diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzbltx" "b/data_all_eng_slimpj/shuffled/split2/finalzzbltx" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzbltx" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe COHERENT experiment has observed coherent elastic neutrino-nucleus scattering\n (CE$\\nu$NS)~\\cite{Akimov:2017ade} 43 years after its theoretical prediction in the standard model (SM)~\\cite{Freedman:1973yd}.\nFor neutrino energies below a few tens of MeV, CE$\\nu$NS occurs when the momentum transfer $Q$ is comparable to the inverse of the nuclear radius $R$, i.e., $QR\\lesssim 1$. Compared to scattering of isolated nucleons, the cross section for coherent scattering off a nucleus is enhanced by the square of the number of neutrons in the nucleus. However, in spite of its large cross section, it is difficult to observe CE$\\nu$NS because of the small momentum transfer \ninvolved.\n\nThe COHERENT experiment measures neutrinos from the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory, using a sodium doped CsI scintillator that can detect nuclear recoil energies down to a few keV. They find 6.7$\\sigma$ CL evidence for CE$\\nu$NS in good\nagreement with SM predictions, after fifteen months of data accumulation. \nThe measurement of CE$\\nu$NS not only completes the SM picture of neutrino interactions, but also provides a tool to study new physics beyond the SM, e.g., nonstandard neutrino interactions (NSI)~\\cite{Barranco:2005yy, Scholberg:2005qs, Coloma:2017egw}, sterile neutrinos~\\cite{Anderson:2012pn}, neutrino magnetic moment~\\cite{Dodd:1991ni}, and light dark matter~\\cite{deNiverville:2015mwa}.\n \nThe total number of events has been used in Refs.~\\cite{Akimov:2017ade} and~\\cite{Coloma:2017ncl} to constrain NSI under the contact interaction approximation. If the momentum transfer of CE$\\nu$NS is comparable to the mediator mass, the shape of the spectrum is also modified. Consequently, constraints obtained using the contact approximation do not apply to the light mediator case. In this Letter, we use the spectrum of the CE$\\nu$NS signal to constrain NSI including mediator effects. In Section~2, we describe our simulation of the COHERENT spectrum. In Section~3, we discuss the sensitivities to the nonstandard parameters for both the light and heavy mediators. We summarize our results in Section~4.\n\n\\section{COHERENT simulation}\nThe expected number of events for neutrino flavor $\\alpha$ of recoil energy $E_r$ is\n\\begin{align}\n\\frac{dN_\\alpha}{dE_r}=n_\\text{N}\\int dE_{\\nu}\\phi_\\alpha(E_\\nu)\\frac{d\\sigma_\\alpha}{dE_r}(E_\\nu)\\,,\n\\end{align} \nwhere the total number of nucleons in the detector is $n_\\text{N}=\\frac{2m_\\text{det}}{M_\\text{CsI}}N_A$, with $m_\\text{det}= 14.6$~kg being the detector mass, $M_\\text{CsI}$ the molar mass of CsI, and $N_A$ the Avogadro constant.\nNeutrinos at the SNS consist of a prompt component of monochromatic $\\nu_\\mu$ from the stopped pion decays, $\\pi^+\\to \\mu^++\\nu_\\mu$, and two delayed components of $\\bar{\\nu}_\\mu$ and $\\nu_e$ from the subsequent muon decays, $\\mu^+\\to e^++\\bar{\\nu}_\\mu+\\nu_e$. The distribution of the total flux for each neutrino flavor is well-known and given