. \nNow choose a stem $\\tau$ such that $\\operatorname{stem}(b^s\\tau)=\\sigma$ by Lemma~\\ref{maponstems}(\\ref{hkm}) (using that the map $h_{\\theta(\\sigma),s}$ is surjective  and so $\\sigma$ must be in the image).  \nThat is, $b^s\\tau = \\sigma b^r$ for some $r \\in \\mathbb{N}$.\nThen\n\\[x\\tau=\\operatorname{stem}(x)b^s\\tau=\\operatorname{stem}(x)\\sigma b^r.\\]\nTherefore\n\\[x\\vee y = \\operatorname{stem}(x)\\sigma b^{\\max(n,r)}=yb^{\\max(n,r)-n}\\]\nso if we let $t=\\max(n,r)-n$, then $x\\vee y = yb^t$.\n\n\nFor part (\\ref{it:iains2}), if $x\\vee y < \\infty$ and $\\theta(y) = \\theta(x)$, then putting $x \\vee y$ into normal form tells us\nthat $\\operatorname{stem}(x) = \\operatorname{stem}(y)$.  The result follows.  \n\\end{proof}\n\n\n\n\n\\section{A presentation for the Toeplitz algebra}\\label{sect:pres}\n\n\n\nWe want to build representations of $C^*(G,P)$. For this we use:\n\n\\begin{prop}\\label{defrel}\nSuppose that $\\pi:C^*(G,P)\\to B$ is a homomorphism. Then $U:=\\pi(T_b)$ and $V:=\\pi(T_a)$ are isometries, and satisfy\n\\begin{enumerate}\n\\item\\label{t1} $VU^c=U^dV$;\n\\item\\label{t4} $U^*V=U^{d-1}VU^{*c}$;\n\\item\\label{t5} $V^*U^jV=0$ for $1\\leq j<d$.\n\\end{enumerate}\nConversely, if $U$ and $V$ are isometries in a $C^*$-algebra $B$ satisfying \\eqref{t1}, \\eqref{t4} and \\eqref{t5}, then there is a Nica covariant representation $S:P\\to B$ such that $S_a=V$ and $S_b=U$, and a homomorphism $\\pi_{U,V}:C^*(G,P)\\to B$ such that $U=\\pi_{U,V}(T_b)$ and $V=\\pi_{U,V}(T_a)$. \n\\end{prop}\n\nLet $\\pi:C^*(G,P)\\to B$ be a homomorphism. Then $\\pi\\circ T$ is a Nica covariant representation of $(G,P)$.\nThe relation \\eqref{t1} follows because $ab^c=b^da$ in $P$. The relation \\eqref{t4} follows from Nica covariance for the pair $(b,a)$, which has $b\\vee a=ab^c$, so that\n\\[\nU^*V=\\pi(T_b^*T_a)=\\pi(T_{b^{-1}(b\\vee a)}T_{a^{-1}(b\\vee a)}^*)=\\pi(T_{b^{d-1}a})\\pi(T_{b^c})^*=U^{d-1}VU^{*c}.\n\\]\nThe relation \\eqref{t5} is Nica covariance for $(a,b^ja)$, for which we have $a\\vee b^ja=\\infty$. So it remains for us to prove the converse.\n\n\n\n\\begin{rmk}\\label{r1} The relation \\eqref{t5} is equivalent to saying that $\\{U^jV:0\\leq j<d\\}$ is a Toeplitz-Cuntz family: in other words, the $U^jV$ are isometries satisfying\n\\[\n1\\geq \\sum_{j=0}^{d-1} (U^jV)(U^jV)^*.\n\\]\nFor $k\\geq 1$, the stems of height $k$ are precisely the words of length $k$ in the alphabet $\\{b^ja:0\\leq j<d\\}$, and for $\\sigma=b^{j_0}ab^{j_1}a \\dots b^{j_{k-1}}a$ we have\n\\[\n\\pi(T_\\sigma)=(U^{j_0}V)(U^{j_1}V) \\dots (U^{j_{k-1}}V).\n\\]\nThus $\\{\\pi(T_\\sigma):\\sigma\\in \\Sigma_k\\}$ is also a Toeplitz-Cuntz family for each $k\\geq 1$.\n\\end{rmk}\n\n\n\n\\begin{rmk}\\label{r2}\nSuppose that $U,V$ satisfy relations \\eqref{t1} and \\eqref{t5} of Proposition~\\ref{defrel} and $U$ is unitary. Then multiplying \\eqref{t1} on the left by $U^*$ and the right by $U^{*c}$ gives \\eqref{t4}. (This argument  uses the extra relation $UU^*=1$, so it does not work when $U$ is just an isometry.) Our relation \\eqref{t1} is relation (3) in \\cite[Theorem~3.23]{SBS}. In \\cite[Remark~3.24]{SBS}, Spielberg suggests that (3) and the Toeplitz-Cuntz relation equivalent to \\eqref{t5} give a presentation of his Toeplitz algebra $\\mathcal{T}(G,P)$. However, we think that his $\\mathcal{T}(G,P)$ is intended to be $C^*(G,P)$, and that the extra relation \\eqref{t4} is required for that. \n\\end{rmk}\n\nThe hard bit in Proposition~\\ref{defrel} is proving that a pair $(U,V)$ of isometries satisfying the relations gives us a Nica-covariant isometric representation $S$ of $P$ in $B$. It is clear how to define $S$: write $x\\in P$ in  normal form $b^{s_0}ab^{s_1}a\\cdots ab^{s_m}$, and define\n\\[\nS_x:=U^{s_0}VU^{s_1}V\\cdots V U^{s_m};\n\\]\nwe also set $S_e:=1$.\nFor $x,y\\in P$, the product of normal forms is not necessarily a normal form, so to see that $S$ is multiplicative, we need to put the product\n\\[\nxy=(b^{s_0}ab^{s_1}a\\cdots ab^{s_m})(b^{t_0}ab^{t_1}a\\cdots ab^{t_n})\n\\]\nin normal form. However, this entails pulling any factors of the form $b^{kd}$ in $b^{s_m}b^{t_0}$ to the right, using the relation $b^da=ab^c$ to pull any such factors across each $a$ in turn. We can perform exactly the same calculations in \n\\[\nS_xS_y=U^{s_0}VU^{s_1}V\\cdots V U^{s_m}U^{t_0}VU^{t_1}V\\cdots V U^{t_n}\n\\]\nusing the relation \\eqref{t1}, arriving at the formula for $S_{xy}$. So $S$ is multiplicative. \n\nTo see that $S$ is Nica covariant, we begin with a special case. To avoid losing detail in subscripts, we say that a pair $(x,y)$ in $P$ is \\emph{Nica covariant} when $S_x$, $S_y$ satisfy the Nica covariance relation \\eqref{altNica}.\n\n\\begin{lemma}\\label{height1Nica}\nFor every $s,t\\in \\mathbb{N}$, the pair $(b^s,ab^t)$ is Nica covariant.\n\\end{lemma}\n\n\\begin{proof}\nWrite $x=b^s$ and $y=ab^t$, and write $s=(n-1)d+j$ with $1\\leq j\\leq d$. Then we have\n\\[\nx\\vee y=\n\\begin{cases}ab^{nc}=xb^{d-j}a=yb^{nc-t}&\\text{if $nc>t$}\\\\\ny=xb^{d-j}ab^{t-nc}&\\text{if $nc\\leq t$,}\n\\end{cases}\n\\]\nand \n\\[\nS_{x^{-1}(x \\vee y)}S_{y^{-1}(x \\vee y)}^*\n=\\begin{cases}U^{d-j}VU^{*(nc-t)}&\\text{if $nc>t$}\\\\\nU^{d-j}VU^{t-nc}&\\text{if $nc\\leq t$.}\n\\end{cases}\n\\]\nNext we observe that \\eqref{t4} implies\n\\begin{equation}\\label{t4'}\nU^{*r}U^{d-1}VU^{*c}=U^{*(r+1)}V\\quad\\text{for every integer $r\\geq 0$.}\n\\end{equation}\nUsing this, we compute\n\\begin{align*}\nS_x^*S_y&=U^{*((n-1)d+j)}VU^t\\\\\n&=U^{*j}U^{*(n-1)d}VU^t\\\\\n&=U^{*j}VU^{*(n-1)c}U^t\\quad\\text{by \\eqref{t4'} with $r=d-1$, $n-1$ times}\\\\\n&=(U^{d-j}VU^{*c})U^{*(n-1)c}U^t\\quad\\text{by \\eqref{t4'} with $r=j-1$,}\n\\end{align*}\nwhich is $U^{d-j}VU^{*(nc-t)}$ if $nc>t$ and $U^{d-j}VU^{t-nc}$ if $nc\\leq t$.\n\\end{proof}\n\nThe next lemma will allow us to bootstrap Lemma~\\ref{height1Nica} to longer words.\n\n\\begin{lemma}\\label{height2Nica}\nSuppose that $(x,y)$ is a Nica-covariant pair with $x\\vee y<\\infty$ and $\\theta(x)\\leq \\theta(y)$. If $w$ has the form $ab^t$, then $(x,yw)$ is a Nica-covariant pair.\n\\end{lemma}\n\n\\begin{proof}\nWe have\n\\[\nS_x^*S_{yw}=(S_x^*S_{y})S_w=(S_{x^{-1}(x \\vee y)}S_{y^{-1}(x \\vee y)}^*)S_{w}.\n\\]\nThe assumption $\\theta(x)\\leq \\theta(y)$ implies that $x\\vee y=$ has the form $yb^s$ (see Lemma~\\ref{lem:iains}), and hence Lemma~\\ref{height1Nica} implies that $(y^{-1}(x\\vee y),w)=(b^s,w)$ is Nica covariant. Thus\n\\begin{align*}\nS_x^*S_{yw}&=S_{x^{-1}(x \\vee y)}(S_{y^{-1}(x \\vee y)}^*S_{w})\\\\\n&=S_{x^{-1}(x \\vee y)}\\big(S_{(y^{-1}(x\\vee y))^{-1}(y^{-1}(x\\vee y)\\vee w)}S^*_{w^{-1}(y^{-1}(x\\vee y)\\vee w)}\\big)\\\\\n&=S_{x^{-1}y(y^{-1}(x\\vee y)\\vee w)}S^*_{w^{-1}(y^{-1}(x\\vee y)\\vee w)}.\n\\end{align*}\nNow we recall that the partial order on $(G,P)$ is left invariant, and hence\n\\begin{align*}\n&y(y^{-1}(x\\vee y)\\vee w)=(yy^{-1}(x\\vee y))\\vee yw=(x\\vee y)\\vee yw=x\\vee yw\\\\\n\\intertext{and}\n&y^{-1}(x\\vee y)\\vee w=y^{-1}(x\\vee y)\\vee y^{-1}yw=y^{-1}((x\\vee y)\\vee yw)=y^{-1}(x\\vee yw).\n\\end{align*}\nThus\n\\[\nS_x^*S_{yw}=S_{x^{-1}(x\\vee yw)}S^*_{w^{-1}y^{-1}(x\\vee yw)}=S_{x^{-1}(x\\vee yw)}S^*_{(yw)^{-1}(x\\vee yw)},\n\\]\nas required.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{defrel}] It remains for us to prove  that the representation $S$ is Nica covariant. Suppose that $x,y\\in P$. It suffices to prove \\eqref{altNica} when $\\theta(x)\\leq \\theta(y)$ (otherwise take adjoints). First we suppose that $x\\vee y=\\infty$. We claim that $\\operatorname{stem}(x)$ is not an initial segment of $\\operatorname{stem}(y)$. To see this, suppose to the contrary that $\\operatorname{stem}(y)=\\operatorname{stem}(x)p$ and $x=\\operatorname{stem}(x)b^t$. Then Lemma~\\ref{maponstems}\\eqref{hkm} implies that there is a stem $q$ such that $b^tq$ has the form $pb^s$. But them $xq=\\operatorname{stem}(x)pb^s$ and $y$ has the same stem, and we can find a common upper bound for $x$ and $y$ of the form $\\operatorname{stem}(x)pb^r$. Thus we have a contradiction, and the claim is proved. So there are distinct stems $\\sigma, \\tau$ in $\\Sigma_{\\theta(x)}$ such that $x$ has the form $x=\\sigma b^s$ and $y=\\tau p$. Then because $\\{S_\\rho=\\pi(T_\\sigma):\\rho\\in\\Sigma_{\\theta(x)}\\}$ \nis a Toeplitz-Cuntz family (Remark~\\ref{r1}), we have\n\\[\nS_x^*S_y=S_{b^s}^*S_{\\sigma}^*S_\\tau S_p=0,\n\\]\nas required in \\eqref{altNica}. \n\nNext we suppose that $x\\vee y<\\infty$, in which case we have $x=\\sigma b^s$ for $\\sigma=\\operatorname{stem}(x)$, and $y$ has the form $\\sigma w$ for some $w\\in P$ by uniqueness of the normal form. Then\n\\[\nS_x^*S_y=S_{b^s}^*S_{\\sigma}^*S_\\sigma S_w=S_{b^s}^*S_w;\n\\]\nsince left invariance of the partial order gives \\[x^{-1}(x\\vee y)=b^{-s}(b^s\\vee w)\\text{\\ and\\ }y^{-1}(x\\vee y)=w^{-1}(b^s\\vee w),\\] it suffices to prove the result for $x=b^s$. Now we trivially have Nica covariance for $(b^s,b^r)$, and Lemma~\\ref{height2Nica} gives Nica covariance for $(b^s,b^rab^t)$. Now an induction argument using Lemma~\\ref{height2Nica} gives Nica covariance of $(b^r,w)$ for all $w$. Thus $S$ is Nica covariant.\n\nThe universal property of $(C^*(G,P),T)$ now gives us the homomorphism $\\pi_{U,V}:=\\pi_S:C^*(G,P)\\to B$ with the required properties.\n\\end{proof}\n\n\n\\section{A characterisation of KMS states}\\label{sect:char}\n\nThe height map $\\theta$ gives  a strongly continuous gauge action $\\gamma:\\mathbb{T}\\to \\operatorname{Aut} C^*(G,P)$ such that $\\gamma_z(T_x)= z^{\\theta(x)}T_x$. We then define $\\alpha:\\mathbb{R}\\to \\operatorname{Aut} C^*(G,P)$ by $\\alpha_t= \\gamma_{e^{i t}}$, and aim to study the KMS states of the dynamical system $(C^*(G,P), \\alpha)$.  \nFor $x,y\\in P$  we have $\\alpha_t(T_xT_y^*) = e^{i t(\\theta(x) - \\theta(y))}T_xT_y^*$, and thus each $T_xT_y^*$ is analytic, with \n$\n\\alpha_z(T_xT_y^*)=e^{i z(\\theta(x) - \\theta(y))}T_xT_y^*\n$.\nSince the $T_xT_y^*$ span a dense subspace of $C^*(G,P)$, it follows from \\cite[Proposition~8.12.3]{P} that a state $\\psi$ of $C^*(G,P)$ is a KMS$_\\beta$ state of $(C^*(G,P),\\alpha)$ for some $\\beta\\in \\mathbb{R}\\setminus\\{0\\}$ if and only if\n\\begin{equation}\\label{eq:KMScondition}\n\\psi((T_xT_y^*)(T_pT_q^*))=\\psi((T_pT_q^*)\\alpha_{i\\beta}(T_xT_y^*))=e^{-\\beta(\\theta(x)-\\theta(y))}\\psi((T_pT_q^*)(T_xT_y^*))\n\\end{equation}\nfor all $x,y,p,q\\in P$.\n\n\n\\begin{prop}\\label{charKMS}\nLet $\\psi$ be a state on $(C^*(G,P),\\alpha)$.  Then $\\psi$ is a KMS$_\\beta$ state if and only if for all $x,y\\in P$ we have\n\\begin{equation}\n \\label{eq:KMS}\n\\psi(T_xT_y^*) = \\begin{cases}\n                  e^{-\\beta \\theta(x)}\\psi(T_{y^{-1}x}) & \\text{if  $\\theta(x)=\\theta(y)$ and  $x \\vee y = x$}\\\\\n\t\t  e^{-\\beta \\theta(x)}\\psi(T_{x^{-1}y}^*) & \\text{if  $\\theta(x)=\\theta(y)$ and  $x \\vee y = y$}\\\\\n\t\t  0 & \\text{otherwise.}\n                 \\end{cases}\n\\end{equation}\n\\end{prop}\n\n\\begin{proof}\n Suppose $\\psi$ is a KMS$_\\beta$ state on $(C^*(G,P),\\alpha)$ and fix $x,y\\in P$. Nica covariance of $T$ gives\n\\[\nT_y^*T_x =  \n\\begin{cases}\nT_{y^{-1}(y\\vee x)}T^*_{x^{-1}(y\\vee x)} &\\text{if $x\\vee y<\\infty$}\\\\\n0&\\text{if $x\\vee y=\\infty$.}\n\\end{cases}\\]\nThe KMS condition says\n\\begin{equation*}\n \\psi(T_xT_y^*) =  e^{-\\beta \\theta(x)} \\psi(T_y^*T_x), \n\\end{equation*}\nand hence $ \\psi(T_xT_y^*)= 0$ unless $x\\vee y < \\infty$.\nApplying the KMS condition again gives\n\\[\n  \\psi(T_xT_y^*) = e^{-\\beta (\\theta(x)-\\theta(y))}\\psi(T_xT_y^*),\n\\] \nand hence also $\\psi(T_xT_y^*) =0$ unless  $\\theta(x) = \\theta(y)$. \n\nNow suppose that $\\theta(x) = \\theta(y)$ and $x\\vee y < \\infty$.\nThen \n\\[\n\\psi(T_xT_y^*) =  e^{-\\beta \\theta(x)} \\psi(T_{y^{-1}(x\\vee y)} T_{x^{-1}(x\\vee y)}^*),\n\\]\nand we recover~(\\ref{eq:KMS}) since either $x\\vee y = x$ or $x\\vee y = y$ by Lemma~\\ref{lem:iains}(\\ref{it:iains2}).\n\nConversely, suppose $\\psi$ is a state satisfying \\eqref{eq:KMS}. We fix $x,y,p,q\\in P$ and aim to show the KMS condition \\eqref{eq:KMScondition} holds.   We will show that if $\\psi(T_xT_y^*T_pT_q^*)\\neq 0$, then $\\psi (T_pT_q^*T_xT_y^*)\\neq 0$ also and the KMS condition holds. Then, by symmetry, $\\psi(T_xT_y^*T_pT_q^*)\\neq 0$ if and only if $\\psi (T_pT_q^*T_xT_y^*)\\neq 0$, and so if $\\psi(T_xT_y^*T_pT_q^*)=0$ then the KMS condition holds with both sides zero. So we  assume that $\\psi(T_xT_y^*T_pT_q^*) \\neq 0$.   \n\n\nBy Nica covariance $y\\vee p<\\infty$ and  \n\\begin{equation}\\label{eq-usedcovariance}\n0\\neq \\psi(T_xT_y^*T_pT_q^*)= \\psi(T_{xy^{-1}(y \\vee p)} T^*_{qp^{-1}(y \\vee p)}).\n\\end{equation}\nWe now argue that it suffices to show the KMS condition when  $\\theta(y)\\geq \\theta(p)$ and $y\\vee p=yb^m$ for some $m\\in\\mathbb{N}$.  If $\\theta(y)> \\theta(p)$, then there exists $m\\in\\mathbb{N}$  such that $y\\vee p=yb^m$ by Lemma~\\ref{lem:iains}\\eqref{it:iains1}.  If $\\theta(y)=\\theta(p)$, then there exists $m\\in\\mathbb{N}$  such that  $y\\vee p=yb^m$   by Lemma~\\ref{lem:iains}\\eqref{it:iains2} ($m=0$ is allowed). If $\\theta(y)< \\theta(p)$, then we take the adjoint of $T_xT_y^*T_pT_q^*$ and use that $\\psi(a^*)=\\overline{\\psi(a)}$.\nSo we assume that $\\theta(y)\\geq \\theta(p)$ and $y\\vee p=yb^m$ for some $m\\in\\mathbb{N}$.\n\nSet \\[M:= xy^{-1}(y \\vee p)=xy^{-1}yb^m=xb^m\\quad\\text{and}\\quad N:= qp^{-1}(y \\vee p)=qp^{-1}yb^m.\\]\nThen \\eqref{eq-usedcovariance} and the equation for $\\psi$  at \\eqref{eq:KMS} implies that   $\\theta(M)=\\theta(N)$, and either $M\\vee N=M$ or $M\\vee N=N$. Thus \n\\[\n\\theta(x)-\\theta(y)=\\theta(q)-\\theta(p)\n\\]\nand then $\\theta(x)\\geq \\theta(q)$. By Lemma~\\ref{lem:iains}\\eqref{it:iains2}  there exists $n \\in \\mathbb{Z}$ such that $M = Nb^n$. For future use we note here that $M = Nb^n$ implies\n\\begin{equation}\\label{eq-x}\nx=qp^{-1}yb^n.\n\\end{equation}\nUsing \\eqref{eq:KMS} we have\n\\begin{equation} \\label{eq:tobereconcile}\n\\psi(T_xT_y^*T_pT_q^*)= \\psi(T_{M}T_N^*) \n=\\begin{cases}\ne^{-\\beta \\theta(N)} \\psi(T_{b^n}) &\\text{if $n\\geq 0$}\\\\\ne^{-\\beta \\theta(N)} \\psi(T_{b^{-n}}^*) &\\text{if $n<0$.}\n\\end{cases} \n\\end{equation}\n\n\nNext we consider $\\psi(T_pT_q^*T_xT_y^*)$. Since $x\\leq M$ and $q\\leq N$ we have $x\\vee q\\leq M\\vee N<\\infty$, and $\\psi(T_pT_q^*T_xT_y^*)=\\psi(T_{pq^{-1}(x\\vee q)}T_{yx^{-1}(x\\vee q)}^*)$.  