by~\\cite{Coloma:2017egw}\n\\begin{align}\n\\phi_{\\nu_\\mu}(E_\\nu)&={\\cal{N}}\\delta\\left(E_\\nu-\\frac{m_\\pi^2-m_\\mu^2}{2m_\\pi}\\right)\\,,\n\\nonumber\\\\\n\\phi_{\\bar{\\nu}_\\mu}(E_\\nu)&={\\cal{N}}\\frac{64E_\\nu^2}{m_\\mu^3}\\left(\\frac{3}{4}-\\frac{E_\\nu}{m_\\mu}\\right)\\,,\n\\nonumber\\\\\n\\phi_{\\nu_e}(E_\\nu)&={\\cal{N}}\\frac{192E_\\nu^2}{m_\\mu^3}\\left(\\frac{1}{2}-\\frac{E_\\nu}{m_\\mu}\\right)\\,,\n\\end{align}\nwhere the normalization factor is ${\\cal{N}}=\\frac{rN_\\text{POT}}{4\\pi L^2}$. Here $r=0.08$ is the number of neutrinos per flavor that are produced for each proton on target~\\cite{Akimov:2017ade}. The total number of protons delivered to the mercury target is $N_\\text{POT}=1.76\\times 10^{23}$ and the distance between the source and the CsI detector is $L=19.3$~m~\\cite{Akimov:2017ade}.\n\nThe differential cross section for a given neutrino flavor $\\nu_\\alpha$ in the SM is\n\\begin{align}\n\\frac{d\\sigma_{\\alpha }}{dE_r}=\\frac{G_F^2}{2\\pi}Q_{\\alpha }^2F^2(2ME_r)M\n\\left(2-\\frac{ME_r}{E_\\nu^2}\\right)\\,,\n\\label{eq:CC}\n\\end{align}\nwhere $M$ is the mass of the target nucleus, $F(Q^2)$ is the nuclear form factor, and the radiative corrections are neglected.\nWe take the nuclear form factor from Ref.~\\cite{Klein:1999gv}. The effective charge in the SM is \n\\begin{align}\nQ_{\\alpha,\\text{SM} }^2=&\\left(Zg_p^V+Ng_n^V\\right)^2\\,,\n\\end{align}\nwhere $Z$ and $N$ are the number of protons and neutrons in the nucleus, and $g_p^V=\\frac{1}{2}-2\\sin^2\\theta_W$ and $g_n^V=-\\frac{1}{2}$ are the SM couplings of the $Z^0$ boson to the proton and neutron, with $\\theta_W$ the weak mixing angle. \n\nWe ignore the contribution to the cross section from the sodium dopant because of its extremely small fractional mass ($10^{-4}-10^{-5}$) in the CsI detector~\\cite{Collar:2014lya}. Also, since the responses of Cs and I to a given neutrino flavor are almost identical due to very similar nuclear masses~\\cite{Collar:2014lya}, we do not distinguish between Cs and I in our analysis. We adopt a simple relation between the observed number of photoelectrons (PE) and the nuclear recoil energy~\\cite{Akimov:2017ade}:\n\\begin{align}\nn_\\text{PE}=1.17 \\left(\\frac{E_r}{ \\text{keV}}\\right)\\,.\n\\end{align} \nAfter taking into account the surviving fraction of the CE$\\nu$NS signals as a function of the number of photoelectrons given in Fig.~S9 in Ref.~\\cite{Akimov:2017ade}, \nwe show the expected CE$\\nu$NS events as a function of the number of photoelectrons for the SM in Fig.~\\ref{fig:spectra}. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\textwidth]{spectra.pdf}\n\\caption{The expected CE$\\nu$NS events as a function of the number of photoelectrons. The dashed lines correspond to the SM, and the solid lines correspond to the NSI case with $M_{Z'}=10$ MeV and $g=10^{-4}$. The blue (red) [black] lines correspond to the $\\nu_\\mu$ ($\\nu_\\mu+\\bar{\\nu}_\\mu$) [$\\nu_\\mu+\\bar{\\nu}_\\mu+\\nu_e$] contributions. }\n\\label{fig:spectra}\n\\end{figure}\n\n\\section{Constraints on nonstandard neutrino interactions}\nWe now analyze the COHERENT spectrum to constrain vector NSI parameters that lead to an effective potential for neutrino propagation in matter. In principle, COHERENT data also constrain axial-vector NSI, but because large nuclei approximately conserve parity, the constraints are weak.