By Lemma~\\ref{lem:iains}, there exists $s\\in\\mathbb{N}$ such that either $x\\vee q=xb^s$ or $x\\vee q=qb^s$.\nFirst, suppose that $x\\vee q=xb^s$. Then \n\\begin{gather*}\nyx^{-1}(x\\vee q)=yb^s\\text{\\ and}\\\\\npq^{-1}(x\\vee q)=pq^{-1}xb^s=yb^nb^s= yx^{-1}(x\\vee q)b^n\n\\end{gather*}\nusing \\eqref{eq-x}.\nSecond, suppose that $x\\vee q=qb^s$. Since $\\theta(x)\\geq \\theta(q)$, we have  $x=x\\vee q$. Thus\n\\begin{gather*}\npq^{-1}(x\\vee q)=pb^s\\text{\\ and}\\\\\nyx^{-1}(x\\vee q)=y=pq^{-1}xb^{-n}=pb^sb^{-n}=pq^{-1}(x\\vee q)b^{-n}\n\\end{gather*}\nusing \\eqref{eq-x}. In either case, $pq^{-1}(x\\vee q)= yx^{-1}(x\\vee q)b^n$, and \n\\begin{align*}\n\\psi(T_pT_q^*T_xT_y^*) = \\psi(T_{pq^{-1}(q \\vee x)}T_{yx^{-1}(q \\vee x)}^*)=\\begin{cases}\ne^{-\\beta \\theta(pq^{-1}(x\\vee q))} \\psi(T_{b^n}) &\\text{if $n\\geq 0$}\\\\\ne^{-\\beta \\theta(pq^{-1}(x\\vee q))} \\psi(T_{b^{-n}}^*) &\\text{if $n<0$.}\n\\end{cases} \\end{align*}\nBut $\\theta(pq^{-1}(x\\vee q))=\\theta(p)-\\theta(q)+\\theta(x)=\\theta(y)$ and $\\theta(N)=\\theta(x)$. Thus\n\\[\ne^{-\\beta(\\theta(x)-\\theta(y))}\\psi(T_pT_q^*T_xT_y^*) =\\begin{cases}\ne^{-\\beta\\theta(x)} \\psi(T_{b^n}) &\\text{if $n\\geq 0$}\\\\\ne^{-\\beta\\theta(x)} \\psi(T_{b^{-n}}^*) &\\text{if $n<0$}\n\\end{cases} \n\\]\nis the same as \\eqref{eq:tobereconcile}, as required.\nAs we argued above, this suffices to show that $\\psi$ is a KMS$_\\beta$ state.\n\\end{proof}\n\n\\begin{cor}\\label{btsuffices}\nSuppose that $\\phi$ and $\\psi$ are KMS$_\\beta$ states on $(C^*(G,P),\\alpha)$ and $\\phi(T_{b^t})=\\psi(T_{b^t})$ for all $t\\in \\mathbb{N}$. Then $\\phi=\\psi$.\n\\end{cor}\n\n\\begin{proof}\nBoth states vanish on generators $T_xT_y^*$ unless $\\theta(x)=\\theta(y)$ and $x\\vee y$ is $x$ or $y$, in which case Lemma~\\ref{lem:iains} implies that either $y^{-1}x$ or $x^{-1}y$ has the form $b^t$. Thus $\\phi(T_xT_y^*)=\\psi(T_xT_y^*)$ for all $x,y\\in P$, and $\\phi=\\psi$.\n\\end{proof}\n\n\\begin{cor}\\label{restonbeta}\nConsider the dynamical system $(C^*(G,P),\\alpha)$ as above and take $\\beta\\in \\mathbb{R}$. \n\\begin{enumerate}\n\\item\\label{factor1} Every KMS$_\\beta$ state of $(C^*(G,P),\\alpha)$ factors through the quotient by the ideal generated by $1-T_bT_b^*$.\n\\item\\label{lowerbd} If $\\beta<\\ln d$, then $(C^*(G,P),\\alpha)$ has no KMS$_\\beta$ states.\n\\item\\label{factor2} Let $I$ be the ideal generated by the element\n\\[\n1-\\sum_{j=0}^{d-1}T_{b^ja}T_{b^ja}^*.\n\\]\nThen a KMS$_{\\beta}$ state factors through the quotient $\\mathcal{O}(G,P):=C^*(G,P)\/I$ if and only if $\\beta=\\ln d$.\n \\end{enumerate}\n\\end{cor}\n\n\n\\begin{proof}\nFor \\eqref{factor1}, suppose that $\\psi$ is a KMS$_\\beta$ state of $(C^*(G,P),\\alpha)$. Then\n\\[\n\\psi(T_bT_b^*)=\\psi(T_b^*\\alpha_{i\\beta}(T_b))=\\psi(T_b^*T_b)=\\psi(1)=1.\n\\]\nThus $\\psi(1-T_bT_b^*)=0$. The projection $1-T_bT_b^*$ is invariant for the dynamics, and the elements $T_xT_y^*$ are analytic elements such that $\\alpha_z(T_xT_y^*)$ is the product of $T_xT_y^*$ by the scalar-valued function $z\\mapsto e^{iz(\\theta(x)-\\theta(y))}$. So we apply Lemma 2.2 of \\cite{aHLRS} with $P=\\{1-T_bT_b^*\\}$ and $\\mathcal{F}=\\{T_xT_y^*\\}$, and deduce that $\\psi$ factors through a state of the quotient, as claimed.\n\nFor \\eqref{lowerbd}, we again suppose that $\\psi$ is a KMS$_\\beta$ state of $(C^*(G,P),\\alpha)$. Then since \\[\\{T_{b^ja}:0\\leq j<d\\}\\] is a Toeplitz-Cuntz family, we have $1\\geq\\sum_{j=0}^{d-1}T_{b^ja}T_{b^ja}^*$ in $C^*(G,P)$. Thus \\eqref{eq:KMS} gives\n\\begin{equation}\\label{applyTCK}\n1=\\psi(1)\\geq \\sum_{j=0}^{d-1}\\psi(T_{b^ja}T_{b^ja}^*)=\\sum_{j=0}^{d-1}e^{-\\beta}\\psi(1)=e^{-\\beta}d,\n\\end{equation}\nwhich is equivalent to $\\beta\\geq \\ln d$.\n\nWe now prove \\eqref{factor2}. If $\\psi$ is a KMS$_{\\ln d}$ state, then $1=e^{-\\beta}d$ forces equality throughout \\eqref{applyTCK}, and \n\\begin{equation}\\label{Cuntzrel}\n\\psi\\Big(1-\\sum_{j=0}^{d-1}T_{b^ja}T_{b^ja}^*\\Big)=0.\n\\end{equation}\nNow another application of \\cite[Lemma~2.2]{aHLRS} shows that $\\psi$ factors through the quotient $\\mathcal{O}(G,P)$. Conversely, if $\\psi$ is a KMS$_\\beta$ state which factors through the quotient, then $\\psi$ satisfies \\eqref{Cuntzrel}, we have equality in \\eqref{applyTCK}, and $\\beta=\\ln d$.\n\\end{proof}\n\n\n\n\n\n\n\\section{KMS states for large inverse temperatures}\\label{sect:large}\n\n\n\n\\begin{thm}\\label{KMSToe}\nSuppose that $\\beta>\\ln d$ and $\\mu$ is a probability measure on $\\mathbb{T}$. Then there is a KMS$_\\beta$ state $\\psi_{\\beta,\\mu}$ on $(C^*(G,P),\\alpha)$ such that\n\\begin{equation}\\label{psivsmoments}\n\\psi_{\\beta,\\mu}(T_{b^t})=(1-e^{-\\beta}d)\\Big(\\int_{\\mathbb{T}}z^t\\,d\\mu(z)+\\sum_{\\{k\\geq 1\\,:d\\,\\mid\\, c^jd^{-j}t\\text{ for }0\\leq j<k\\}}e^{-\\beta k}d^k\\int_{\\mathbb{T}}z^{c^kd^{-k}t}\\,d\\mu(z)\\Big),\n\\end{equation}\nwhere the sum is interpreted as $0$ if there are no such integers $k$. Every KMS$_\\beta$ state has this form. If $d$ does not divide $c$, then the map $\\mu\\mapsto \\psi_{\\beta,\\mu}$ is an affine continuous isomorphism  of the simplex $P(\\mathbb{T})$ of probability measures on $\\mathbb{T}$ onto the KMS$_\\beta$ simplex of $(C^*(G,P),\\alpha)$.\n\\end{thm}\n\nThe proof of this theorem will occupy the rest of this section. Our first task is to show existence of such states, and for this we need some concrete representations of $C^*(G,P)$.\n\n\nWe consider the subgroup $K:=\\{b^t:t\\in \\mathbb{Z}\\}$ of the Baumslag-Solitar group $G=BS(c,d)$, and let $W:K\\to U(H)$ be a unitary representation. We choose a section $c:G\/K\\to G$ for the quotient map, and write $c(g)$ for $c(gK)$. Then we can realise the induced representation $\\operatorname{Ind}_K^GW$ as acting in the space $\\ell^2(G\/K,H)$ according to the formula\n\\[\n\\big((\\operatorname{Ind}_K^GW)_l\\xi\\big)(gK)=W_{c(g)^{-1}lc(l^{-1}g)}\\big(\\xi(l^{-1}gK)\\big)\\quad\\text{for $\\l\\in G$, $\\xi\\in \\ell^2(G\/K,H)$}.\n\\]\n(See, for example, \\cite[page~50]{KT}.) \n\n\\begin{prop}\\label{indrepconst}\nChoose a section $c:G\/K\\to G$ such that $c(xK)=\\operatorname{stem} (x)$ for every $x\\in P$, and use $c$ to pull over the usual orthonormal basis $\\{e_{k,\\sigma}:\\sigma\\in\\Sigma_k\\}$ for $\\ell^2(\\Sigma_k)$ to an orthonormal set in $\\ell^2(G\/K)$. Then the subspace\n\\[\nH_0:=\\overline{\\operatorname{span}}\\{e_{k,\\sigma}\\otimes h:k\\in \\mathbb{N},\\ \\sigma\\in \\Sigma_k, h\\in H\\}\n\\]\nof $\\ell^2(G\/K,H)$ is invariant for every $(\\operatorname{Ind}_K^GW)_x$ with $x\\in P$, and we then have\n\\begin{equation}\\label{formpi(Tx)}\n(\\operatorname{Ind}_K^GW)_x(e_{k,\\sigma}\\otimes h)=e_{k+\\theta(x),\\operatorname{stem}(x\\sigma)}\\otimes W_{b^t}h\\quad\\text{where $x\\sigma=\\operatorname{stem}(x\\sigma)b^t$.}\n\\end{equation}\nThe map $x\\mapsto (\\operatorname{Ind}_K^GW)_x\\big|_{H_0}$ is a Nica-covariant isometric representation of $P$, and the corresponding representation $\\pi$ of $C^*(G,P)$ on $H_0$ satisfies\n\\begin{equation}\\label{formforpi}\n\\pi(T_x)(e_{k,\\sigma}\\otimes h)=e_{k+\\theta(x),\\operatorname{stem}(x\\sigma)}\\otimes W_{b^t}h\\quad\\text{where $x\\sigma=\\operatorname{stem}(x\\sigma)b^t$.}\n\\end{equation}\nThe operator $\\pi(T_b)$ is unitary.\n\\end{prop}\n\n\\begin{proof}\nIt suffices to check invariance for the generators $x=a$ and $x=b$ of $P$. First we take $x=a$. We are viewing functions in $\\ell^2(\\Sigma_k)$ as functions on $G\/K$ by viewing $\\Sigma_k$ as the subset $\\{\\sigma K:\\sigma\\in \\Sigma_k\\}$ of $G\/K$ and extending the functions to be $0$ off $\\Sigma_k$. Thus\n\\[\ne_{k,\\sigma}(gK)=\\begin{cases}\n1&\\text{if $gK=\\sigma K$}\\\\\n0&\\text{otherwise},\n\\end{cases}\n\\]\nand\n\\begin{align*}\n(\\operatorname{Ind}_K^GW)_a(e_{k,\\sigma}\\otimes h)(gK)\n&=W_{c(g)^{-1}ac(a^{-1}g)}(e_{k,\\sigma}(a^{-1}gK)h)\\\\\n&=\\begin{cases}\nW_{c(g)^{-1}ac(a^{-1}g)}h&\\text{if $a^{-1}gK=\\sigma K$}\\\\\n0&\\text{otherwise}.\n\\end{cases}\n\\end{align*}\nNow we have\n\\[\na^{-1}gK=\\sigma K\\Longleftrightarrow a^{-1}g=\\sigma b^n\\text{ for some $n$}\\Longleftrightarrow g=a\\sigma b^n\\text{ for some $n$;}\n\\]\nthen, since $a\\sigma$ is a stem, and $c(\\tau K)=\\tau$ for $\\tau\\in \\bigcup_k\\Sigma_k$, we have $c(g)^{-1}ac(a^{-1}g)=(a\\sigma)^{-1}a\\sigma=e$. So \n\\begin{align}\\label{formindWa}\n(\\operatorname{Ind}_K^GW)_a(e_{k,\\sigma}\\otimes h)(gK)&=\\begin{cases}\nh&\\text{if $gK=a\\sigma K$}\\\\\n0&\\text{otherwise}\n\\end{cases}\\\\\n&=(e_{k+1,a\\sigma}\\otimes h)(gK).\\notag\n\\end{align}\nThis gives invariance of $H_0$ for $(\\operatorname{Ind}_K^GW)_a$. \n\nFor $x=b$, similar considerations give\n\\[\n(\\operatorname{Ind}_K^GW)_b(e_{k,\\sigma}\\otimes h)(gK)\n=\\begin{cases}\nW_{c(g)^{-1}bc(b^{-1}g)}h&\\text{if $b^{-1}gK=\\sigma K$}\\\\\n0&\\text{otherwise,}\n\\end{cases}\n\\]\nand \n\\[\nb^{-1}gK=\\sigma K\\Longleftrightarrow b^{-1}g=\\sigma b^n\\text{ for some $n$}\\Longleftrightarrow g=b\\sigma b^n\\text{ for some $n$.}\n\\]\nWhile $b\\sigma$ need not be a stem, it is certainly in $P$, and hence $c(b\\sigma K)=\\operatorname{stem}(b\\sigma)$. Then $b\\sigma=\\operatorname{stem}(b\\sigma)b^t$ for some $t\\in \\mathbb{N}$, and\n\\[\nc(g)^{-1}bc(b^{-1}g)=(\\operatorname{stem}(b\\sigma))^{-1}b\\sigma=b^t.\n\\]\nThus\n\\begin{equation}\\label{formindWb}\n(\\operatorname{Ind}_K^GW)_b(e_{k,\\sigma}\\otimes h)\n=e_{k,\\operatorname{stem}(b\\sigma)}\\otimes W_{b^t}h,\n\\end{equation}\nwhich is back in $\\ell^2(\\Sigma_k)\\otimes H\\subset H_0$.\n\nSince both $U:=(\\operatorname{Ind}_K^GW)_b|_{H_0}$ and $V:=(\\operatorname{Ind}_K^GW)_a|_{H_0}$ are the restrictions of unitary operators, they are isometries.  We next prove \\eqref{formpi(Tx)}. First, we prove it by induction on $i$ for $x$ of the form $b^i$. It is by definition true for $i=1$. If it is true for $i$, then\n\\begin{align*}\n(\\operatorname{Ind}_K^GW)_{b^{i+1}}&(e_{k,\\sigma}\\otimes h)\\\\\n&=(\\operatorname{Ind}_K^GW)_{b^{i}}(e_{k,\\operatorname{stem}(b\\sigma)}\\otimes W_{b^t}h)\\quad\\text{where $b\\sigma=\\operatorname{stem}(b\\sigma)b^t$}\\\\\n&=e_{k,\\operatorname{stem}(b^i\\operatorname{stem}(b\\sigma))}\\otimes W_{b^s}W_{b^t}h\\quad\\text{where $b^i\\operatorname{stem}(b\\sigma)=\\operatorname{stem}(b^i\\operatorname{stem}(b\\sigma))b^s$.}\n\\end{align*}\nLemma~\\ref{forlisa} implies that $\\operatorname{stem}(b^{i+1}\\sigma)=\\operatorname{stem}(b^i\\operatorname{stem}(b\\sigma))$ and $b^{i+1}\\sigma=\\operatorname{stem}(b^{i+1}\\sigma)b^{s+t}$, and we have proved the result for $x=b^i$. It is quite easy to check that it works for $x=b^ia$, and then a similar argument by induction on the height of $x$ (again using Lemma~\\ref{forlisa}) gives  the general result.\n\nWe aim to prove that $U$ and $V$ satisfy the relations of Proposition~\\ref{defrel}. Since we know from Lemma~\\ref{maponstems} that $\\sigma\\mapsto \\operatorname{stem}(b\\sigma)$ is a bijection, Equation \\eqref{formindWb} implies that $U$ is surjective. Thus $U$ is unitary, and it suffices to verify that $U$ and $V$ satisfy relations \\eqref{t1} and \\eqref{t5} of Proposition~\\ref{defrel}.\n\nFor \\eqref{t1} we have on the one hand\n\\begin{align*}\nVU^c(e_{k,\\sigma}\\otimes h)&=V(e_{k,\\operatorname{stem}(b^c\\sigma)}\\otimes W_{b^t}h)\\quad\\text{where $b^c\\sigma=\\operatorname{stem}(b^c\\sigma)b^t$}\\\\\n&=e_{k+1,a\\operatorname{stem}(b^c\\sigma)}\\otimes W_{b^t}h\\quad\\text{where $b^c\\sigma=\\operatorname{stem}(b^c\\sigma)b^t$;}\n&\\end{align*}\non the other hand\n\\[\nU^dV(e_{k,\\sigma}\\otimes h)=e_{k+1,\\operatorname{stem}(b^da\\sigma)}\\otimes W_{b^s}h\\quad\\text{where $b^da\\sigma=\\operatorname{stem}(b^da\\sigma)b^s$.}\n\\]\nNow Lemma~\\ref{forlisa} gives $a\\operatorname{stem}(b^c\\sigma)=\\operatorname{stem}(ab^c\\sigma)=\\operatorname{stem} (b^da\\sigma)$, and \n\\[\nb^t=\\operatorname{stem}(b\\sigma)^{-1}b^c\\sigma=(a\\operatorname{stem}(b\\sigma))^{-1}ab^c=\\operatorname{stem}(b^da\\sigma)^{-1}b^da\\sigma=b^s,\n\\]\nand we have \\eqref{t1}. For \\eqref{t5}, we observe that \n$U^jV(e_{k,\\sigma}\\otimes h)$ has the form $e_{k+1,\\operatorname{stem}(b^ja\\sigma)}\\otimes W_{b^t}h$. For $0\\leq j<d$ and $\\sigma\\in \\Sigma_k$, $b^ja\\sigma$ is itself a stem, and for $1\\leq j<d$ it is distinct from $a\\sigma$. Thus for $1\\leq j<d$ the vector $U^jV(e_{k,\\sigma}\\otimes h)$ is always orthogonal to $V(e_{k,\\sigma}\\otimes h)=e_{k+1,a\\sigma}\\otimes h$, and hence to the range of $V$. Thus $V^*U^jV(e_{k,\\sigma}\\otimes h)$ is always $0$, and we have~\\eqref{t5}. \n\nNow Proposition~\\ref{defrel} implies that $S:x\\mapsto (\\operatorname{Ind}_K^GW)_x\\big|_{H_0}$ is Nica covariant, and gives a homomorphism $\\pi=\\pi_S$ such that $\\pi(T_{b^t})=S_{b^t}=U^t$ is unitary.\n\\end{proof} \n\n\\begin{prop}\\label{abstractKMSconst}\nIn the situation of Proposition~\\ref{indrepconst}, fix a unit vector $h\\in H$. Then for every $\\beta>\\ln d$ there is a KMS$_\\beta$ state $\\psi_h$ on $(C^*(G,P),\\alpha)$ such that\n\\begin{equation}\\label{defphih}\n\\psi_h(a)=(1-e^{-\\beta}d)\\sum_{k=0}^\\infty\\sum_{\\sigma\\in\\Sigma_k}e^{-\\beta k}\\big(\\pi(a)(e_{k,\\sigma}\\otimes h)\\,|\\,e_{k,\\sigma}\\otimes h\\big)\\quad\\text{for $a\\in C^*(G,P)$.}\n\\end{equation}\n\\end{prop}\n\n\\begin{proof}\nWe begin by checking that the series converges. Indeed, since $|\\Sigma_k|=d^k$, and $e^\\beta>d$, we have\n\\[\n\\sum_{k=0}^\\infty\\sum_{\\sigma\\in\\Sigma_k}e^{-\\beta k}\\big(e_{k,\\sigma}\\otimes h\\,|\\,e_{k,\\sigma}\\otimes h\\big)=\\sum_{k=0}^\\infty e^{-\\beta k}d^k=\\frac{1}{1-e^{-\\beta}d}.\n\\]\nIn particular, $\\psi_h(1)=1$, and we have a well-defined state.\n\nWe now want to verify that $\\psi_h$ satisfies Equation~\\eqref{eq:KMS} in Proposition~\\ref{charKMS}. So we take $x,y\\in P$ and consider \n\\begin{equation}\\label{singleip}\n\\big(\\pi(T_xT_y^*)(e_{k,\\sigma}\\otimes h)\\,|\\,e_{k,\\sigma}\\otimes h\\big)\n=\\big(\\pi(T_y^*)(e_{k,\\sigma}\\otimes h)\\,|\\,\\pi(T_x^*)(e_{k,\\sigma}\\otimes h)\\big).\n\\end{equation}\nFor each $k\\geq 0$, we identify\n\\[\n\\overline{\\operatorname{span}}\\{e_{k,\\sigma}\\otimes g:\\sigma\\in \\Sigma_k, g\\in H\\}\n\\]\nwith $\\ell^2(\\Sigma_k)\\otimes H$. Then the subspaces $\\{\\ell^2(\\Sigma_k)\\otimes H:k\\in \\mathbb{N}\\}$ are mutually orthogonal, and we have \n\\[\nH_0=\\textstyle{\\bigoplus_{k=0}^{\\infty}\\ell^2(\\Sigma_k)\\otimes H}.