\n\\subsection{Light mediator}\nWe first consider the case that NSI are induced by a light vector mediator $Z'$ with mass $M_{Z'}$ that is comparable to the square root of the momentum transfer. For simplicity, we assume that the $Z'$ has purely vector universal flavor-conserving couplings to neutrinos, first generation quarks and the muon. Then the effective charge in Eq.~(\\ref{eq:CC}) can be written as\n\\begin{align}\nQ_{\\alpha,\\text{NSI} }^2=\\left[Z\\bigg(g_p^V+\\frac{3g^2}{2\\sqrt{2}G_F(Q^2+M_{Z'}^2)}\\bigg)\n+N\\bigg(g_n^V+\\frac{3g^2}{2\\sqrt{2}G_F(Q^2+M_{Z'}^2)}\\bigg)\\right]^2\\,,\n\\label{eq:propagator}\n\\end{align}\nwhere $Q^2=2ME_r$ is the square of the momentum transfer. \n\nTo evaluate the statistical significance, we define\n\\begin{align}\n\\chi^2=\\sum_i\\left[\\frac{N_\\text{exp}^i-N_\\text{NSI}^i(1+\\alpha)}{\\sigma_{\\text{stat}}^{i}}\\right]^2+\\left(\\frac{\\alpha}{\\sigma_\\alpha}\\right)^2\\,,\n\\end{align}\nwhere $N_\\text{exp}^i$ ($N_\\text{NSI}^i$) is the number of observed (predicted) events per bin, $\\sigma_{\\text{stat}}^{i}$ is the statistical uncertainty, and the total normalization uncertainty is $\\sigma_\\alpha=0.28$, which incorporates the neutrino flux, form factor, quenching factor and signal acceptance uncertainties~\\cite{Akimov:2017ade}. We extract $N_\\text{exp}^i$ and $\\sigma_{\\text{stat}}^{i}$ from the top right panel of Fig. 3 in Ref.~\\cite{Akimov:2017ade}, and consider 12 bins in the $6\\leq \\text{PE}<30$ range, and ignore the small background from prompt neutrons. \n\nWe scan over possible values of the coupling $g$ and the mediator mass $M_{Z'}$, and show the $2\\sigma$ limits in the $(M_{Z'}, g)$ plane in Fig.~\\ref{fig:lm}. The $2\\sigma$ allowed region that explains the discrepancy in the anomalous magnetic moment of the muon~\\cite{jeg} is also shown for comparison. We see that a light mediator that can explain the discrepancy in the anomalous magnetic moment of the muon is disfavored.\n\nThe shape of the limit curve in Fig.~\\ref{fig:lm} can be understood from the propagator in Eq.~(\\ref{eq:propagator}), in which the NSI contribution is proportional to $\\frac{g^2}{2ME_r+M_{Z'}^2}$. For a very light mediator, i.e., $M_{Z'}\\ll \\sqrt{2ME_r}\\sim 50$~MeV, the limit is only sensitive to the coupling $g$. Note that since the momentum transfer in coherent forward scattering is zero, the NSI matter effect for neutrino propagation is sensitive to $\\frac{g^2}{M_{Z'}^2}$~\\cite{Farzan:2015hkd}, and the constraint does not apply to matter NSI induced by a very light mediator. For a heavy mediator, i.e., $M_{Z'}\\gg \\sqrt{2ME_r}$, NSI do not change the shape of the spectra, and the limit is dependent on the ratio $\\frac{g}{M_{Z'}}$. There is also a degenerate region that is not excluded by current data.\nSince the data are consistent with the SM, the degenerate region can be understood by the relation, $Q_{\\alpha,\\text{NSI}}=-Q_{\\alpha,\\text{SM}}$, i.e.,\n\\begin{align}\n\\frac{g^2}{M_{Z'}^2}=-\\frac{4\\sqrt{2}(Zg_p^V+Ng_n^V)}{3(Z+N)}G_F \\,,\n\\end{align} \nwhich holds for all $E_r$ bins when $M_{Z'}\\gg \\sqrt{2ME_r}$. For a light mediator, the spectral shapes are modified by NSI (see the solid lines in Fig.~\\ref{fig:spectra} for example), which breaks the degeneracy.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{light_mediator.pdf}\n\\caption{The $2\\sigma$ exclusion region in the $(M_{Z'}, g)$ plane from the COHERENT data. The $2\\sigma$ allowed region that explains the discrepancy in the anomalous magnetic moment of the muon ($\\Delta a_\\mu=(29\\pm 9)\\times 10^{-10}$~\\cite{jeg}) is shown for comparison.