\n\\]\nThe operator $\\pi(T_x)$ maps each summand $\\ell^2(\\Sigma_k)\\otimes H$ into $\\ell^2(\\Sigma_{k+\\theta(x)})\\otimes H$, and hence the adjoint $T_x^*$ vanishes on $\\ell^2(\\Sigma_k)\\otimes H$ for $k<\\theta(x)$, and maps the other $\\ell^2(\\Sigma_k)\\otimes H$ into $\\ell^2(\\Sigma_{k-\\theta(x)})\\otimes H$. Thus when $\\theta(x)\\not=\\theta(y)$,  $T_x^*$ and $T_y^*$ map $\\ell^2(\\Sigma_k)\\otimes H$ into orthogonal summands in $H_0=\\bigoplus \\ell^2(\\Sigma_k)\\otimes H$. Thus if $\\theta(x)\\not=\\theta(y)$, we have $\\psi_h(T_xT_y^*)=0$. \n\nIt remains to consider $x,y$ satisfying $\\theta(x)=\\theta(y)$. Then by Lemma~\\ref{lem:iains} we have one of $x\\vee y=x$, $x\\vee y=y$ or $x\\vee y=\\infty$. If $x\\vee y=\\infty$, then Nica covariance of $T$ implies that $T_x^* T_y=0$, so that the range of $\\pi(T_x)$ is orthogonal to the range of $\\pi(T_y)$; since $\\pi(T_x)^*(e_{k,\\sigma}\\otimes h)=0$ unless $e_{k,\\sigma}\\otimes h$ is in the range of $\\pi(T_x)$, all the inner products \\eqref{singleip} are $0$, and $\\psi(T_xT_y^*)=0$. Since \n\\[\n\\big(\\pi(T_yT_x^*)(e_{k,\\sigma}\\otimes h)\\,|\\,e_{k,\\sigma}\\otimes h\\big)=\n\\overline{\\big(\\pi(T_xT_y^*)(e_{k,\\sigma}\\otimes h)\\,|\\,e_{k,\\sigma}\\otimes h\\big)},\n\\]\nit remains for us to compute $\\psi(T_xT_y^*)$ when $x\\vee y=x$ (if $x\\vee y=y$ switch $x$ and $y$). So suppose $x\\vee y=x$. Then Lemma~\\ref{lem:iains} implies that $x=yb^t$ for some $t\\in \\mathbb{N}$.\n\nWe begin by fixing $\\sigma\\in \\Sigma_k$ and computing\n\\begin{align}\\label{firstcalc}\n\\big(\\pi(T_xT_y^*)(e_{k,\\sigma}\\otimes h)\\,|\\,e_{k,\\sigma}\\otimes h\\big)\n&=\\big(\\pi(T_yT_{b^t}T_y^*)(e_{k,\\sigma}\\otimes h)\\,|\\,e_{k,\\sigma}\\otimes h\\big)\\\\\n&=\\big(\\pi(T_{b^t})\\pi(T_y^*)(e_{k,\\sigma}\\otimes h)\\,|\\,\\pi(T_y^*)(e_{k,\\sigma}\\otimes h)\\big).\n\\notag\n\\end{align}\n\n\nNotice that because $\\pi(T_b)$ is unitary, the operator $\\pi(T_xT_y^*)$ is not changed if we replace $y$ by its stem. So we assume that $y$ is a stem. Then  $y\\sigma$ is also a stem, so the formula \\eqref{formforpi} implies that $\\pi(T_y)(e_{k,\\sigma}\\otimes h)=e_{k+\\theta(y),y\\sigma}\\otimes h$. Since each $e_{k,\\sigma}\\otimes h $ is either in the range of $\\pi(T_y)$ or orthogonal to it, $\\pi(T_y)^*(e_{k,\\sigma}\\otimes h)$ vanishes unless $k\\geq \\theta(y)$ and $\\sigma$ has the form $y\\tau$ for some $\\tau\\in \\Sigma_{k-\\theta(y)}$. Then\n\\[\n\\big(\\pi(T_xT_y^*)(e_{k,\\sigma}\\otimes h)\\,|\\,e_{k,\\sigma}\\otimes h\\big)\n=\\big(\\pi(T_{b^t})(e_{k-\\theta(y),\\tau}\\otimes h)\\,|\\,e_{k-\\theta(y),\\tau}\\otimes h\\big).\n\\]\nNext we observe that because $y$ is a stem, $\\tau\\mapsto y\\tau$ is an injection of $\\Sigma_j$ into $\\Sigma_{j+\\theta(y)}$ for every $j\\geq 0$. Thus\n\\begin{align*}\n\\psi_h(T_x T_y^*)&=(1-e^{-\\beta}d)\\sum_{k=\\theta(y)}^\\infty\\sum_{\\tau\\in \\Sigma_{k-\\theta(y)}}e^{-\\beta k}\\big(\\pi(T_{b^t})(e_{k-\\theta(y),\\tau}\\otimes h)\\,|\\,e_{k-\\theta(y),\\tau}\\otimes h\\big)\\\\\n&=(1-e^{-\\beta}d)e^{-\\beta\\theta(y)}\\sum_{j=0}^\\infty\\sum_{\\tau\\in \\Sigma_{j}}e^{-\\beta j}\\big(\\pi(T_{b^t})(e_{j,\\tau}\\otimes h)\\,|\\,e_{j,\\tau}\\otimes h\\big)\\\\\n&=e^{-\\beta\\theta(y)}\\psi_h(T_{b^t})\\\\\n&=e^{-\\beta\\theta(y)}\\psi_h(T_{y^{-1}x}).\n\\end{align*}\nThus $\\psi_h$ satisfies \\eqref{eq:KMS}, and Proposition~\\ref{charKMS} implies that $\\psi_h$ is a KMS$_\\beta$ state.\n\\end{proof}\n\n\n\nThe subgroup $K$ is a copy of the additive group $\\mathbb{Z}$, written multiplicatively because it sits inside the nonabelian group $G$. Thus $C^*(K)$ is isomorphic to the algebra $C(\\mathbb{T})$, and states on $C^*(K)$ are given by probability measures $\\mu$ on $\\mathbb{T}$. For such a measure $\\mu$, we consider the representation $W=W(\\mu)$ of $K$ on $H=L^2(\\mathbb{T},d\\mu)$ given by \n\\begin{equation}\\label{defW}\n(W_{b^t}f)(z)=z^tf(z),\n\\end{equation}\nand the unit vector $h=1_\\mu$ in $L^2(\\mathbb{T},d\\mu)$ associated to the constant function $1$. Then Proposition~\\ref{abstractKMSconst} gives us a KMS$_\\beta$ state $\\psi_{\\beta,\\mu}:=\\psi_{1_\\mu}$, and we need to calculate the values of this state on the elements $T_{b^t}$, which by Corollary~\\ref{btsuffices} determine the state. For $k=0$, we have just the trivial stem $e$, and \n\\begin{equation}\\label{k=0}\n\\big(\\pi(T_{b^t})(e_{k,e}\\otimes 1_\\mu)\\,\\big|\\,e_{k,e}\\otimes 1_\\mu\\big)=(W_{b^t}1_\\mu\\,|\\,1_\\mu)=\\int_{\\mathbb{T}} z^t\\,d\\mu(z).\n\\end{equation}\nFor $k\\geq 1$ and each stem $\\sigma\\in \\Sigma_k$, we have \n\\begin{equation}\\label{calcpiT}\n\\pi(T_{b^t})(e_{k,\\sigma}\\otimes 1_\\mu)=e_{k,\\operatorname{stem}(b^t\\sigma)}\\otimes W_{b^s}1_\\mu\\quad\\text{where $b^t\\sigma=\\operatorname{stem}(b^t\\sigma)b^s$.}\n\\end{equation}\nThus \n\\[\n\\big(\\pi(T_{b^t})(e_{k,\\sigma}\\otimes 1_\\mu)\\,\\big|\\,e_{k,\\sigma}\\otimes 1_\\mu\\big)\n\\]\nvanishes unless $\\operatorname{stem}(b^t\\sigma)=\\sigma$, and hence we need to know when this happens.\n\n\\begin{lemma}\\label{idsummands}\nSuppose that $\\sigma\\in \\Sigma_k$ for some $k\\geq 1$ and $b^t\\in K$. Then $\\operatorname{stem}(b^t\\sigma)=\\sigma$ if and only if $d$ divides $c^jd^{-j}t$ for every $j$ such that $1\\leq j<k$. If so, we have\n\\begin{equation}\\label{carrythrough}\nb^t\\sigma =\\sigma b^{c^kd^{-k}t}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nSuppose first that $\\operatorname{stem}(b^t\\sigma)=\\sigma$, and write $\\sigma=b^{t_0}ab^{t_1}a\\cdots b^{t_{k-1}}a$ in normal form. Then since $\\operatorname{stem}(b^t\\sigma)$ begins with $\\operatorname{stem}(b^tb^{t_0}a)$, we have $\\operatorname{stem}(b^tb^{t_0}a)=b^{t_{0}}a$. Write $t+t_0=r+nd$ with $0\\leq r<d$, and then $b^ra=\\operatorname{stem}(b^tb^{t_0}a)=b^{t_0}a$. Now uniqueness of the normal form gives $r=t_0$, $t=nd$  and $b^tb^{t_0}a=b^{t_0}ab^{nc}=b^{t_0}ab^{cd^{-1}t}$. Now running a similar argument on $b^{cd^{-1}t}b^{t_1}a$ shows that $d$ divides $cd^{-1}t$, and $b^{cd^{-1}t}b^{t_1}a=b^{t_1}ab^{c^2d^{-2}t}$. Continuing this way shows that $d$ divides $c^jd^{-j}t$ for every $j<k$, and gives the formula \\eqref{carrythrough}.\n\nFor the converse, just note that the condition on $d$ allows us to pull $b^t$  through $\\sigma$ without changing the powers $b^{t_j}$ for $j\\leq k$.\n\\end{proof}\n\n\\begin{rmk}\nWhen $c$ and $d$ are coprime, $d$ divides $c^jd^{-j}t$ if and only if $d^{j+1}$ divides $t$, and hence left multiplication by  $b^t$ fixes all the stems in $\\Sigma_k$ if and only if $d^k$ divides $t$. However, in general it is possible that $d$ divides $c^jd^{-j}t$ but does not divide $c^{j-1}d^{-(j-1)}$. For example, consider $c=8$ and $d=12$. Then $cd^{-1}t=2t\/3$ and $c^2d^{-2}t=4t\/9$. Thus with $t=27$, $d=12$ divides $c^2d^{-2}t$ but not $cd^{-1}t$.\n\\end{rmk}\n\nEquation~\\eqref{k=0} tells us where the first integral in \\eqref{psivsmoments} comes from. For $k\\geq 1$, Lemma~\\ref{idsummands} tells us why only the summands described in \\eqref{psivsmoments} survive. Suppose that $k\\in \\mathbb{N}$ and $d$ divides $c^jd^{-j}t$ whenever $1\\leq j<k$. Then \\eqref{carrythrough} tells us that the $s$ in \\eqref{calcpiT} is $c^kd^{-k}t$. Thus the $k$th summand in the formula \\eqref{defphih} for $\\psi_{\\beta,\\mu}(T_{b^t})=\\phi_{1_\\mu}(T_{b^t})$ is\n\\begin{align*}\n(1-e^{-\\beta}d)\\sum_{\\sigma\\in \\Sigma_k}e^{-\\beta k}&\\big(\\pi(T_{b^t})(e_{k,\\sigma}\\otimes 1_\\mu)\\,\\big|\\,e_{k,\\sigma}\\otimes 1_\\mu\\big)\\\\\n&=(1-e^{-\\beta}d)\\sum_{\\sigma\\in \\Sigma_k}e^{-\\beta k}\\big(e_{k,\\sigma}\\otimes W_{b^{c^kd^{-k}t}}1_\\mu)\\,\\big|\\,e_{k,\\sigma}\\otimes 1_\\mu\\big)\\\\\n&=(1-e^{-\\beta}d)\\sum_{\\sigma\\in \\Sigma_k}e^{-\\beta k}\\int_{\\mathbb{T}} z^{c^kd^{-k}t}\\,d\\mu(z)\\\\\n&=(1-e^{-\\beta}d)|\\Sigma_k|e^{-\\beta k}\\int_{\\mathbb{T}} z^{c^kd^{-k}t}\\,d\\mu(z)\\\\\n&=(1-e^{-\\beta}d)d^ke^{-\\beta k}\\int_{\\mathbb{T}} z^{c^kd^{-k}t}\\,d\\mu(z).\n\\end{align*}\nThus we recover the formula \\eqref{psivsmoments} for $\\psi_{\\beta,\\mu}(T_{b^t})$. \n\n\nNext we have to prove that every KMS$_\\beta$ state has the form $\\psi_{\\beta,\\mu}$. So we suppose that $\\phi$ is a KMS$_\\beta$ state. Recall that the set\n\\[\n\\{T_x:x\\in \\Sigma_1\\}=\\{U^iV: 0\\leq i<d\\}\n\\]\nis a Toeplitz-Cuntz-Krieger family, and set\n\\[\nP:=1-\\sum_{x\\in \\Sigma_1} T_xT_x^*.\n\\]\nThen the KMS condition implies that $\\phi(P)=1-e^{-\\beta}d$, so we may consider the \\emph{conditioned state} \n\\[\n\\phi_P:a\\mapsto (1-e^{-\\beta}d)^{-1}\\phi(PaP).\n\\]\nThis is in particular a state on the $C^*$-subalgebra $C^*(K)=C^*(T_b)\\cong C(\\mathbb{T})$, and hence is given by a measure $\\mu$ on $\\mathbb{T}$: we choose the measure that satisfies\n\\[\n\\phi_P(T_{b^n})=\\int_{\\mathbb{T}} z^n\\,d\\mu(z)\\quad\\text{ for $n\\in \\mathbb{N}$.}\n\\]\nWe aim to prove that $\\phi=\\psi_{\\beta,\\mu}$. The argument follows closely that of \\cite[Proposition~7.1]{LRR}, though the details in the calculations are quite different.\n\nWe begin by claiming that \n\\[\n\\big\\{T_xPT_x^*:x\\in \\textstyle{\\bigcup_{k=0}^{\\infty}}\\Sigma_k\\big\\}\n\\]\nis a family of mutually orthogonal projections. To see this, take $x\\in \\Sigma_j$ and $y\\in \\Sigma_k$ and consider $PT_x^*T_yP$. If $j=k$, then either $x=y$ or $x\\vee y=\\infty$, and Nica covariance gives $PT_x^*T_yP=\\delta_{x,y}P$. If $j\\not=k$, then Nica covariance leaves a factor of the form $PT_\\sigma$ or $T_\\sigma^* P$ with $\\theta(\\sigma)>0$. Now we write $\\sigma=w\\sigma'$ with $w\\in \\Sigma_1$, and then\n\\begin{align*}\nPT_\\sigma&=\\big(1-\\sum_{z\\in \\Sigma_1} T_zT_z^*\\Big)T_wT_{\\sigma'}\\\\\n&=\\big(T_w-\\sum_{z\\in \\Sigma_1} T_zT_z^*T_w\\Big)T_{\\sigma'}\\\\\n&=(T_w-T_w)T_{\\sigma'}=0.\n\\end{align*} \nThus $PT_x^*T_yP=0$ when $j\\neq k$, and the claim follows.\n\nNow for each $n$,  \n\\[\nP_n:=\\sum_{k=0}^n\\sum_{x\\in \\Sigma_k} T_xPT_x^*\n\\]\nis a projection. Then as in the proof of \\cite[Proposition~7.2]{LRR}, the KMS condition implies that\n\\begin{align*}\n\\phi(P_n)&=\\sum_{k=0}^n\\sum_{x\\in \\Sigma_k}\\phi(T_xPT_x^*)\n=\\sum_{k=0}^n\\sum_{x\\in \\Sigma_k}e^{-\\beta k}\\phi(PT_x^*T_x)\\\\\n&=\\phi(P)\\sum_{k=0}^ne^{-\\beta k}d^k=(1-e^{-\\beta} d)\\sum_{k=0}^ne^{-\\beta k}d^k\n\\end{align*}\nconverges to $1$ as $n\\to \\infty$. It follows from \\cite[Lemma~7.3]{LRR} that for each $c\\in C^*(G,P)$, we have $\\phi(P_ncP_n)\\to \\phi(c)$ as $n\\to \\infty$.\nWe now use the KMS condition to simplify\n\\begin{align*}\n\\phi(c)=\\lim_{n\\to\\infty}\\phi(P_ncP_n)&=\\lim_{n\\to\\infty}\\sum_{j,k=0}^n\\;\\sum_{x\\in \\Sigma_j,\\;y\\in \\Sigma_k}\\phi\\big((T_xPT_x^*)c(T_yPT_y^*)\\big))\\\\\n&=\\lim_{n\\to\\infty}\\sum_{j,k=0}^n\\;\\sum_{x\\in \\Sigma_j,\\;y\\in \\Sigma_k}e^{-\\beta j}\\phi\\big(PT_x^*cT_yPT_y^*T_xP\\big))\\\\\n&=\\lim_{n\\to\\infty}\\sum_{k=0}^n\\sum_{x\\in \\Sigma_k}e^{-\\beta k}\\phi(PT_x^*cT_xP)\\\\\n&=\\lim_{n\\to\\infty}(1-e^{-\\beta}d)\\sum_{k=0}^n\\sum_{x\\in \\Sigma_k}e^{-\\beta k}\\phi_P(T_x^*cT_x).\n\\end{align*}\nThis is an analogue of the reconstruction formula of \\cite[Proposition~7.2]{LRR}.\n\nTo finish off, we take $c=T_{b^t}$ in the reconstruction formula. Then for $x\\in \\Sigma_k$, \n$T_x^*T_{b^t}T_x$ has the form $T_x^*T_{\\operatorname{stem}(b^tx)}T_{b^s}$, which vanishes unless $\\operatorname{stem}(b^tx)=x$. In that case Lemma~\\ref{idsummands} implies that $d$ divides $c^jd^{-j}t$ for all $j<k$, $b^tx =xb^{c^kd^{-k}t}$ and $T_x^*T_{\\operatorname{stem}(b^tx)}T_{b^{c^kd^{-k}t}}=T_{b^{c^kd^{-k}t}}$. (We have to worry separately about $k=0$, though.)\nThus  we have\n\\[\n\\phi(T_{b^t})=(1-e^{-\\beta}d)\\Big(\\phi_P(T_{b^t})+\\sum_{\\{k\\geq 1\\,:d\\,\\mid\\, c^jd^{-j}t\\text{ for }0\\leq j<k\\}}e^{-\\beta k}\\phi_P(T_{b^{c^kd^{-k}t}})\\Big),\n\\]\nwhich by  definition of the measure $\\mu$ is the same as $\\psi_{\\beta,\\mu}(T_{b^t})$. Thus Corollary~\\ref{btsuffices} implies that $\\phi=\\phi_{\\beta,\\mu}$. \n\n\n\nNow we add the assumption that $d$ does not divide $c$, and aim to prove that $\\mu\\mapsto\\psi_{\\beta,\\mu}$ is injective.\nSo we suppose that $\\mu$ and $\\nu$ are probability measures on $\\mathbb{T}$ satisfying $\\psi_{\\beta,\\mu}=\\psi_{\\beta,\\nu}$. To simplify the notation, we observe that the integrals appearing in our formulas are the moments\n\\[\nM_n(\\mu):=\\int_{\\mathbb{T}} z^n\\,d\\mu\n\\]\nof the measures. Since the non-negative moments characterise a probability measure, we will prove that $M_t(\\mu)=M_t(\\nu)$ for all $t\\in\\mathbb{N}$.\n\nWe prove by induction on $k$ that, if $t\\in \\mathbb{N}$, $d$ divides $c^jd^{-j}t$ for all $j<k$ and $d$ does not divide $c^kd^{-k}t$, then\n\\begin{equation}\\label{indhyp}\nM_{c^jd^{-j}t}(\\mu)=M_{c^jd^{-j}t}(\\nu)\\quad\\text{for all $j$ satisfying $0\\leq j\\leq k$.}\n\\end{equation} \nFor $k=0$, which we interpret to mean that $d$ does not divide $t$, the sum in \\eqref{psivsmoments} is absent, and we have\n\\[\n(1-e^{-\\beta}d)M_t(\\mu)=\\psi_{\\beta,\\mu}(T_{b^t})=\\psi_{\\beta,\\nu}(T_{b^t})=(1-e^{-\\beta}d)M_t(\\nu).\n\\]\nSince $e^{\\beta}>d$, we deduce that $M_t(\\mu)=M_t(\\nu)$.\n\nSuppose that the inductive hypothesis is true for $k$, and we have $s\\in \\mathbb{N}$ such that $d$ divides $c^jd^{-j}s$ for $j<k+1$ and $d$ does not divide $c^{k+1}d^{-(k+1)}s$. Then \\eqref{psivsmoments} gives\n\\[\n\\psi_{\\beta,\\mu}(T_{b^s})=(1-e^{-\\beta} d)\\big(M_s(\\mu)+e^{-\\beta}dM_{cd^{-1}s}(\\mu)+\\cdots +e^{-\\beta (k+1)}d^{k+1}M_{c^{k+1}d^{-(k+1)}s}(\\mu)\\big),\n\\]\nand hence $\\psi_{\\beta,\\mu}=\\psi_{\\beta,\\nu}$ implies that\n\\begin{align}\\label{eqmoments}\nM_s(\\mu)+e^{-\\beta}d&M_{cd^{-1}s}(\\mu)+\\cdots +e^{-\\beta (k+1)}d^{k+1}M_{c^{k+1}d^{-(k+1)}s}(\\mu)\\\\\n&=M_s(\\nu)+e^{-\\beta}dM_{cd^{-1}s}(\\nu)+\\cdots +e^{-\\beta (k+1)}d^{k+1}M_{c^{k+1}d^{-(k+1)}s}(\\nu).\\notag\n\\end{align}\nNotice that $t=cd^{-1}s$ has the property that $d$ divides $c^jd^{-j}t$ for all $j<k$ and $d$ does not divide $c^kd^{-k}t$. Thus the inductive hypothesis implies that\n\\[\nM_{c^{j+1}d^{-(j+1)}s}(\\mu)=M_{c^jd^{-j}t}(\\mu)=M_{c^jd^{-j}t}(\\nu)=M_{c^{j+1}d^{-(j+1)}s}(\\nu) \\text{ for $0\\leq j<k$.}\n\\]\nBut this says precisely that \\eqref{indhyp} holds for $1\\leq j\\leq k+1$. Now cancelling terms in \\eqref{eqmoments} gives $M_s(\\mu)=M_s(\\nu)$, which is \\eqref{indhyp} for the remaining case $j=0$.  \n Thus we have proved the inductive hypothesis for $k+1$, and this completes the proof by induction.