}\n\\label{fig:lm}\n\\end{figure}\n\n\\subsection{Heavy mediator}\nFor a heavy mediator, matter NSI can be described by four-fermion contact operators of the form~\\cite{Wolfenstein:1977ue}\n\\begin{eqnarray}\n \\label{eq:NSI}\n \\mathcal{L}_\\text{NSI} = - \\sqrt{2}G_F\n \\epsilon^{fV}_{\\alpha\\beta} \n \\left[ \\overline{\\nu}_{\\alpha L} \\gamma^{\\rho} \\nu_{\\beta L} \\right] \n \\left[ \\bar{f} \\gamma_{\\rho}f \\right]\\,,\n\\end{eqnarray}\nwhere $\\alpha, \\beta=e, \\mu, \\tau$, $f=u,d$, and the strength of the new interaction\n$\\epsilon^{fV}_{\\alpha\\beta}$ is parameterized in units of $G_F$. As before, we consider NSI couplings to first generation quarks but not to electrons. We take the phases of the off-diagonal NSI parameters to be 0. For a heavy mediator, the effective charge in Eq.~(\\ref{eq:CC}) is\n\\begin{align}\nQ_{\\alpha }^2=\\left[Z(g_p^V+2\\epsilon_{\\alpha\\alpha}^{uV}+\\epsilon_{\\alpha\\alpha}^{dV})\n+N(g_n^V+\\epsilon_{\\alpha\\alpha}^{uV}+2\\epsilon_{\\alpha\\alpha}^{dV})\\right]^2\n+\\sum_{\\beta\\neq\\alpha}\\left[Z(2\\epsilon_{\\alpha\\beta}^{uV}+\\epsilon_{\\alpha\\beta}^{dV})\n+N(\\epsilon_{\\alpha\\beta}^{uV}+2\\epsilon_{\\alpha\\beta}^{dV})\\right]^2\\,.\n\\end{align}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth]{eedeeu.pdf}\n\\includegraphics[width=0.4\\textwidth]{mmdmmu.pdf}\n\\includegraphics[width=0.4\\textwidth]{eedmmd.pdf}\n\\includegraphics[width=0.4\\textwidth]{eedetd.pdf}\n\\caption{The $90\\%$ CL regions in the NSI parameter space allowed by COHERENT data. The NSI parameters not shown in each graph are assumed to be zero. }\n\\label{fig:nsi}\n\\end{figure}\nWe consider four cases with only two nonzero NSI parameters for simplicity:\n\\begin{itemize}\n\\item[(a)] $\\epsilon_{ee}^{uV}\\neq 0$, $\\epsilon_{ee}^{dV}\\neq 0$. \nIn this case, only the electron neutrinos are affected, and the $90\\%$~CL allowed region in the ($\\epsilon_{ee}^{dV}$, $\\epsilon_{ee}^{uV}$) plane is shown in the top-left panel of Fig.~\\ref{fig:nsi}. Since the data are consistent with the SM, the allowed regions can be understood by the relation, $Zg_p^V+Ng_n^V=\\pm \\left[Z(g_p^V+2\\epsilon_{ee}^{uV}+\\epsilon_{ee}^{dV})\n+N(g_n^V+\\epsilon_{ee}^{uV}+2\\epsilon_{ee}^{dV})\\right]$, which yields two linear bands in the ($\\epsilon_{ee}^{dV}$, $\\epsilon_{ee}^{uV}$) parameter space~\\cite{Scholberg:2005qs}. Because the $\\nu_e$ contribution to the total neutrino flux is small, the two bands merge into a single band. In principle, multiple detector elements should break the degeneracy, but since the Cs and I nuclei have very similar nucleon masses, the degeneracy is unbroken. \n\\item[(b)] $\\epsilon_{\\mu\\mu}^{uV}\\neq 0$, $\\epsilon_{\\mu\\mu}^{dV}\\neq 0$. \nThe $90\\%$ CL allowed regions in this case are shown in the top-right panel of Fig.~\\ref{fig:nsi}.\nThis case is similar to the electron neutrino case, except that both $\\nu_\\mu$ and $\\bar{\\nu}_\\mu$ are affected. Since the $\\nu_\\mu+\\bar{\\nu}_\\mu$ flux contribution is more than twice that of $\\nu_e$, the two bands do not overlap. \n\\item[(c)] $\\epsilon_{ee}^{dV}\\neq 0$, $\\epsilon_{\\mu\\mu}^{dV}\\neq 0$. \nThe $90\\%$ CL allowed region in this case is shown in the bottom-left panel of Fig.~\\ref{fig:nsi}. All three neutrino components are affected. The results for $\\epsilon_{ee}^{uV}= 0$ and for $\\epsilon_{\\mu\\mu}^{dV}=0$ are consistent with those in case (a) and case (b), respectively.