\n\nNow we fix $t\\in \\mathbb{N}$. Since $d$ does not divide $c$, there is a first $k$ such that $d$ does not divide $c^kd^{-k}t$, and then we have \\eqref{indhyp} for this $k$. But now taking $j=0$ in \\eqref{indhyp} shows that $M_t(\\mu)=M_t(\\nu)$. Thus $\\mu$ and $\\nu$ have the same moments, and $\\mu=\\nu$. Thus $\\mu\\mapsto \\psi_{\\beta,\\mu}$ is injective.\n\n\n\nSince the sum in \\eqref{psivsmoments} is always finite, the assignment $\\mu\\mapsto \\psi_{\\beta,\\mu}$ is affine and continuous for the weak* topologies. We have shown that it is a bijection of the compact space $P(\\mathbb{T})$ onto the simplex of KMS$_\\beta$ states. Hence it is a homeomorphism, and this completes the proof of Theorem~\\ref{KMSToe}.\n\n\\section{KMS states at the critical inverse temperature}\n\nRecall from Corollary~\\ref{restonbeta} that every KMS$_{\\ln d}$ state of $C^*(G,P)$ factors through the quotient map of $C^*(G,P)$ onto the Cuntz algebra $\\mathcal{O}(G,P)$. We write $\\bar T_x$ for the image of $T_x\\in C^*(G,P)$ in $\\mathcal{O}(G,P)$.\n\n\\begin{prop}\\label{KMScrit}\nThere is a KMS$_{\\ln d}$ state $\\psi$ on $(\\mathcal{O}(G,P),\\alpha)$ such that\n\\[\n\\psi(\\bar T_x\\bar T_y^*)=\\delta_{x,y}e^{-\\beta\\theta(x)}.\n\\]\nIf $d$ does not divide $c$, then this is the only KMS state on $(\\mathcal{O}(G,P),\\alpha)$. \n\\end{prop}\n\n\n\\begin{lemma}\\label{offdiag0}\nSuppose that $d$ does not divide $c$ and $\\phi$ is a KMS$_{\\ln d}$ state of the Cuntz system $(\\mathcal{O}(G,P),\\alpha)$. Then $\\phi(\\bar T_{b^t})=0$ for all $t\\not=0$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $\\phi(\\bar T_{b^t})\\not=0$.\nSince $\\{S_j:=\\bar T_{b^ja}:0\\leq j<d\\}$ is a Cuntz family, so is \n\\[\n\\{S_\\mu:=S_{\\mu_1}S_{\\mu_2}\\cdots S_{\\mu_k}: 0\\leq\\mu_i<d\\}=\\{\\bar T_{\\sigma}:\\sigma\\in \\Sigma_k\\}.\n\\]\nThus for every $k$, \n\\[\n\\phi(\\bar T_{b^t})=\\phi\\Big(\\bar T_{b^t}\\sum_{\\sigma\\in \\Sigma_k}\\bar T_{\\sigma}\\bar T_{\\sigma}^*\\Big)=\\sum_{\\sigma\\in \\Sigma_k}\\phi(\\bar T_{b^t\\sigma}\\bar T_{\\sigma}^*)\n\\]\nand there exists $\\sigma\\in \\Sigma_k$ such that $\\phi(\\bar T_{b^t\\sigma}\\bar T_{\\sigma}^*)\\not=0$. Proposition~\\ref{charKMS} then implies that $b^t\\sigma\\vee \\sigma<\\infty$, and since $b^t\\sigma$ and $\\sigma$ have the same height, we must have $\\operatorname{stem}(b^t\\sigma)=\\sigma$. Thus we can apply Lemma~\\ref{idsummands} for every $k$, and deduce that $d$ divides $c^jd^{-j}t$ for all $j\\geq 1$. Write $g:=\\gcd(c,d)$, and define $c_1:=g^{-1}c$ and $d_1:=g^{-1}d$. Then $\\gcd(c_1,d_1)=1$ and for all $j$ we have\n\\begin{align*}\nd\\text{ divides }c^jd^{-j}t&\\Longrightarrow d_1\\text{ divides }c_1^jd_1^{-j}t\\\\\n&\\Longrightarrow d_1\\text{ divides }d_1^{-j}t\\\\\n&\\Longrightarrow d_1^{j+1}\\text{ divides }t.\n\\end{align*} \nSince $d$ does not divide $c$, we have $d_1>1$, and hence $t=0$.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{KMScrit}]\nWe choose a decreasing sequence $\\{\\beta_n\\}$ such that $\\beta_n\\to \\ln d$, and take $\\mu$ to be the Haar measure on $\\mathbb{T}$. Then by passing to a subsequence, we may assume that $\\{\\psi_{\\beta_n,\\mu}\\}$ converges weak* to a state $\\phi$ of $(C^*(G,P),\\alpha)$. Then it follows from \\cite[Proposition~5.3.23]{BR} that $\\phi$ is a KMS$_{\\ln d}$ state of $(C^*(G,P),\\alpha)$. Corollary~\\ref{restonbeta}  implies that $\\phi$ factors through a state $\\psi$ of $(\\mathcal{O}(G,P),\\alpha)$. Since the non-zero moments of the Haar measure all vanish, we have $\\psi(\\bar T_{b^t})=0$ for all $t\\not=0$, and then the formula for $\\psi$ follows.\n\nFor uniqueness, we use Lemma~\\ref{offdiag0} to see that any other KMS$_{\\ln d}$ state agrees with $\\psi$ on the elements $\\bar T_{b^t}$, and hence by Corollary~\\ref{btsuffices} on all of $\\mathcal{O}(G,P)$.\n\\end{proof}\n\n\\begin{ex}\\label{ddividesc}\nSuppose that $d$ divides $c$, that $\\beta>\\ln d$, and that $\\mu\\in P(\\mathbb{T})$. Then either $d$ divides $c^jd^{-j}t$ for all $j\\geq 0$, or $d$ does not divide $t$. Thus\n\\[\n\\psi_{\\beta,\\mu}(T_{b^t})=\\begin{cases}(1-e^{-\\beta}d)\\big(M_t(\\mu)+\\sum_{k=1}^\\infty e^{-\\beta k}d^kM_{c^kd^{-k}t}(\\mu)\\big)&\\text{if $d$ divides $t$}\\\\\n(1-e^{-\\beta}d)M_t(\\mu)&\\text{if $d$ does not divide $t$.}\n\\end{cases}\n\\]\n\n\nIf $d=c$, then the only moment appearing in our formula is $M_t(\\mu)$, and summing the geometric series shows that\n\\[\n\\psi_{\\beta,\\mu}(T_{b^t})=\\begin{cases}M_t(\\mu)&\\text{if $d$ divides $t$}\\\\\n(1-e^{-\\beta}d)M_t(\\mu)&\\text{otherwise.}\n\\end{cases}\n\\]\nNow the procedure in the proof of Proposition~\\ref{KMScrit} gives a KMS$_{\\ln d}$ state on $(\\mathcal{O}(G,P),\\alpha)$, and measures with different moments $M_{dn}$ will give different states.\n\nIf $d\\not=c$, then we write $\\delta_z$ for the point mass at $z\\in\\mathbb{T}$, and take \n\\[\n\\mu=\\frac{1}{cd^{-1}}\\sum_{w^{cd^{-1}}=1}\\delta_w.\n\\]\nThen $\\mu$ is a probability measure with moments\n\\[\nM_n(\\mu)=\\begin{cases}\n1&\\text{if $cd^{-1}$ divides $n$}\\\\\n0&\\text{otherwise.}\n\\end{cases}\n\\]\nThus\n\\[\n\\psi_{\\beta,\\mu}(T_{b^t})=\\begin{cases}1&\\text{if $cd^{-1}$ and $d$ divide $t$}\\\\\n(1-e^{-\\beta}d)&\\text{if $cd^{-1}$ divides $t$ and $d$ does not}\\\\\n0&\\text{if $cd^{-1}$ does not divide $t$.}\n\\end{cases}\n\\]\nIf we now choose $\\beta_n$ decreasing to $\\ln d$ as in the proof of Proposition~\\ref{KMScrit}, we get a KMS$_{\\ln d}$ state $\\psi_\\mu$ on $(\\mathcal{O}(G,P),\\alpha)$ such that\n\\[\n\\psi_{\\mu}(\\bar T_{b^t})=\\begin{cases}1&\\text{if $cd^{-1}$ and $d$ divide $t$}\\\\\n0&\\text{otherwise}.\\end{cases}\n\\]\n\nSo uniqueness of the KMS$_{\\ln d}$ state fails whenever $d$ divides $c$.\n\\end{ex}\n\n\\begin{ex}\nThe case $d=c$ is quite different. Then the unitary element $\\bar T_{b^c}$ commutes with everything. It and the Cuntz family $\\{\\bar T_{b^ja}:0\\leq j<c\\}$ generate $\\mathcal{O}(G,P)$: indeed, $\\bar T_a$ is a member of the Cuntz family, and we can recover $\\bar T_b$ using the formula\n\\[ \n\\sum_{j=0}^{c-2}\\bar T_{b^{j+1}a}\\bar T_{b^ja}^*+ \\bar T_{b^c}\\bar T_a\\bar T_{b^{c-1}a}=\\bar T_b\\Big(\\sum_{j=0}^{c-1}\\bar T_{b^ja}\\bar T_{b^ja}^*\\Big)=\\bar T_b.\n\\]\nThe Cuntz family gives us a homomorphism $\\pi:\\mathcal{O}_c=C^*(S_j)\\to \\mathcal{O}(G,P)$ such that $\\pi(S_j)=\\bar T_{b^ja}$, and the unitary $\\bar T_{b^c}$ gives us a homomorphism $\\rho:C(\\mathbb{T})\\to \\mathcal{O}(G,P)$ such that $\\rho(\\iota)=\\bar T_{b^c}$. Since they have commuting ranges, they induce a homomorphism $\\pi\\otimes \\rho$ of $\\mathcal{O}_c\\otimes C(\\mathbb{T})$ onto $\\mathcal{O}(G,P)$. This is in fact an isomorphism because the universal property of $\\mathcal{O}(G,P)$ gives an inverse.\n\\end{ex}\n\n\\begin{rmk}\\label{sortleftright}\nIn the definition of the Baumslag-Solitar group $\\operatorname{BS}(c,d)$, the integers $c$ and $d$ play an equal role. So at first sight it seems strange that the critical inverse temperature is dictated by $d$ alone. However, right at the beginning of his theory, Nica made a critical choice: his covariance relation was modelled on the behaviour of the \\emph{left}-regular representation. If he had started with the right-regular representation, his theory would have looked quite different.\n\nThe right-regular representation $\\rho$ of a group $G$ is characterised in terms of the usual basis $\\{e_g:g\\in G\\}$ for $\\ell^2(G)$ by $\\rho_he_g=e_{gh^{-1}}$ (the inverse has to be there to ensure that $\\rho_g\\rho_h=\\rho_{gh}$). So the natural representation of $P$ by operators $\\{R_x:x\\in P\\}$ on $\\ell^2(P)$ is given by\n\\[\nR_xe_y=\\begin{cases}0&\\text{if $y\\notin Px$}\\\\\ne_{yx^{-1}}&\\text{if $y\\in Px$.}\n\\end{cases}\n\\]\nEach $R_x$ is a \\emph{coisometry}: $R_x^*$ is an isometry. In other words, $R_x$ is a partial isometry with initial projection $R_x^*R_x:\\ell^2(P)\\to \\ell^2(Px)$ and range projection $R_xR_x^*=1$.\n\nThe analogue of Nica's partial order is the right-invariant order defined by $x\\leq_{\\textrm{r}} y\\Longleftrightarrow y\\in Px$ (and for the duration of this remark, we'll write $\\leq_{\\textrm{l}}$ for the usual left-invariant one). There is an analogous notion of ``right-quasi-lattice ordered'' involving least upper bounds $x\\vee_{\\textrm{r}}y$ with respect to $\\leq_{\\textrm{r}}$, and the analogue of Nica covariance is the relation\n\\begin{equation}\\label{rtNica}\n(R_x^*R_x)(R_y^*R_y)=\\begin{cases}R_{x\\vee_{\\textrm{r}} y}^*R_{x\\vee_{\\textrm{r}} y}&\\text{if $x\\vee_{\\textrm{r}}y<\\infty$}\\\\\n0&\\text{if $x\\vee_{\\textrm{r}}y=\\infty$,}\n\\end{cases}\n\\end{equation}\nwhich is satisfied by the right-regular representation. One can then get a universal $C^*$-algebra, which we will denote by $C^*(G,P,\\leq_{\\textrm{r}})$.\n\nFortunately, there is a device for studying this $C^*$-algebra (for which we thank Ilija Tolich). Consider the opposite group $G^{\\textrm{op}}=\\{g^\\flat:g\\in G\\}$ with $g^\\flat h^\\flat=(hg)^\\flat$, and the corresponding subsemigroup $P^\\textrm{op}$. Then the usual partial order $\\leq_{\\textrm{l}}$ on $G^\\textrm{op}$ satisfies\n\\[\ng^\\flat\\leq_{\\textrm{l}} h^\\flat\\Longleftrightarrow h^\\flat\\in g^\\flat P^\\textrm{op}=(Pg)^\\flat\\Longleftrightarrow h\\in Pg\\Longleftrightarrow g\\leq_{\\textrm{r}} h;\n\\]\nwe deduce that $(G^\\textrm{op},P^\\textrm{op})$ is quasi-lattice ordered in the usual sense if and only if $(G,P,\\leq_\\textrm{r})$ is right-quasi-lattice ordered, with $g^\\flat\\vee_{\\textrm{l}} h^\\flat =(g\\vee_{\\textrm{r}} h)^\\flat$. One can check quite easily that $R^T:x\\mapsto T_{x^\\flat}^*$ is a covariant coisometric representation of $P$ in the sense that \\eqref{rtNica} holds if and only if $T$ is a Nica-covariant representation of $P^{\\textrm{op}}$. Thus $(C^*(G^\\textrm{op},P^\\textrm{op}), R^T)$ is universal for coisometric representations satisfying \\eqref{rtNica}.\n\nWhen $G=\\operatorname{BS}(c,d)$, the opposite group is \n\\[\nG^\\textrm{op}=\\langle a, b : b^ca = ab^d \\rangle=\\operatorname{BS}(d,c).\n\\]\nThus, had we chosen to work with the partial order $\\leq_{\\textrm{r}}$, we would have found a system with a phase transition at inverse temperature $\\ln c$. (There is a minor wrinkle: because passing from the isometric representation $T$ to the coisometric representation $R^T$ involves an adjoint, the dynamics satisfies $\\alpha^\\textrm{r}_t(R_x)=e^{-i\\theta(x)}R_x$. However, one can argue that this is the natural one, because both this and the usual dynamics are implemented spatially on $\\ell^2(P)$ by the unitary representation $U$ such that $U_te_x=e^{it\\theta(x)}e_x$.)\n\\end{rmk}\n\n\\section{Ground states}\n\nA state $\\phi$ is   a \\emph{ground state} of $(C^*(G,P),\\alpha)$ if, for all analytic elements $a$ and $b$,  the entire function $z\\mapsto \\phi(a\\alpha_z(b))$ is bounded in the upper half-plane ${\\operatorname{Im}} z>0$. The KMS$_\\infty$ states are the weak* limits of sequences of KMS$_{\\beta_n}$ states as $\\beta_n\\to \\infty$. Every KMS$_\\infty$ state is a ground state, but a ground state need not be a KMS$_\\infty$ state by \\cite[Proposition~5.3.23]{BR} and \\cite[Proposition~3.8]{CM}.\n\n\n\\begin{thm}\\label{thmground}\nSuppose that $\\omega$ is a state of the Toeplitz algebra $\\mathcal{T}(\\mathbb{N})=C^*(S)$. Then there is a ground state $\\psi_\\omega$ of $(C^*(G,P),\\alpha)$ such that\n\\begin{equation}\\label{defground}\n\\psi_\\omega(T_xT_y^*)=\n\\begin{cases}\n0&\\text{if $\\theta(x)\\not=0$ or $\\theta(y)\\not=0$}\\\\\n\\omega(S^sS^{*t})&\\text{if $x=b^s$ and $y=b^t$.}\n\\end{cases}\n\\end{equation}\nThe state $\\psi_\\omega$ is a KMS$_\\infty$ state if and only if $\\omega$ factors through the quotient map $q:\\mathcal{T}(\\mathbb{N})\\to C(\\mathbb{T})$. The map $\\omega\\mapsto \\psi_\\omega$ is an affine isomorphism of the state space of $\\mathcal{T}(\\mathbb{N})$ onto the compact convex set of ground states.\n\\end{thm}\n\nThere are many states of $\\mathcal{T}(\\mathbb{N})$ which do not factor through $q:\\mathcal{T}(\\mathbb{N})\\to C(\\mathbb{T})$: for example, the vector states given by unit vectors in $\\ell^2$. Thus Theorem~\\ref{thmground} implies that the system $(C^*(G,P),\\alpha)$ has many ground states which are not KMS$_\\infty$ states. Thus (in the terminology of \\cite{CM}) the system admits a second phase transition at $\\beta=\\infty$.\n\nWe now do some preparation for the the proof of Theorem~\\ref{thmground}. First we need to be able to recognise ground states.\n\n\\begin{lemma}\\label{idground}\nA state $\\psi$ of $(C^*(G,P),\\alpha)$ is a ground state if and only if\n\\begin{equation}\\label{charground}\n\\psi(T_xT_y^*)\\not=0\\Longrightarrow \\theta(x)=\\theta(y)=0.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $\\psi$ is a ground state and $\\psi(T_xT_y^*)\\not=0$. Then\n\\[\n\\big|\\psi(T_x\\alpha_{r+is}(T_y^*))\\big|=\\big| e^{-i(r+is)\\theta(y)}\\psi(T_xT_y^*)\\big|=e^{s\\theta(y)}|\\psi(T_xT_y^*)|\n\\]\nis bounded on the upper half-plane $s\\geq 0$, and hence $\\theta(y)=0$.  Since $\\psi(T_yT_x^*)=\\overline{\\psi(T_xT_y^*)}\\not=0$, we also deduce that $\\theta(x)=0$.\n\nNext suppose that $\\psi$ is a state satisfying \\eqref{charground}. Let $X=T_xT_y^*$ and $Y$ be analytic elements for $\\alpha$. Then the Cauchy-Schwarz inequality gives\n\\begin{align}\\label{CSground}\n|\\psi(Y\\alpha_{r+is}(X))|^2&=e^{-2s(\\theta(x)-\\theta(y))}|\\psi(YT_xT_y^*)|^2\\\\\n&\\leq e^{-2s(\\theta(x)-\\theta(y))}\\psi(Y^*Y)\\psi(T_yT_x^*T_xT_y^*)\\notag\\\\\n&= e^{-2s(\\theta(x)-\\theta(y))}\\psi(Y^*Y)\\psi(T_yT_y^*).\\notag\n\\end{align}\nIf $\\theta(y)\\not= 0$ then \\eqref{charground} implies that $\\psi(T_yT_y^*)=0$, and the right-hand side of \\eqref{CSground} is trivially bounded. If $\\theta(y)=0$, then the right-hand side of \\eqref{CSground} is bounded by $\\psi(Y^*Y)\\psi(T_yT_y^*)$. So either way, $|\\psi(Y\\alpha_{r+is}(X))|$ is bounded for $s\\geq 0$, and $\\psi$ is a ground state.\n\\end{proof}\n\nNext we need a good supply of representations. Our basic construction was inspired by our earlier one using induced representations. \n\nWe continue to use the orthonormal basis $\\{e_{k,\\sigma}:\\sigma\\in \\Sigma_k\\}$ for $\\ell^2(\\Sigma_k)$.\n\n\\begin{lemma}\\label{defUV}\nSuppose that $W$ is an isometry of a Hilbert space $H$. Then there are isometries $U$ and $V$ on $\\bigoplus_{k=0}\\ell^2(\\Sigma_k)\\otimes H$ such that\n\\begin{align*}\nU(e_{k,\\sigma}\\otimes h)&=e_{k,\\operatorname{stem}(b\\sigma)}\\otimes W^sh\\quad\\text{where $b\\sigma=\\operatorname{stem}(b\\sigma)b^s$, and }\\\\\nV(e_{k,\\sigma}\\otimes h)&=e_{k+1,a\\sigma}\\otimes h.