\n\\item[(d)] $\\epsilon_{ee}^{dV}\\neq 0$, $\\epsilon_{e\\tau}^{dV}\\neq 0$. \nIn this case, only the electron neutrinos are affected, and the $90\\%$~CL allowed region is shown in the bottom-right panel of Fig.~\\ref{fig:nsi}. The expected allowed region is given by the relation, $\\left(Zg_p^V+Ng_n^V\\right)^2= \\left[Z(g_p^V+\\epsilon_{ee}^{dV})+N(g_n^V+2\\epsilon_{ee}^{dV})\\right]^2+\\left[Z\\epsilon_{e\\tau}^{dV}\n+2N\\epsilon_{e\\tau}^{dV}\\right]^2$, which yields a region between two ellipses (an annulus)~\\cite{Scholberg:2005qs}. We see a single ellipse due to the small $\\nu_e$ flux. \n\\end{itemize}\n\nDegeneracies between different combinations of NSI parameters, especially the cancellation between the NSI coupling to up and down quarks, permit large values of NSI parameters. However, the effective NSI paremeters in Earth matter are dependent on the sum of the up-type and down-type NSI parameters, i.e.,~\\cite{rk}\n\\begin{align}\n\\epsilon_{\\alpha\\alpha}\\approx 3(\\epsilon_{\\alpha\\alpha}^{uV}+\\epsilon_{\\alpha\\alpha}^{dV})\\,.\n\\end{align}\nThus, COHERENT constraints on the effective NSI parameters do not depend on the cancellation between the up-type and down-type NSI parameters. As an illustration, we scan over all possible values of $\\epsilon_{ee}^{dV}$, $\\epsilon_{ee}^{uV}$, $\\epsilon_{\\mu\\mu}^{dV}$, $\\epsilon_{\\mu\\mu}^{uV}$, and show the projected $90\\%$ CL allowed regions in the ($\\epsilon_{ee}$, $\\epsilon_{\\mu\\mu}$) plane in Fig.~\\ref{fig:eemm}. At $90\\%$ CL, the effective NSI parameters lie in the ranges, \n\\begin{equation}\n-0.95 \\leq \\epsilon_{ee} \\leq 1.95\\,,\\ \\ \\ \\ \\ \\ -0.66 \\leq \\epsilon_{\\mu\\mu} \\leq 1.57\\,.\n\\end{equation}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{eemm.pdf}\n\\caption{The $90\\%$ allowed regions in the ($\\epsilon_{ee}$, $\\epsilon_{\\mu\\mu}$) plane from the COHERENT data. }\n\\label{fig:eemm}\n\\end{figure}\n\n\\section{Summary}\nWe analyzed the spectrum of coherent elastic neutrino-nucleus scattering observed by the COHERENT experiment to constrain nonstandard neutrino interactions. For NSI induced by a vector mediator lighter than 50~MeV, COHERENT data only constrain the mediator coupling $g$. Since the NSI matter effect in neutrino propagation depends on $\\frac{g^2}{M_{Z'}^2}$, the constraint does not apply to matter NSI induced by a very light mediator. For a heavier mediator, the COHERENT constraints are weakened by degeneracies between different combinations of NSI parameters. In particular, a cancellation between the NSI couplings to up and down quarks allows very large NSI parameters. However, COHERENT data place meaningful constraints on the effective NSI parameters in Earth matter since they depend on the sum of the up-type and down-type NSI parameters.\n\n\\vspace{0.1 in}\n{\\it Acknowledgments.} We thank K.~Scholberg for helpful correspondence. This research was supported in part by the\nU.S. DOE under Grant No. DE-SC0010504.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\noindent In this paper we extend our previous resut \\cite{BM} concerning the\nhomogenization of integral functionals with linear growth involving manifold valued mappings.