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nLemma~\\ref{maponstems} implies that $\\sigma\\mapsto \\operatorname{stem}(b\\sigma)$ is a bijection of $\\Sigma_k$ onto $\\Sigma_k$. Thus if $\\{h_i:i\\in I\\}$ is an orthonormal basis for $H$, then $\\{e_{k,\\operatorname{stem}(b\\sigma)}\\otimes W^sh_i:\\sigma\\in \\Sigma,\\,i\\in I\\}$ is an orthonormal set in $\\ell^2(\\Sigma_k)\\otimes H$ for each $k$. Thus there is an isometry $U$ as claimed. Since each $a\\sigma$ is already a stem, Lemma~\\ref{maponstems} also implies that $\\sigma\\mapsto a\\sigma$ is an injection of $\\Sigma_k$ in $\\Sigma_{k+1}$ for each $k$. Thus $\\{e_{k+1,a\\sigma}\\otimes h_i\\}$ is also orthonormal, and there is an isometry $V$ with the required property.\n\\end{proof}\n\n\\begin{prop}\nSuppose that $W$ is an isometry of a Hilbert space $H$, and $U$, $V$ are as in Lemma~\\ref{defUV}. Then $U$ and $V$ satisfy the relations \\eqref{t1}, \\eqref{t4} and \\eqref{t5} of Proposition~\\ref{defrel}.\n\\end{prop}\n\n\\begin{proof}\nThe calculations in the fourth paragraph of the proof of Proposition~\\ref{indrepconst} show that $U$ and $V$ satisfy \\eqref{t1} and \\eqref{t5}. To verify \\eqref{t4}, we need a formula for $U^*$. We claim that \n\\begin{equation}\\label{formU*}\nU^*(e_{k,\\tau}\\otimes h)=e_{k,\\rho}\\otimes W^{*t}h \\quad\\text{where $\\rho\\in \\Sigma_k$ satisfies $\\tau b^t=b\\rho$;}\n\\end{equation}\nLemma~\\ref{maponstems} implies that there is a unique stem $\\rho$ such that $b\\rho$ begins with $\\tau$, and then $t$ is uniquely determined by $\\tau b^t=b\\rho$. To prove the claim, we compare\n\\begin{equation}\\label{lhs}\n\\big(e_{k,\\rho}\\otimes W^{*t}h\\,|\\,e_{k,\\sigma}\\otimes g\\big)=\\delta_{\\rho,\\sigma}(h\\,|\\,W^tg)\\quad\\text{where $\\tau b^t=b\\rho$}\n\\end{equation}\nwith \n\\begin{equation}\\label{rhs}\n\\big(e_{k,\\tau}\\otimes h\\,|\\,U(e_{k,\\sigma}\\otimes g)\\big)\n=\\delta_{\\tau,\\operatorname{stem}(b\\sigma)}(h\\,|\\,W^sg)\\quad\\text{where $b\\sigma=\\operatorname{stem}(b\\sigma)b^s$.}\n\\end{equation}\nFirst, suppose that $\\rho=\\sigma$. Then $b\\sigma=\\operatorname{stem}(b\\sigma)b^s=\\operatorname{stem}(b\\rho)b^s=\\operatorname{stem}(\\tau b^t)b^s=\\tau b^s$ because $\\tau$ is a stem. Thus $\\tau=\\operatorname{stem}(b\\sigma)$. Now $\\tau b^t=b\\rho=b\\sigma=\\operatorname{stem}(b\\sigma)b^s=\\tau b^s$, and hence $s=t$. Thus \\eqref{lhs} and \\eqref{rhs} agree.\nSecond, suppose that $\\rho\\neq \\sigma$. By Lemma~\\ref{maponstems}\n\\eqref{hkm}, $x\\mapsto \\operatorname{stem}(bx)$ is a bijection on $\\Sigma_k$, and hence $\\operatorname{stem}(b\\sigma)\\neq\\operatorname{stem}(b\\rho)=\\operatorname{stem}(\\tau b^t)=\\tau$, and both \\eqref{lhs} and \\eqref{rhs} are $0$. This proves the claim.\n\n\n\n\nWe now compute the right-hand side of \\eqref{t4}:\n\\begin{align*}\nU^{d-1}VU^{*c}(e_{k,\\sigma}\\otimes h)&=U^{d-1}V(e_{k,\\mu}\\otimes W^{*t}h)\\quad\\text{where $\\mu\\in \\Sigma_k$ satisfies $\\sigma b^t=b^c\\mu$}\\\\\n&=e_{k+1,\\operatorname{stem}(b^{d-1}a\\mu)}\\otimes W^sW^{*t}h\\quad\\text{where $b^{d-1}a\\mu=\\operatorname{stem}(b^{d-1}a\\mu)b^s$}\\\\\n&=e_{k+1,b^{d-1}a\\mu}\\otimes W^{*t}h\n\\end{align*}\nbecause $b^{d-1}a\\mu$ is a stem. The left-hand side of \\eqref{t4} is\n\\[\nU^*V(e_{k,\\sigma}\\otimes h)=e_{k+1,\\rho}\\otimes W^{*r}h \\quad\\text{where $\\rho\\in \\Sigma_{k+1}$ satisfies $a\\sigma b^r=b\\rho$.} \n\\]\nNow the equation\n\\[\nb(b^{d-1}a\\mu)=b^da\\mu=ab^c\\mu=a\\sigma b^t\n\\]\nimplies that $\\rho=b^{d-1}a\\mu$ (because $\\rho$ is the unique stem such that $b\\rho$ begins with $a\\sigma$) and then $r=t$. Thus \\eqref{t4} follows.\n\\end{proof}\n\n\n\\begin{cor}\\label{absground}\nSuppose that $W$ is an isometry on a Hilbert space $H$, and $U$, $V$ are the isometries described in Lemma~\\ref{defUV}. Let $\\pi_{U,V}$ be the corresponding representation of $C^*(G,P)$ on $\\bigoplus_{k\\geq 0}\\ell^2(\\Sigma_k)\\otimes H$. Then for every unit vector $h$ in $H$, there is a ground state  $\\psi_{h,W}$ of $(C^*(G,P),\\alpha)$ such that\n\\[\n\\psi_{h,W}(a)=\\big(\\pi_{U,V}(a)(e_{0,e}\\otimes h)\\,|\\,e_{0,e}\\otimes h\\big).\n\\]\n\\end{cor}\n\n\\begin{proof}\nSince $T_x$ maps $\\ell^2(\\Sigma_0)\\otimes H$ into $\\ell^2(\\Sigma_{\\theta(x)})\\otimes H$, \n\\[\n\\psi_{h,W}(T_xT_y^*)=\\big(\\pi_{U,V}(T_y)^*(e_{0,e}\\otimes h)\\,|\\,\\pi_{U,V}(T_x)^*(e_{0,e}\\otimes h)\\big)\n\\]\nvanishes unless $\\theta(x)=0=\\theta(y)$. So Lemma~\\ref{idground} implies that $\\psi_{h,W}$ is a ground state.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thmground}]\nSuppose $\\omega$ is a state of $\\mathcal{T}(\\mathbb{N})=C^*(S)$. We consider the GNS representation $\\pi_\\omega$ of $\\mathcal{T}(\\mathbb{N})$ on $H_\\omega$ with cyclic vector $\\xi_\\omega$, from which we can recover $\\omega$ via the formula $\\omega(c)=(\\pi_\\omega(c)\\xi_\\omega\\,|\\,\\xi_\\omega)$. Applying Corollary~\\ref{absground} with $W=\\pi_\\omega(S)$, $U$ and $V$ the isometries of Lemma~\\ref{defUV},  and $h=\\xi_\\omega$ gives a ground state $\\psi_\\omega:=\\psi_{\\xi_\\omega, \\pi_\\omega(S)}$ of $(C^*(G,P),\\alpha)$ such that \n\\[\n\\psi_\\omega(a)=\\big(\\pi_{U,V}(a)(e_{0,e}\\otimes \\xi_\\omega)\\,|\\,e_{0,e}\\otimes \\xi_\\omega\\big).\n\\]\nWe need to verify the formula \\eqref{defground}.\n\nSince $V=\\pi_{U,V}(T_a)$ maps $\\ell^2(\\Sigma_0)\\otimes H_\\omega=\\mathbb{C} e_{0,e}\\otimes H_\\omega$ into $\\ell^2(\\Sigma_1)\\otimes H_\\omega$, we have\n\\[\n\\psi_\\omega(T_xT_y^*)=0 \\text{\\ \\ unless $\\theta(x)=0=\\theta(y)$.}\n\\]\nIf $\\theta(x)=0=\\theta(y)$, then $x=b^s$ and $y=b^t$ for some $s,t\\in \\mathbb{N}$, and\n\\begin{align*}\n\\psi_\\omega(T_xT_y^*)&=\\psi_\\omega(T_b^sT_b^{*t})\\\\\n&=\\big(\\pi_{U,V}(T_b^sT_b^{*t})(e_{0,e}\\otimes \\xi_\\omega)\\,|\\,e_{0,e}\\otimes \\xi_\\omega\\big)\\\\\n&=\\big(U^sU^{*t}(e_{0,e}\\otimes \\xi_\\omega)\\,|\\,e_{0,e}\\otimes \\xi_\\omega\\big)\\\\\n&=\\big(U^{*t}(e_{0,e}\\otimes \\xi_\\omega)\\,|\\,U^{*s}(e_{0,e}\\otimes \\xi_\\omega)\\big).\n\\end{align*}\nSince $\\Sigma_0=\\{e\\}$, the formula \\eqref{formU*} for $U^*$ collapses to $U^*(e_{0,e}\\otimes h)=e_{0,e}\\otimes W^*h$, and we have\n\\begin{align*}\n\\psi_\\omega(T_xT_y^*)&=\\big(e_{0,e}\\otimes \\pi_{\\omega}(S)^{*t}\\xi_\\omega\\,|\\,e_{0,e}\\otimes \\pi_{\\omega}(S)^{*s}\\xi_\\omega\\big)\\\\\n&=\\big(\\pi_{\\omega}(S)^{*t}\\xi_\\omega\\,|\\,\\pi_{\\omega}(S)^{*s}\\xi_\\omega\\big)\\\\\n&=\\big(\\pi_{\\omega}(S^sS^{*t})\\xi_\\omega\\,|\\,\\xi_\\omega\\big)\\\\\n&=\\omega(S^sS^{*t}),\n\\end{align*}\nas in \\eqref{defground}.\n\nNext we suppose that $\\psi_\\omega$ is a KMS$_\\infty$ state. Then there are an increasing sequence $\\beta_n\\to \\infty$ and KMS$_{\\beta_n}$ states $\\phi_n$ such that $\\phi_n$ converges weak* to $\\psi_\\omega$. Corollary~\\ref{restonbeta} implies that each $\\phi_n$ factors through the quotient by the ideal generated by $1-T_bT_b^*$, and hence so does the limit $\\psi_\\omega$. The kernel of $q$ is spanned by the elements $S^m(1-SS^*)S^{*n}$ (they are a family of matrix units spanning $\\ker q=\\mathcal{K}(\\ell^2)$), and the formula \\eqref{defground} implies that\n\\[\n\\omega(S^m(1-SS^*)S^{*n})=\\psi_\\omega(T_{b^m}(1-T_bT_b^*)T_{b^n}^*)=0.\n\\]\nThus $\\omega$ factors through $q$.\n\nConversely, suppose that $\\omega$ factors through $q$. Then there is a probability measure $\\mu$ on $\\mathbb{T}$ such that $\\omega(c)=\\int q(c)\\,d\\mu$ for all $c\\in \\mathcal{T}(\\mathbb{N})$. Choose a sequence $\\beta_n$ with $\\beta_n\\to \\infty$. Then for each $n$, the state $\\psi_{\\beta_n,\\mu}$ is determined by Corollary~\\ref{btsuffices} and the formula \\eqref{psivsmoments} for $\\psi_{\\beta_n,\\mu}(T_{b^t})$ in Proposition~\\ref{KMSToe}. The sum on the right-hand side of \\eqref{psivsmoments} is finite, and for each $k$ we have\n\\[\ne^{-\\beta_nk}d^k\\int_{\\mathbb{T}} z^{c^kd^{-k}t}\\,d\\mu(z)\\to 0\\quad\\text{as $n\\to \\infty$.}\n\\]\nSince we also have $1-e^{-\\beta_n}d\\to 1$, we deduce that\n\\[\n\\psi_{\\beta_n,\\mu}(T_b^t)\\to \\int z^t\\,d\\mu(z)=\\int q(S^t)\\,d\\mu=\\psi_\\omega(T_{b^t}).\n\\]\nThus $\\psi_\\omega$ is a KMS$_\\infty$ state.\n\nThe formula \\eqref{defground} shows that $\\omega\\mapsto \\psi_\\omega$ is affine, weak* continuous and one-to-one. To see that is is onto, suppose $\\phi$ is a ground state. Since $T_b$ is a non-unitary isometry, Coburn's theorem implies that there is an isomorphism $\\pi_{T_b}$ of $\\mathcal{T}(\\mathbb{N})$ into $C^*G,P)$ such that $\\pi_{T_b}(S)=T^b$, and then $\\omega:=\\psi\\circ\\pi_{T_b}$ is a state of $\\mathcal{T}(\\mathbb{N})$. Lemma~\\ref{idground} implies that $\\phi$ vanishes on all spanning elements except those of the form $T_{b^s}T_{b^t}^*$, and formula \\eqref{defground} shows that $\\phi$ agrees with $\\psi_{\\omega}$ on all spanning elements. Thus $\\phi=\\psi_\\omega$, and $\\omega\\mapsto \\psi_\\omega$ is onto. Now we can deduce that it is a homeomorphism of the compact state space of $\\mathcal{T}(\\mathbb{N})$ onto the compact set of ground states.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n    \n    \n   \n    Complex adaptive systems \\cite{hollandComplexAdaptiveSystems1992} \n    often arise when interacting nodes (or agents)\n    adapt their dynamics (or strategies)\n    due to external influences.\n   \n    This occurs for example in socio-economic systems, such as\n    epidemic and information spreading \n    \\cite{funkModellingInfluenceHuman2010,pastor-satorrasEpidemicProcessesComplex2015},\n    markets \\cite{sornetteCriticalMarketCrashes2003},\n    evolution of languages \\cite{steelsModelingCulturalEvolution2011},\n    or evolutionary games with environmental feedback\n    \\cite{tilmanEvolutionaryGamesEnvironmental2020},\n   \n    but also in automation problems\n    including machine learning \\cite{caoSelfAdaptiveEvolutionaryExtreme2012} or control theory\n    \\cite{astromTheoryApplicationsAdaptive1983}.\n   \n    In standard network models adaptive dynamics is often imposed by\n    creation, deletion or rewiring of edges between pairs of nodes\n    \\cite{grossAdaptiveCoevolutionaryNetworks2008},\n    some aspects of which have been extended\n    to higher-order networks\n    \\cite{battistonNetworksPairwiseInteractions2020,\n        horstmeyerAdaptiveVoterModel2020,\n        schlagerStabilityAnalysisMultiplayer2021}.\n   \n    From a more general perspective adaptivity is characterized by a\n    set of strategies which are selected either by the system itself or by individual agents in the system,\n    each leading to potentially different dynamical rules and future evolution.\n   \n    \n    The adaption of strategies may be characterized by a change of parameters.\n   \n    This occurs for example in piece-wise deterministic Markov processes (PDMPs)\n    \\cite{benaimStabilityPlanarRandomly2014,\n        hieuDynamicalBehaviorStochastic2015,\n        liuAnalysisSIRSEpidemic2017,\n        hurthRandomSwitchingBifurcations2020}, where\n    some parameter of the system\n    evolves according to an additional, independent stochastic process.\n   \n    Assuming that the parameter space represents a possible set of strategies,\n    this is equivalent to a system where some decision maker (randomly) \n    changes the strategy.\n   \n    If this parameter crosses a bifurcation the stability of the system\n    changes abruptly, typically leading to an entirely different dynamical evolution.\n   \n    On the other hand, if parameter changes are influenced by the history of the system and\n    occur with some temporal delay usually\n    delay equations \\cite{kuangDelayDifferentialEquations2012} are considered.\n   \n    Examples are biomedical models of cancer evolution \n    \\cite{bakerModellingAnalysisTimelags1998,\n        villasanaDelayDifferentialEquation2003},\n    population dynamics \\cite{gopalsamyStabilityOscillationsDelay1992}\n    or\n    machine learning \\cite{grigoryevaOptimalNonlinearInformation2015}.\n   \n    It is reasonable to assume that in real world systems both\n    effects occur: delayed adaptation depending on the history of the state and\n    piece-wise deterministic evolution coupled to sudden\n    strategy changes.\n   \n    Still, these approaches have so far not been connected.\n   \n    \n    \n    \n    \n    Particularly interesting are systems with policy makers\n    who adapt either the rules of the system or their own behavior\n    according to some, usually complex, evaluation mechanism.\n   \n    For example, restrictions and lockdown laws have been enforced and lifted\n    during the Sars-Cov-2 pandemics depending on the recent course of the pandemic\n    \\cite{schlickeiserReasonableLimiting7Day2021},\n    and are regularly changed, e.g.,  due to the emergence of new variants\n    or available vaccinations.\n   \n    Therefore it is reasonable to ask how basic epidemic models behave in the context\n    of adaptive policies.\n    This is related to recent models for\n    the influence of individual risk perception on the epidemic spreading of Sars-Cov-2\n    \\cite{dongesInterplayRiskPerception2022}, where direct feedback\n    mechanisms are considered.\n    \n    \n   \n    Epidemic modeling has a long history \\cite{pastor-satorrasEpidemicProcessesComplex2015},\n    the most famous examples being the SIS and SIR models\n    \\cite{kermackContributionMathematicalTheory1927,andersonInfectiousDiseasesHumans1992}.\n    In these compartmental models the population is divided into \n    susceptible ($\\suscept$) and infected individuals ($\\infect$), which after recovery become either susceptible again or recover ($\\recov$), for SIS and SIR systems, respectively.\n   \n    These models are often considered on contact networks,\n    where existing links determine possible infections between individuals, see Ref.~\\cite{pastor-satorrasEpidemicProcessesComplex2015} and references therein.\n   \n    Many extensions have been studied,\n    from additional compartements \\cite{brauerCompartmentalModelsEpidemiology2008},\n    to adaptive rewiring of edges\n    \\cite{grossEpidemicDynamicsAdaptive2006,shawFluctuatingEpidemicsAdaptive2008},\n    adaptive force of infection\n    \\cite{capassoMathematicalStructuresEpidemic1993,\n        donofrioInformationrelatedChangesContact2009,fenichelAdaptiveHumanBehavior2011}, and\n    including vaccination\n    \\cite{britoExternalitiesCompulsaryVaccinations1991,\n        shulginPulseVaccinationStrategy1998, donofrioVaccinatingBehaviourInformation2007,\n        zamanStabilityAnalysisOptimal2008}.