\nMore precisely, we are interested in energies of the form\n\\begin{equation}\\label{mainfunct}\\int_\\O f\\left(\\frac{x}{\\e},\\nabla\nu\\right)dx\\,,\\quad u : \\O \\to \\mathcal{M}\\subset{\\mathbb{R}}^d\\,,\\end{equation}\nwhere $\\O \\subset {\\mathbb{R}}^N$ is a bounded open set,\n$f:{\\mathbb{R}}^N \\times {\\mathbb{R}}^{d \\times N} \\to [0,+\\infty)$ is a periodic integrand\nin the first variable with linear growth in the second one, and ${\\mathcal{M}}$ is a smooth submanifold. Our main goal is to\nfind an effective description of such energies as $\\varepsilon\\to 0$. To this aim we perform a $\\Gamma$-convergence analysis\nwhich is an appropriate approach to study asymptotics in variational problems (see \\cite{DM} for a detailed description of this subject).\nFor energies with superlinear growth, the most general homogenization result has been obtained independently in \\cite{Br,M} in the nonconstrained case, and in\n\\cite{BM} in the setting of manifold valued maps.\n\nThe functional in \\eqref{mainfunct} is naturally defined for maps in the Sobolev class $W^{1,1}$. However if one wants to apply\nthe Direct Method in the Calculus of Variations, it becomes necessary to extend the original energy to a larger class of functions (possibly singular) in which the existence\nof minimizers is ensured. In the nonconstrained case, this class is exactly the space of functions of bounded variation and the problem of finding an\nintegral representation for the extension, the so-called {\\it ``relaxed functional\"}, has been widely invetigated, see {\\it e.g.}, \\cite{Serr,GMSbv,DMsc,AMT,AmPal,FR,ADM,FM,FM2,BFF}\nand \\cite{Bouch,DAG} concerning homogenization in $BV$-spaces.\n\nMany models from material science involve vector fields taking their values into a manifold.\nThis is for example the case in the study of equilibria\nfor liquid crystals, in ferromagnetism or for magnetostrictive\nmaterials. It then became necessary to understand the\nbehaviour of integral functionals of the type (\\ref{mainfunct})\nunder this additional constraint.\nIn the framework of Sobolev spaces, it was the object of\n\\cite{DFMT,AL,BM}. For $\\varepsilon$ fixed, the complete analysis in the linear growth case has been performed in\n\\cite{AEL} assuming that the manifold is the unit sphere of ${\\mathbb{R}}^d$.\nUsing a different approach, the arbitrary manifold case has been recently treated in\n\\cite{Mucci} where a further isotropy assumption on\nthe integrand is made. We will present in the Appendix the analogue result to \\cite{AEL}\nfor a general integrand and a general manifold.\n\nWe finally mention that the topology of ${\\mathcal{M}}$ does not play an important role here. This is in\ncontrast with a slightly different problem originally introduced in \\cite{BCL,BBC}, where the starting\nenergy is assumed to be finite only for smooth maps. In this direction,\nsome recent results in the linear growth case can be found in \\cite{GM,GM1} where the study\nis performed within the framework of Cartesian Currents \\cite{GMS}.\nWhen the manifold ${\\mathcal{M}}$ is topologically nontrivial, it shows the emergence in the relaxation process of non\nlocal effects essentially related to the non density of\nsmooth maps (see \\cite{B,BZ}).