\n   \n    Depending on the specific model and parameters, the system state\n    typically either approaches\n    a stable disease-free or endemic equilibrium,\n    oscillates, or shows more complex dynamics.\n   \n    Apparently such adaptive models often become analytically unfeasible,\n    even if the high-dimensional network dynamics is reduced to the level of differential equations by appropriate closure techniques. \n   \n    \n    \n    \n    \n    \n    \n    In this paper we propose a framework for history\n    dependent strategy adaption with piece-wise deterministic dynamics.\n   \n    This is applied to simple models of adaptive epidemics\n    for which an analytical treatment is possible.\n   \n    In particular we consider a threshold-based activation and deactivation of\n    edges within a fixed contact network for the examples of SIS and SIR systems.\n   \n    We observe and describe a large parameter region with stable oscillations, as well as bifurcations towards stable disease-free and stable endemic states for the case\n    of SIS.\n   \n    We derive analytic expressions for the period and observe that\n    the network topology significantly influences the agreement by\n    comparing different types of random networks.\n   \n    \n    \n    The paper is structured as follows.\n    In Sec.\\ref{sec:dynamics} we propose a model of strategy adaption.\n    Section~\\ref{sec:EoN} recapitulates SIS and SIR models on networks.\n    In Sec.~\\ref{sec:epidemics-strategy} we apply the adaptive framework\n    to SIS and SIR epidemics on networks.\n    A summary and outlook is presented in Sec.~\\ref{sec:outlook}.\n    \n    \n    \\section{Dynamics with strategy adaption}\n    \\label{sec:dynamics}\n    \n    There are many ways to include adaptivity into dynamical\n    systems, e.g., by rewiring rules \\cite{grossEpidemicDynamicsAdaptive2006}\n    on the network level or\n    by state-dependent parameter changes \\cite{donofrioInformationrelatedChangesContact2009} on the\n    level of differential equations.\n   \n    In contrast, here we extend a system with a strategy\n    space $\\strat$ such that the dynamics\n    depends on the currently active strategy $S_i \\in \\strat$.\n   \n    For example, in a system with infectious dynamics, the strategies $S_i$ could refer\n    to different combinations of reduced numbers of average contacts $\\degavg_i < \\degavg_0$,\n    and\/or increased measures against transmission, $\\tau_i < \\tau$.\n   \n    Alternatively, such a strategy could impose the emergence\n    (or deletion) of additional compartments $X \\in \\{\\mathrm{Q}, \\mathrm{E}, \\mathrm{\\tilde{I}}, \\dots\\}$ into the model, e.g.,\n    enforcing quarantine, separately counting exposed but not infectious agents,\n    or introducing new variants of the disease.\n   \n    \n    It is natural to assume that the chosen strategy depends on some\n    observable function $g : \\Gamma \\rightarrow \\mathbb{R}^n$, which evaluates the\n    state $x\\in\\Gamma$ of the system.\n   \n    This is in particular relevant if\n    the strategies are chosen by some decision maker(s)\n    based on their evaluation metric.\n   \n    Note that in real social systems often many different metrics are used\n    to evaluate the same state, e.g., \n    incidence, hospitalization rate, and vaccination rate are all possible options \n    to evaluate the state of an epidemic.\n   \n    For the sake of simplicity we consider a one-dimensional metric, $n=1$, $g(x)\\in\\mathbb{R}$,\n    in the following.\n   \n    \n    In general, the history and the current state are important for\n    the evaluation of future strategies.\n   \n    In order to include the recent history of the state\n    we average the observable $g$ over time\n    \\cite{donofrioInformationrelatedChangesContact2009},\n    \\begin{equation}\n        J(t) = \\int_{0}^\\infty \\rho(t') g[x(t - t')] \\ud t'\n        = \\int_{-\\infty}^t \\rho(t - t') g[x(t')] \\ud t',\n        \\label{eq:def-indicator}\n    \\end{equation}\n    with a suitable integration kernel $\\rho$, also known as delay kernel,\n    satisfying $\\int_{t = 0}^\\infty \\rho(t)\\ud t = 1$.\n   \n    The function $J$ at the time $t$ acts as a measure on how the\n    strategy of the system is adapted due to the history of\n    the state. We call $J$ strategy function in the following.\n   \n    Additionally let us define the adaption function $A$ which specifies\n    which strategy $S\\in\\strat$ is currently chosen.\n   \n    For observable $g$ and strategy function $J$ defined as above,\n    the adaption function is defined as a mapping from\n    $A : T \\times \\mathbb{R} \\times \\strat \\rightarrow \\strat$,\n    where $T$ is the time domain and $A(t, J(t), S) = S'\\in\\strat$.\n\n    \n    \n\n   \n    We define an adaptive dynamical system with discrete strategies as follows.\n    Let $\\Phi$ be a map from\n    the strategy space $\\strat$ to a flow on $\\Gamma$,\n    i.e., $\\Phi_S := \\Phi(S)$ is a mapping\n    $\\Phi_S : T\\times \\Gamma \\rightarrow \\Gamma$ for all $S\\in\\strat$\n    with $\\Phi_S(0, x) = x$ and\n    $\\Phi_S[t_2, \\Phi_S(t_1, x)] = \\Phi_S(t_1 + t_2, x)$\n    for all $t_1, t_2 \\in T$ and $x\\in\\Gamma$.\n   \n    An adaptive dynamical system is then defined as the tuple\n    $(T, \\Gamma, \\Phi, \\strat, A)$, such that for each $S\\in\\strat$\n    the triple $(T, \\Gamma, \\Phi(S))$ is a dynamical system and\n    where the strategy $S_t$ evolves according to\n    the adaption function $A$ as defined above.\n\n    \n    \n    We emphasize that the time evolution of the system\n    depends on the current strategy $S_{t}$, and simultaneously the\n    current strategy depends on the time evolution of the system.\n   \n    This leads to a nontrivial feedback mechanism and possibly complex and\n    interesting dynamics.\n   \n    In order to introduce such an adaptive mechanism to some dynamical system,\n    it is therefore necessary to specify a set of strategies $\\strat$ and their\n    implications on the internal dynamics,\n    some observable $g$ and strategy function $J$, and how\n    the system adapts its dynamics to changes in $J$ through the function $A$.\n   \n    \n    This framework connects piecewise deterministic\n    Markov processes and delay equations:\n   \n    If the switching of strategies $A$\n    is governed by a Markov process\n    the full system is equivalent to PDMPs,\n    see e.g. \\cite{benaimStabilityPlanarRandomly2014,\n        hieuDynamicalBehaviorStochastic2015,\n        liuAnalysisSIRSEpidemic2017,\n        hurthRandomSwitchingBifurcations2020}.\n   \n    On the other hand,\n    if the selected strategy depends on some parameter \n    evaluated at a delayed time $\\tau$ in the past,\n    i.e., the integration kernel is\n    $\\rho(t) = \\delta(t - \\tau)$,\n    the resulting system can be reduced to delay equations\n    \\cite{busenbergDelayDifferentialEquations1991}.\n   \n    In the following we apply this adaptive framework\n    to epidemiological models on networks.\n   \n    \n    \n    \\begin{figure}[b!]\n        \\includegraphics[scale=0.9]{fig1}\n        \\caption{Evolution of SIS and SIR epidemics.\n            (a) Infection along $\\suscept\\infect$-edges with rate $\\tau$.\n            (b) Recovery of infected nodes with rate $\\gamma$\n            for SIS (left) and SIR (right).            \n        }\n        \\label{fig:sketch-SIRS}\n    \\end{figure}\n    \\section{Epidemics on networks}\\label{sec:EoN}\n\n    \n    One of the simplest epidemiological models on networks is the SIS model, see \\cite{pastor-satorrasEpidemicProcessesComplex2015}.\n   \n    Let $\\graph = (\\nodes, \\edges)$ be a graph consisting of nodes $\\nodes$ and\n    edges $\\edges$.\n    The state of each node $n \\in \\nodes$\n    is either susceptible ($X(n) = \\suscept$) or infected ($X(n) = \\infect$).\n   \n    The epidemic is transmitted through the edges $(i,j)\\in\\edges$\n    from infected nodes to adjacent susceptible nodes with a fixed\n    transmission rate $\\tau$, see Fig.~\\ref{fig:sketch-SIRS}.\n   \n    Recovery of infected nodes occurs with rate $\\gamma$. For SIS\n    recovered nodes become susceptible again.\n   \n    The full SIS dynamics on a network is described by the unclosed\n    differential equations for the number of infected $\\ninf$ and\n    the number of susceptible $\\nsus$ individuals \\cite{kissMathematicsEpidemicsNetworks2017},\n    \\begin{align}\n        \\begin{split}\n            \\ninf' &= - \\gamma \\ninf + \\tau \\nsi,  \\\\\n            \\nsus' &= \\phantom{+} \\gamma \\ninf - \\tau \\nsi,\n        \\end{split}\\label{eq:sis-unclosed}\n    \\end{align}\n    where $\\nsi$ denotes the number of $\\suscept\\infect$-edges.\n   \n    The differential equation for $\\nsi$ depends on the higher-order\n    moments $n_{abc}$ with $a,b,c \\in \\{\\suscept,\\infect\\}$, which\n    in general leads to an infinite series of differential equations.\n   \n    Only for specific network types, such as the complete graph, or by\n    applying closure relations to Eq.~\\eqref{eq:sis-unclosed}\n    one obtains a finite system of differential equations, which in general only approximate\n    the full network dynamics, see e.g.\\ \\cite{kuehnMomentClosureBrief2016,kissMathematicsEpidemicsNetworks2017}.\n   \n    \n    In the following we focus on the simplest closure relation, given\n    by the pairwise approximation\n    $\\nsi \\approx \\frac{\\degavg_0}{N} \\ninf \\nsus$,\n    which is based on the assumption that connections\n    between $\\suscept$ and $\\infect$ individuals are spread homogeneously\n    through the network.\n    Here, $\\degavg_0$ denotes the average degree of nodes in the network and\n    $N = |\\nodes|$ is the number of nodes.\n   \n    Without birth and death-processes the number of individuals remains\n    constant and is given by\n    $N = \\ninf + \\nsus$.\n   \n    Altogether, introducing the proportion of infected\n    individuals $\\xinf = \\ninf \/ N \\in [0, 1]$\n    the dynamics reduces to the one-dimensional ODE\n   \n    \\begin{align}\n        \\xinf' = - \\gamma \\xinf + \\tau \\degavg_0 (1 - \\xinf)\\xinf.\n        \\label{eq:ode-pairwise}\n    \\end{align}\n   \n    Stability analysis reveals two different regimes \\cite{kissMathematicsEpidemicsNetworks2017}:\n    For $\\beta := \\tau \\degavg_0  \/ \\gamma < 1$ the disease free state\n    $x_0 = 0$ is stable and it is the only equilibrium within the reasonable interval $\\xinf \\in [0, 1]$.\n   \n    For $\\beta > 1$ a transcritical bifurcation occurs and\n    $x_0$ becomes unstable, while the endemic state\n    $x_e := 1 - 1\/\\beta \\in [0, 1]$ emerges as a stable equilibrium.\n    \n    \n    Another simple model for the spreading of a\n    single epidemic wave through a population is the\n    SIR model,  see e.g. \\cite{kissMathematicsEpidemicsNetworks2017}.\n   \n    Here, infected individuals which recover\n    are fully immune and removed ($\\recov$) from the system,\n   \n    \\begin{align}\n        \\begin{split}\n            \\ninf' &= - \\gamma \\ninf + \\tau \\nsi,  \\\\\n            \\nsus' &= \\phantom{+ \\gamma \\ninf} - \\tau \\nsi,\\\\\n            \\nrec' &= \\phantom{+} \\gamma \\ninf.\n        \\end{split}\\label{eq:sir-unclosed}\n    \\end{align}\n   \n    The corresponding pairwise closed moment equations reduce to the\n    two-dimensional system\n    \\begin{align}\n        \\xinf' &= -\\gamma\\xinf + \\tau \\degavg_0 \\xinf \\xsus\\\\\n        \\xsus' &= \\phantom{+\\gamma\\xinf} -\\tau \\degavg_0 \\xinf \\xsus.\n    \\end{align}\n    The  fraction of recovered individuals follows from\n    $\\xrec = 1 - \\xinf -\\xsus$ for all times.\n    \n    \n    \\section{Epidemics with strategy adaption}\n    \\label{sec:epidemics-strategy}\n    We consider an  epidemic system on a network $\\graph = (\\nodes, \\edges)$ with transmission rate $\\tau$ and recovery rate $\\gamma$.\n   \n    The basic network structure of $\\graph$ is assumed to be fixed, i.e., the set of edges\n    $\\edges$ does not change.\n   \n    Adaptive activation and deactivation of edges occurs within this framework by\n    assigning a weight\n    $w :\\edges \\rightarrow \\{0, 1\\}$ to each edge $e\\in\\edges$,\n    which determines if the edge is active, $w(e) = 1$, or not $w(e) = 0$.\n   \n    More generally, one could assume transmission rates $\\tau(e) = w(e) \\tau$,\n    which are  relevant in the context of different preventive measures, such as quarantine\n    \\cite{maierEffectiveContainmentExplains2020,kucharskiEffectivenessIsolationTesting2020} or social distancing \\cite{giordanoModellingCOVID19Epidemic2020}.\n   \n   \n    In order to make this system adaptive we consider the\n    simplest case with only two strategies, $\\strat = \\{S_0, S_-\\}$, which determine the\n    number of active edges.\n   \n    Here, the null-strategy $S_0$ corresponds to case where all (existing) edges in the network\n    are active, $w(e) = 1\\, \\forall e \\in \\edges$.\n   \n    The strategy $S_-$ corresponds to a lockdown strategy, where\n    each edge is deactivated with probability $\\pcut$, \n    i.e.,  $\\mathbb{P}[w(e)\\mapsto 0] = \\pcut$.\n   \n    This means that the average degree in the adaptive network becomes time dependent\n    with $\\degavg_- = \\degavg_0(1 - \\pcut)$.\n    \n    \n    For the adaptive mechanism $A$ we consider prevalence and incidicence as suitable observable\n    functions, i.e.,\n    $g_1(t, \\xinf) = \\xinf$ and\n    $g_2(t, \\xinf) = \\tau \\nsi\/N \\approx\n    \\tau \\degavg(t)\\xinf\\xsus$.\n   \n    Furthermore, the kernel in Eq.~\\eqref{eq:def-indicator} is chosen such that $g$ is averaged\n    over fixed time intervals\n    $[t- \\Delta t, t]$ for some time span $\\Delta t$, i.e.,\n    $\\rho(t) = \\frac{1}{\\Delta t}[\\Theta(t) - \\Theta(t + \\Delta t)]$\n    with Heaviside function $\\Theta$.\n   \n    This leads to\n    $J_{1,2}(t) =\\int_{t - \\Delta t}^t\\,g_{1,2}[t', \\xinf(t')]\\,\\ud t'$.\n   \n    We emphasize that such an average has been a commonly used measure during\n    the Corona pandemic, e.g., in terms of the so-called $7$-day incidence, see e.g.,\n    \\cite{schlickeiserReasonableLimiting7Day2021}.\n   \n    Let us emphasize that $J$ measures the seriousness of the\n    current pandemic situation, where more infections or larger\n    prevalence manifest in  larger values of $J$.\n    \n    \n    The simplest possible adaptive function $A$ for two levels is\n    based on fixed thresholds.\n   \n    For this, we define two thresholds   $\\xi_+ > \\xi_-$ and the following strategy adaptions,\n   \n    \\begin{align}\n        \\begin{split}\n            A(t, J(t), S_0) = \\begin{cases}\n                S_- & \\text{for } J(t) \\geq \\xi_+, \\\\\n                S_0 & \\text{else,}\n            \\end{cases}\\\\\n            A(t, J(t), S_-) = \\begin{cases}\n                S_0 & \\text{for } J(t) \\leq \\xi_-, \\\\\n                S_- & \\text{else.}\n            \\end{cases}\n        \\end{split}\n       \n    \\end{align}\n   \n    This means, whenever the strategy function hits the predefined\n    thresholds\n    from below or from above, respectively, the system automatically\n    enters the state of social distancing or goes back to the usual state.\n   \n    Let us emphasize that the memory dependence\n    is hidden in the strategy function $J$.\n   \n    This function is continuously evaluated\n    for all $t$ and thereby determines the current strategy\n    $S\\in \\strat$ of the system.\n   \n    \n   \n    Note, that there are numerous possibilities for defining the\n    adaption function, which can either deterministically\n    or also stochastically select the future strategy.\n    In the latter case the system contains two distinct sources of stochasticity,\n    one from the dynamics and one from the adaption mechanism, which adds another\n    layer of complexity. Here we focus on the deterministic case.\n    \n    \n    We emphasize that our proposed model differs quite substantially from recent stochastic SIRS models \n    \\cite{hieuDynamicalBehaviorStochastic2015,\n        liuAnalysisSIRSEpidemic2017,\n        hurthRandomSwitchingBifurcations2020},\n    which are piecewise deterministic Markov processes (PDMPs).\n    In these systems, the switching of parameters is guided by\n    an additional random stochastic process, which is\n    independent of the (deterministic or stochastic) dynamics.\n   \n    Here, in contrast, the dynamical evolution of the system feeds back into\n    the adaptive mechanism.\n    If the underlying dynamics is stochastic,\n    the adaption of strategies is guided by the (stochastic)\n    feedback process implemented with the adaptive function.\n   \n    On the other hand, in a purely deterministic setting\n    the proposed system can be seen as an extension \n    of the phase-space with the set of strategies,\n    $\\mathcal{X} \\times \\strat$, where switching between\n    different branches occurs deterministically according\n    to the adaptive function $A$.\n   \n    \n    \n\n    \n    \n    \n    \\subsection{Results for SIS}\n    For the SIS epidemics on a network\n    one expects an initial growth of the prevalence $\\xinf$\n    until $J$ exceeds $\\xi_+$. At this point the social distancing\n    strategy $S_-$ is applied by removing a proportion of $p_\\mathrm{cut}$\n    edges from the system, see illustration in\n    Fig.~\\ref{fig:skizze-bifurcation}(a).\n   \n    If sufficienlty many edges are removed, the prevalence decreases\n    and $J$ becomes smaller than $\\xi_-$, such that the null-strategy\n    $S_0$ is reapplied. Within this regime we expect\n    periodic behaviour.\n   \n    \n   \n    In the following we first derive different dynamical\n    regimes in the adaptive pairwise closed SIS model\n    and secondly compare this to numerical results\n    obtained from simulation on networks.\n   \n    Recall that\n    for each strategy, $S_0$ and $S_-$, there is one\n    stable equilibrium $x_\\ast(\\beta)$ with\n    $x_\\ast(\\beta) = x_0 = 0$ for $\\beta = \\degavg \\tau \/\\gamma < 1$ and\n    $x_\\ast(\\beta) = \\xen(\\beta) = 1 - \\beta^{-1}$ for $\\beta >1$, see red curve\n    in Fig.~\\ref{fig:skizze-bifurcation}(a).\n   \n    In the following we assume arbitrary but fixed\n    thresholds $0 < \\xi_- < \\xi_+ < 1$ and specify how\n    the dynamics depends on the choice of $(\\beta_0, \\beta_-)$. For an illustration see Fig.~\\ref{fig:skizze-bifurcation}(b).\n   \n    For simplicity, we also assume that $\\Delta t = 0$,\n    such that $J(t) = g(\\xinf(t))$.\n   \n    First, if $\\beta_0 < 1$ the state of the system\n    converges to the stable equilibrium\n    $x_0 = 0$ and the strategy remains in $S_0$ for all times.\n    Secondly, for $\\beta_0 > 1$ the state of the system\n    approaches the endemic state $\\xen(\\beta_0)$.\n   \n    The strategy switches, if $g(\\xinf) \\geq \\xi_+$, which implies the limit $\\xen(\\beta_0) = 1 - \\beta_0^{-1} =  g^{-1}(\\xi_+)$.\n   \n    In particular,\n    for $1 < \\beta_0 < \\frac{1}{1 - g^{-1}(\\xi_+)}$ the\n    threshold $\\xi_+$ is too large and the state of the system converges to $\\xen(\\beta_0)$.\n   \n    On the other hand, for\n    $\\beta_0 > \\frac{1}{1 - g^{-1}(\\xi_+)}$ \n    the strategy will eventually switch to $S_-$ for\n    some $t > 0$.\n   \n    After switching, the state converges to the stable\n    equilibrium $x_\\ast(\\beta_-)$.\n   \n    Similar considerations as above lead to the following\n    limits depending on the threshold $\\xi_-$. If $\\beta_- > \\frac{1}{1 - g^{-1}(\\xi_-)}$\n    the state approaches the stable endemic equilibrium\n    $\\xen(\\beta_-)$, without reaching the lower threshold,\n    thus remaining in strategy $S_-$ for all times.\n   \n    If $\\beta_- < \\frac{1}{1 - g^{-1}(\\xi_-)}$ the\n    stable equilibrium is below the threshold, such that\n    after some finite time the system switches back\n    to $S_0$ again. Hence, in this regime the dynamics\n    is periodic.\n    \n    \n     \\begin{figure}[t!]\n        \\begin{overpic}[scale=1.]\n            {fig_sketch}\n            \\put(-3, 97)  {(a)}\n            \\put(-3, 47)  {(b)}\n        \\end{overpic}\n        \n        \\caption{(a) Bifurcation diagram for pairwise closed SIS model with illustration of threshold based periodic dynamics.\n            (b) Phase diagram in $(\\beta_0, \\beta_-)$-plane\n            illustrating different regimes of the dynamics for $\\Delta t = 0$.}\n        \\label{fig:skizze-bifurcation}\n    \\end{figure}\n    Conversely, if $\\beta_0 > 1$ and $\\beta_-$ are fixed,\n    it is possible to specify limits for the thresholds\n    $\\xi_\\pm$, within which we expect periodic dynamics.\n   \n    In particular, one obtains\n    $\\xi_+ \\leq g(1 - 1\/\\beta_0)$\n   \n    and\n    $\\xi_- \\geq g(x^\\ast(\\beta_-))$.\n   \n    With  $\\beta_- = \\frac{(1 - p_\\mathrm{cut}) \\degavg_0\\tau}{\\gamma}$\n    these conditions similarly imply a lower bound for\n    the cutting probability $p_\\mathrm{cut}$ at given $\\xi_-$,  given by\n    $p_\\mathrm{cut} \\geq 1 - \\frac{\\gamma}{[1 - g^{-1}(\\xi_-)]\\degavg_0\\tau}$.\n   \n    \n    \n    \n     \\begin{figure}[t]\n        \\includegraphics{fig3}\n        \n        \\caption{\n           \n            Relative prevalence $\\xinf = I\/N$ and incidence\n            $\\tau \\nsi \/ N$ for SIS epidemics on adaptive network with\n            $N=2000$, $\\degavg_0=50$, $\\gamma = 0.25$, $\\beta_0=2$,\n            and threshold adaption as in Eq.~\\eqref{eq:def-indicator}\n            using $g_2$ (incidence thresholds)\n            and $\\Delta t =5$, $\\xi_+=0.025$, $\\xi_-=0.005$.\n           \n            The lockdown strategy is given by\n            $p_\\mathrm{cut}(S_-) = 0.8$ ($\\beta_- = 0.4$).\n           \n            Considered are (a) Erd\\\"os-R\\'enyi networks and\n            (b) Barrabasi-Albert networks.}\n        \\label{fig:sis-continuous-threshold}\n    \\end{figure}\n   \n    In Figure~\\ref{fig:sis-continuous-threshold} we illustrate the\n    time dependence of the relative prevalence $\\xinf$ for the adaptive SIS epidemics\n    with strategy function $J_2$\n    on two different network types,\n    networks form the random Erd\\\"os-Renyi (ER)  ensemble\n    $G(N, p)$ \\cite{gilbertRandomGraphs1959,bollobasRandomGraphs2001}\n    and scale-free Barabasi-Albert (BA) networks \\cite{barabasiEmergenceScalingRandom1999}.\n   \n    The parameters are $\\gamma = 0.25$, $\\degavg_0 = 50$,\n    $\\beta_0 = 2$ and $\\pcut = 0.8$\n    with thresholds $\\xi_+ = 0.025$ and $\\xi_- = 0.005$, i.e., \n    the lockdown strategy is enforced when\n    each infected individual infects on average $2.5\\%$ of the population\n    (over the past time-window $\\Delta t = 5$) and it ends below $0.5\\%$.\n   \n    These parameters correspond to the periodic regime in\n    Fig.~\\ref{fig:skizze-bifurcation}(b).\n    \n    \n    For both network types\n    the fraction of infected individuals $\\xinf$ oscillates see top panels in Fig.~\\ref{fig:sis-continuous-threshold}(a) and (b).\n   \n    The corresponding incidence function also oscillates between \n    the predefined thresholds $\\xi_\\pm$, shown as\n    blue curve and red dotted lines, respectively, in the bottom panels.\n   \n    Note that the discontinuities of the incidence function are caused by\n    the sudden (de)activation of edges, which immediately changes the number of\n    $\\suscept\\infect$-edges and thereby the possible number of new infections.\n   \n    For comparison the analytic result of the pairwise closed moment \n    system is shown as a dashed black curve,\n    see App.~\\ref{app:sis-analytic}.\n   \n    This agrees very well with the simulation for the ER-network.\n   \n    In contrast, the initial spreading of the epidemics in the BA network \n    is  much steeper, showing also larger maximal values of $\\xinf$.\n   \n    Consequently, the frequency of fluctuation for these scale-free networks is increased compared to the pairwise closed system.\n    \n    \n    \n    \\begin{figure}[b]\n        \\includegraphics{fig4}\n        \n        \\caption{\n           \n            Period $T$ as a function of $\\xi_+$ for\n            (a, c) Erd\\\"os-R\\'enyi and\n            (b, d) Barrabasi-Albert networks, each with $N = 1000$ nodes and $\\degavg_0 = 50$ and\n            (a, b) $\\beta_0 = 1.6$ and (c, d) $\\beta_0 = 2$.\n           \n            Other parameters are $\\gamma = 0.25$,\n            $\\xi_- = 0.05$, and $p_\\mathrm{cut} = 0.6$.\n           \n            Threshold adaption with\n            $g_1$ (prevalence thresholds) and $\\Delta t = 5$.\n           \n            The period is numerically determined from simulation up to\n            $t = 1000$ as the average over all time-intervals in which the strategy switches from $S_0$ to $S_-$ and back,\n            with $20$ different realizations of the  initial network.\n           \n            For comparison the expectation from pairwise closed adaptive SIS model   is shown (black line), in (d) also for $\\pcut = 0.49$ (gray line).\n           \n            Gray dashed line indicates maximal\n            $\\xi_+$ for periodic regime,\n            see Fig.~\\ref{fig:skizze-bifurcation}.\n           \n        }\n        \\label{fig:sis-continuous-period}\n    \\end{figure}\n    \n    \\paragraph{Period of fluctuations $T$.}\n    We further investigate how the period of fluctuations $T$ depends on the parameters of the system for the prevalence strategy function $J_1$.\n   \n    The period $T$ is the sum of the expected time $T_{0-}$ to switch\n    from $S_0$ to $S_-$ and $T_{-0}$ to switch back.\n   \n    For $J_1$ a derivation of $T$ is possible in the pairwise closed\n    SIS model, see App.~\\ref{app:sis-analytic},\n    which leads to\n    \\begin{align}\n        T = T_{0-}& + T_{-0} \\nonumber\\\\\n       \n        =\\phantom{+}&\\frac{1}{\\gamma}\n        \\log \\left[\\frac{\n            \\xi_+ (\\beta_0 - 1 - \\beta_0 \\xi_-)}{(\\beta_0 - 1) \\xi_- - \\xi_+ \\beta_0 \\xi_-}\\right]^{\\beta_0 - 1}\n        \\\\\n        +&\\frac{1}{\\gamma}\n        \\log \\left[\\frac{\n            \\xi_- (\\beta_- - 1 - \\beta_- \\xi_+)}{(\\beta_- - 1) \\xi_+ - \\xi_- \\beta_- \\xi_+}\\right]^{\\beta_- - 1}.\n        \\nonumber\n    \\end{align}\n   \n    \n    In Figure~\\ref{fig:sis-continuous-period} we show $T$ \n    versus the upper threshold $\\xi_+$ for two different values\n    of $\\beta_0 \\in \\{1.6, 2\\}$ and compare network simulations\n    (green errorbars) to the analytic result (black curve).\n   \n    The upper threshold is chosen in the interval\n    $\\xi_- \\leq \\xi_+ < \\xen(\\beta_0)$.\n    Note that the expected period becomes zero at some $\\xi_+$ smaller than $\\xi_-$,\n    see Fig.~\\ref{fig:t-vs-xi+}, e.g., for $\\Delta t = 0$ this occurs at $\\xi_+ = \\xi_-$.\n    On the other hand, if $\\xi_+ \\rightarrow \\xen(\\beta_0)$ the expected period goes to infinity for the analytic model.\n   \n    This is also observed in the network simulations:\n   \n    For random  ER networks the analytic expression of the pairwise\n    closed model fits very well for both $\\beta_0$, see (a) and (c).\n   \n    For the scale-free BA network and $\\beta_0 = 1.6$\n    the observed period is systematically too\n    large for smaller $\\xi_+$ and too small for larger $\\xi_+$.\n    \n    Intuitively, this is explained as follows. The prevalence increases\n    initially much faster in scale-free networks due to nodes with very\n    large degrees (superspreaders), thereby overshooting the\n    threshold significantly stronger, see Fig.~\\ref{fig:sis-continuous-threshold}(c).\n    Thus, the initial prevalence in the second half of the cycle is increased\n    compared to the random networks, thereby increasing the\n    period significantly.\n   \n    This effect becomes negligible, if the upper threshold gets closer to\n    the endemic equilibrium, since here the prevalence typically flattens\n    for all network types and the overshooting effect vanishes. Instead, the\n    initially much faster speed of infection leads to shorter periods\n    in this regime.\n    \n    \n    The strong dependence on the topology is revealed when considering\n    larger values of $\\beta_0 = 2$ and leaving the other parameters unchanged.\n    Note that with $\\pcut = 0.6$ this still implies $\\beta_- = 0.8$ significantly \n    smaller than one.\n   \n    In the scale-free network the observed periods are up to four times longer than\n    predicted by the moment equations, Fig.~\\ref{fig:sis-continuous-period} (d).