\n\\vskip5pt\n\nThroughout this paper we consider a\ncompact and connected smooth submanifold ${\\mathcal{M}}$ of ${\\mathbb{R}}^d$ without\nboundary. The classes of maps we are interested in are defined as\n$$BV(\\O;{\\mathcal{M}}):=\\big\\{ u \\in BV(\\O;{\\mathbb{R}}^d) : \\; u(x) \\in {\\mathcal{M}} \\text{ for ${\\mathcal{L}}^N$-a.e. }x \\in \\O\\big\\}\\,,$$\nand $W^{1,1}(\\O;{\\mathcal{M}})=BV(\\O;{\\mathcal{M}}) \\cap W^{1,1}(\\O;{\\mathbb{R}}^d)$. For a smooth ${\\mathcal{M}}$-valued map, it is well known that first order derivatives belong to the tangent space of ${\\mathcal{M}}$, and this\nproperty has a natural extension to $BV$-maps with values in ${\\mathcal{M}}$, see Lemma \\ref{manifold}.\n\\vskip5pt\n\nThe function $f : {\\mathbb{R}}^N \\times {\\mathbb{R}}^{d \\times N} \\to [0,+\\infty)$ is\nassumed to be a Carath\\'eodory integrand satisfying\n\\begin{itemize}\n\\item[$(H_1)$] for every $\\xi \\in\n{\\mathbb{R}}^{d \\times N}$ the function $f(\\cdot,\\xi)$ is $1$-periodic, {\\it\ni.e.} if $\\{e_1,\\ldots,e_N\\}$ denotes the canonical basis\nof ${\\mathbb{R}}^N$, one has $f(y+e_i,\\xi)=f(y,\\xi)$ for every $i=1,\\ldots,N$ and $y \\in {\\mathbb{R}}^N$;\\\\\n\\item[$(H_2)$] there exist $0<\\a \\leq \\b < +\\infty$\nsuch that\n$$\\a |\\xi| \\leq f(y,\\xi)\\leq \\b(1+|\\xi|) \\quad \\text{ for a.e. }y \\in {\\mathbb{R}}^N \\text{ and all }\n\\xi \\in {\\mathbb{R}}^{d \\times N}\\,;$$\n\\item[$(H_3)$]there exists $L>0$ such that\n$$|f(y,\\xi)-f(y,\\xi')| \\leq L |\\xi-\\xi'|\\, \\quad \\text{ for a.e. }y \\in {\\mathbb{R}}^N \\text{ and all }\n\\xi,\\, \\xi' \\in {\\mathbb{R}}^{d \\times N}\\,.$$\n\\end{itemize}\nFor $\\e>0$, we define the functionals ${\\mathcal{F}}_\\e:L^1(\\O;{\\mathbb{R}}^d) \\to\n[0,+\\infty]$ by\n$${\\mathcal{F}}_\\e(u):=\\begin{cases} \\displaystyle \\int_\\O f\\left(\\frac{x}{\\e},\\nabla\nu\\right) dx & \\text{if }u \\in\nW^{1,1}(\\O;\\mathcal{M})\\,,\\\\[8pt]\n+\\infty & \\text{otherwise}\\,.\n\\end{cases}$$\n\\vskip5pt\n\nWe have proved in \\cite{BM} the following representation result on\n$W^{1,1}(\\O;{\\mathcal{M}})$.\n\n\\begin{theorem}[\\cite{BM}]\\label{babmilp=1}\nLet ${\\mathcal{M}}$ be a compact and connected smooth submanifold of ${\\mathbb{R}}^d$\nwithout boundary, and $f:{\\mathbb{R}}^N \\times {\\mathbb{R}}^{d \\times N} \\to\n[0,+\\infty)$~be a Carath\\'eodory function satisfying $(H_1)$ to\n$(H_3)$. Then the family $\\{{\\mathcal{F}}_\\e\\}_{\\e>0}$ $\\G$-converges for the\nstrong $L^1$-topology at every $u \\in W^{1,1}(\\O;{\\mathcal{M}})$ to ${\\mathcal{F}}_{\\rm\nhom} : W^{1,1}(\\O;{\\mathcal{M}}) \\to [0,+\\infty)$, where\n$${\\mathcal{F}}_{\\rm hom}(u):= \\int_\\O Tf_{\\rm hom}(u,\\nabla u)\\, dx\\,,$$\nand $Tf_{\\rm hom}$ is the tangentially homogenized energy density\ndefined for every $s\\in {\\mathcal{M}}$ and $\\xi\\in [T_s({\\mathcal{M}})]^N$ by\n\\begin{equation}\\label{Tfhom}\nTf_{\\rm hom}(s,\\xi)=\\lim_{t\\to+\\infty}\\inf_{\\varphi} \\bigg\\{\n- \\hskip -1em \\int_{(0,t)^N} f(y,\\xi+ \\nabla \\varphi(y))\\, dy : \\varphi \\in\nW^{1,\\infty}_0((0,t)^N;T_s(\\mathcal{M})) \\bigg\\}.\n\\end{equation}\n\\end{theorem}\n\\vskip5pt\n\nNote that the previous theorem is not really satisfactory since the domain of the $\\G$-limit is obviously larger than the Sobolev\nspace $W^{1,1}(\\O;{\\mathcal{M}})$. In view of the studies performed in\n\\cite{GM,Mucci}, the domain is exactly given by $BV(\\O;{\\mathcal{M}})$. \nUnder the additional (standard)\nassumption,\n\\begin{itemize}\n\\item[$(H_4)$]there exist $C>0$ and $00\\}\n\\text{ and }\\varphi=b \\text{ on }\\partial (tQ_\\nu)\\cap\\{x\\cdot \\nu \\leq 0\\}\\bigg\\}\\,,\n\\end{multline}\n$Q_\\nu$ being any open unit cube in ${\\mathbb{R}}^N$ centered at the\norigin with two of its faces orthogonal to $\\nu$.