\n   \n    This means that with the same number of deleted edges in scale-free networks\n    the progression of the epidemic spreading is slowed down more efficiently\n    than in random networks. \n    Note that these longer periods are mainly caused \n    by a much flatter slope  during the lockdown phase $S_-$.\n    This corresponds to an effective cutting probability\n    of $\\pcut \\approx 0.49$ (gray line).\n    \n    \n    This shows that the network topology leads to observable differences\n    in adaptive network systems.\n   \n    We believe that such topology induced effects on oscillations can also be observed in more complex models.\n    \n    \n    \\begin{figure}[b!]\n        \\includegraphics{fig5}\n       \n        \\caption{\n           \n            Reproductive number $R_0$ as a function of $t$ for\n            (a) Erd\\\"os-R\\'enyi and\n            (b) Barrabasi-Albert networks, each with $N = 1000$ nodes and $\\degavg_0 \\approx 50$.\n           \n            Other parameters are $\\gamma = 0.25$, $\\beta_0 = 2$, $\\degavg=50$,\n            $\\xi_+=0.025$,\n            $\\xi_- = 0.005$, $p_\\mathrm{cut}(S_-) = 0.6$, $\\Delta t = 5$,\n            and the prevalence strategy function $J_1$ is considered.\n           \n            The reproductive ratio is numerically determined from the simulation (blue), compared to the prediction of the pairwise model (black dashed).\n           \n        }\n        \\label{fig:sis-reproductive}\n    \\end{figure}\n    \\begin{figure}[t]\n       \n        \\begin{overpic}[scale=0.9, percent]\n            {fig_r0_rki_corona}\n        \\end{overpic}\n       \n        \\caption{\n           \n            Estimation for daily new infections $\\Delta \\ninf$ and\n            $R_0$ based on $7$-day nowcast for Sars-Cov-2 epidemics in Germany between 12.03.2020 and 06.03.2022 \\cite{anderheidenSchatzungAktuellenEntwicklung2020,\n                anderheidenSARSCoV2NowcastingUndRSchaetzung2022}.\n           \n            The nowcast estimation for $R_0$\n            is determined by the quotient of new infections for one\n            time-interval and those of the preceding interval, assuming\n            that each interval corresponds to one generation of the virus.\n           \n        }\n        \\label{fig:reproductive-rki}\n    \\end{figure}\n    \\paragraph{Basic reproductive number $R_0$.}\n   \n    The basic reproductive number $R_0$ is commonly defined as the expected number of\n    new infections caused by one infected individual in a fully\n    susceptible community\n    \\cite{fraserEstimatingIndividualHousehold2007,kissMathematicsEpidemicsNetworks2017}.\n   \n    For the pairwise closed SIS model this gives\n    the ratio of total rate of infection to total rate of recovery,\n    $R_0 = \\beta = \\degavg \\tau \/ \\gamma$ \\cite{kissMathematicsEpidemicsNetworks2017}.\n   \n    If for infectious disease $R_0 > 1$ an epidemic outbreak\n    occurs, while for $R_0 < 1$ the disease vanishes.\n   \n    \n    For the SIS epidemics with adaption the average degree on the\n    network depends on the currently active strategy.\n   \n    Hence, the simplest approximation for $R_0$ is given by\n    $\\beta \\in \\{\\beta_0, \\beta_-\\}$ with $\\beta_0 = \\degavg \\tau \/ \\gamma$ and $\\beta_- = \\beta_0 (1 - \\pcut)$.\n   \n    Numerically, the reproductive number is obtained by\n    averaging the number of infections caused by each infected node over time.\n   \n    In particular, let $\\tinf_j$ be the time of the $j$-th infection during\n    the simulation and let $\\Ninf_j$ be the number of infections caused by\n    the infected $j$-th node.\n    With this, we estimate $R_0$ as\n   \n    \\begin{equation}\n        R_0(t) \\approx  \\langle \\{ \\Ninf_j : t - \\delta t < \\tinf_j \\leq t \\}\\rangle,\n    \\end{equation}\n   \n    where $\\langle \\cdot \\rangle$ denotes the average and we choose the time\n    interval according to the recovery time $\\delta t = 1\/\\gamma$.\n    \n    \n    In Fig.~\\ref{fig:sis-reproductive} we illustrate the\n    reproductive ratio $R_0$ as a function of time,\n    comparing numerical values (blue) to the simplest estimation based on $\\beta$ (black).\n   \n    The numerically observed $R_0$ oscillates in a similar pattern around\n    $R_0 \\approx 1$, however, fluctuating on faster scales and typically \n    below $R_0 = \\beta_0 = 2$.\n   \n    We assume that the latter is caused by infectious clusters with a small\n    number of susceptibles, leading to much fewer infections for large prevalence.\n   \n    \n    \n    We emphasize that such fluctuations of the reproductive number\n    around $R_0 \\approx 1$ are also observed in many different countries during the Corona pandemic,\n    e.g., for evolution in Germany see Fig.~\\ref{fig:reproductive-rki}.\n   \n    This indicates that in real epidemics there is\n    some form of self-organized criticality around $R_0 = 1$ \n    due to the conflicting goals\n    of reducing the number of infections and increasing the freedom of individuals. \n   \n    In particular, our simple model with adaptive strategies already mimics\n    the complex adaptive dynamics due to stricter lockdown measures and more general cautious \n    behavior at higher incidence levels.\n   \n    It is therefore reasonable to expect similar observations in more sophisticated models,\n    e.g., when each agent chooses from a set of strategies\n    and adapts in a more complex manner to the development of an epidemic. Furthermore, our \n    results strongly support the claim that epidemic models with self-limiting feedback \n    mechanisms should be viewed through the lens of self-organized\n    criticality~\\cite{stollenwerkEvolutionCriticalityEpidemiological2003}.\n    \n    \\begin{figure}[b]\n        \\includegraphics{fig7}\n        \n       \n        \\caption{\n           \n            Relative prevalence $\\xinf = I\/N$ and incidence\n            $\\tau \\nsi \/ N$ for SIR epidemics on adaptive network with\n            $N=2000$, $\\degavg=50$, $\\gamma = 0.25$, $\\beta_0=2$,\n            and threshold adaption as in Eq.~\\eqref{eq:def-indicator}\n            using $g_2$ (incidence thresholds)\n            and $\\Delta t =5$, $\\xi_+=0.01$, $\\xi_-=0.002$.\n           \n            The lockdown strategy is given by\n            $p_\\mathrm{cut}(S_-) = 0.8$ ($\\beta_- = 0.4$).\n           \n            Considered are (a) Erd\\\"os-R\\'enyi and\n            (b) Barrabasi-Albert networks.}\n        \\label{fig:sir-continuous-threshold}\n    \\end{figure}\n   \n    \\subsection{Results for SIR}\n   \n    We further apply the proposed adaptive strategy approach to the SIR epidemics on networks \\cite{kissMathematicsEpidemicsNetworks2017}.\n   \n    In the SIR model recovered individuals\n    do not contribute to the epidemic spreading anymore.\n   \n    Similar to the SIS model there is a regime where initial\n    transient oscillations are expected.\n   \n    When a significant proportion of individuals is\n    recovered, $\\recov$, the epidemic dies out\n    after a certain number of cycles.\n    \n    \n    This is illustrated in Fig.~\\ref{fig:sir-continuous-threshold}, showing the\n    time dependence of relative prevalence and incidence for the SIR\n    dynamics on two different network types,\n    similar to Fig.~\\ref{fig:sis-continuous-threshold}.\n   \n    The  considered thresholds are $\\xi_+ = 0.01$ and $\\xi_- = 0.002$.\n    Note that for real epidemics these thresholds should be chosen\n    according to the capacity of the health system, which\n    typically can sustain only a very small fraction of infected individuals.\n    Furthermore, choosing larger thresholds in the simulations\n    leads to many infected and recovered individuals already before strategy adaption takes place.\n   \n    For both network types we observe the expected initial oscillatory dynamics\n    in Fig.~\\ref{fig:sir-continuous-threshold}, similar to the SIS system.\n   \n    We observe that Erd\\\"os-R\\'enyi networks show comparable\n    results to the pairwise closed solution (black dashed line),\n    even though individual simulations differ substantially.\n   \n    In contrast, the epidemic spreading in the scale-free network takes place\n    much faster.\n   \n    This is intuitively explained with the power-law\n    degree distribution and the existence of a small number of\n    highly connected nodes. First, if one of these\n    nodes is infected it triggers a lot of subsequent infections.\n    Secondly, after its recovery\n    it efficiently blocks the epidemic spreading.\n    \n    \n    \\begin{figure}[b]\n        \\includegraphics{fig8}\n       \n        \\caption{\n           \n            Times of switching  as a function of $p_\\mathrm{cut}(S_-)$\n            for SIR epidemics\n            on adaptive network with $N=2000$, $\\degavg=50$,\n            $\\gamma = 0.25$, $\\beta_0=2$.\n            Threshold adaption as in Eq.~\\eqref{eq:def-indicator}\n            using $g_2$ (incidence thresholds)\n            and $\\Delta t =5$, $\\xi_+=0.01$, $\\xi_-=0.002$.\n           \n            Considered are (a) Erd\\\"os-R\\'enyi network\n            and (b) Barrabasi-Albert network.\n           \n            Colors reporesent\n            switch $S_0 \\rightarrow S_-$ (violet colors)\n            and\n            $S_- \\rightarrow S_0$ (green colors).\n           \n        }\n        \\label{fig:sir-investigation-pcut}\n    \\end{figure}\n    \n    \n    \\paragraph{Adaption times.}\n    We investigate numerically how the times of strategy adaption \n    depend on the probability to cut edges during the\n    lockdown phase $\\pcut(S_-)$.\n   \n    This is illustrated in Fig.~\\ref{fig:sir-investigation-pcut} for random\n    ER and scale-free BA networks\n    and compared to numerical solutions \n    of the first order moment SIR system with strategy adaption (black lines).\n   \n    For both network types and\n    small $\\pcut \\leq 0.5$ the strategies switch only twice,\n    from $S_0$ to $S_-$ (violet colors) and back (green colors).\n   \n    In this regime the corresponding $\\beta_-  \\geq 1$, such that\n    the epidemic spreading slows down, but does not revert.\n    Note that for $\\pcut = 0$ no edges are removed\n    and time evolution is equivalent to the\n    SIR dynamics on the original networks.\n   \n    For increasing $\\pcut$ the number of adaptions increases\n    and we find very good agreement between simulations on the ER network and the solution of the closed moment equation, see\n    Fig.~\\ref{fig:sir-investigation-pcut}(a).\n    The regime with $\\pcut \\gtrsim 0.8$ is characterized by initial\n    transient oscillations similar to SIS epidemics.\n   \n    We observe for the scale-free network that the time\n    of the first strategy adaption is much smaller than\n    for the closed moment equation.\n    This is caused by the initial faster epidemic spreading\n    observed in these networks.\n    \n    \n    \n    \\section{Conclusion and outlook}\n    \\label{sec:outlook}\n    In this paper a self adaptive mechanism with a finite\n    set of strategies is proposed, which leads to \n    a coupled system of piecewise deterministic dynamics \n    with history dependent strategy switching.\n   \n    Such a modelling approach will be helpful in the description of systems, where the dynamics depends\n    strongly on the currently active rules, e.g.,\n    social systems and opinion formatting, epidemics.\n    It also may be applied to systems, where the full history of the current state changes its evolution,\n    like weaker regeneration of skin cells after multiple exposure with sun light leads to increased probability of  cancer.\n    \n    \n    This adaptive framework is applied to epidemic models\n    on networks.\n    For the SIS system we observe a stable regime of\n    strategy induced oscillations.\n    Their period depends on the network type and\n    the degree distribution.\n    Based on the pairwise closed model an analytic \n    prediction for the period is derived, which fits very well for Erd\\\"os-R\\'enyi and random regular\n    networks and also approximates the period observed in\n    scale free networks.\n    \n    \n    We emphasize that the proposed adaptive mechanism\n    deterministically depends on the state of the system\n    and its history. One promising generalization is the to consider a stochastic process for the adaptive mechanism, with state (and history) dependent\n    transition rates between the strategies.\n    This adaptive mechanism could be coupled with\n    deterministic ODE models (like the pairwise\n    closed SIS), or with stochastic models (like the\n    SIS network model). The latter implies two distinct\n    sources of random behavior, which could lead to\n    interesting phenomena.\n    \n    It would be further interesting to investigate in more detail \n    the observation of the fluctuating reproductive rate around its \n    critical value $R_0=1$. There is a clear connection of this observation \n    to self-organized criticality, which\n    has been observed in a wide variety of adaptive\/co-evolutionary \n    network models~\\cite{bornholdtTopologicalEvolutionDynamical2000,kuehnTimescaleNoiseOptimality2012,meiselAdaptiveSelforganizationRealistic2009}.\n    We conjecture that having an observable controlling a switching mechanism that entails lowering or increasing\n    the infection numbers is the simplest mechanism to obtain self-organized criticality in epidemic\n    dynamics. One could now also investigate power law distributions of epidemic event sizes in various models \n    that could further confirm this conjecture. Yet, for our SIS-based model the underlying bifurcation-theoretic \n    mechanism is the switching around a transcritical bifurcation, where one can calculate mathematically that \n    power laws emerge close to the bifurcation point~\\cite{kuehnMathematicalFrameworkCritical2011,hurthRandomSwitchingBifurcations2020}. Hence,\n    finding evidence for self-organized criticality in very complex epidemic models and\/or long-time data\n    sets with multiple outbreaks are the most challenging aspects. \n    \n    \n    \n    \\acknowledgments\n    KC and CK thank the VolkswagenStiftung for support via the grant\n    ``Self-Dynamics of Self-Adapting Networks'' within a Lichtenberg Professorship awarded to CK. \n    \n    ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}