\n\\end{theorem}\n\n\nThe paper is organized as follows. We first review in Section 2\nstandard facts about of manifold valued Sobolev mappings and\nfunctions of bounded variation that will be used all the way\nthrough. The main properties of the energy densities $Tf_{\\rm hom}$ and\n$\\vartheta_{\\rm hom}$ are the object of\nSection 3. A locality property of the $\\Gamma$-limit is established in Section 4. The upper bound inequality\nin Theorem \\ref{babmil2} is the object of Section~5. The lower bound is obtained\n in Section 6 where the proof of the theorem is completed.\nFinally we state in the Appendix a relaxation\nresult for general manifolds and integrands which extends\n\\cite{AEL} and \\cite{Mucci}.\n\n\n\n\\section{Preliminaries}\n\nLet $\\O$ be a generic\nbounded open subset of ${\\mathbb{R}}^N$. We write ${\\mathcal{A}}(\\O)$ for the family of\nall open subsets of $\\O$, and $\\mathcal B(\\O)$ for the\n$\\sigma$-algebra of all Borel subsets of $\\O$. We also consider a\ncountable subfamily ${\\mathcal{R}}(\\O)$ of ${\\mathcal{A}}(\\O)$ made of all finite unions\nof cubes with rational\nedge length centered at rational points of ${\\mathbb{R}}^N$.\nGiven $\\nu \\in {\\mathbb{S}^{N-1}}$, $Q_\\nu$ stands for an open unit cube in ${\\mathbb{R}}^N$\ncentered at the origin with two of its faces orthogonal to $\\nu$ and\n$Q_\\nu(x_0,\\rho):= x_0 + \\rho \\,Q_\\nu$. Similarly $Q:=(-1\/2,1\/2)^N$\nis the unit cube in ${\\mathbb{R}}^N$ and $Q(x_0,\\rho):= x_0 + \\rho \\,Q$. We\ndenote by $h^\\infty$ the recession function of a generic scalar\nfunction $h$, {\\it i.e.},\n$$h^\\infty(\\xi):=\\limsup_{t\\to+\\infty}\\,\\frac{h(t\\xi)}{t}\\,.$$\n\n\nThe space of vector valued Radon measures in $\\O$ with finite total\nvariation is denoted by ${\\mathcal{M}}(\\O;{\\mathbb{R}}^m)$. We shall follow \\cite{AFP}\nfor the standard notation on functions of bounded variation. We only\nrecall Alberti Rank One Theorem which states that for $|D^c u|$-a.e.\n$x \\in \\O$, $$A(x):=\\frac{dD^cu}{d|D^cu|}(x)$$ is a rank one matrix.\n\n\n\\bigskip\n\nIn this paper, we are interested in Sobolev and $BV$ maps taking\ntheir values into a given manifold. We consider a connected smooth\nsubmanifold ${\\mathcal{M}}$ of ${\\mathbb{R}}^d$ without boundary. The tangent space of\n${\\mathcal{M}}$ at $s \\in {\\mathcal{M}}$ is denoted by $T_s({\\mathcal{M}})$, ${\\rm co}({\\mathcal{M}})$ stands\nfor the convex hull of ${\\mathcal{M}}$, and $\\pi_1({\\mathcal{M}})$ is the\nfundamental group of ${\\mathcal{M}}$. \n\nIt is well known that if $u \\in W^{1,1}(\\O;{\\mathcal{M}})$, then $\\nabla u(x)\n\\in [T_{u(x)}({\\mathcal{M}})]^N$ for ${\\mathcal{L}}^N$-a.e. $x \\in \\O$. The analogue\nstatement for $BV$-maps is given in Lemma \\ref{manifold} below.\n \n \n\\begin{lemma}\\label{manifold}\nFor every $u \\in BV(\\O;{\\mathcal{M}})$,\n\\begin{align}\n\\label{aplimM}&\\tilde u(x) \\in {\\mathcal{M}}\\text{ for every } x \\in \\O\\setminus S_u\\,;\\\\[0.2cm]\n\\label{jumpM}&u^\\pm(x) \\in {\\mathcal{M}}\\text{ for every }x \\in J_u\\,;\\\\[0.2cm]\n\\label{gradM}&\\nabla u(x) \\in [T_{u(x)}({\\mathcal{M}})]^N \\text{ for } {\\mathcal{L}}^N\\text{-a.e. }x \\in \\O\\,;\\\\[0.2cm]\n\\label{cantM}&\\displaystyle A(x):=\\frac{dD^c u}{d|D^c u|}(x) \\in [T_{\\tilde u(x)}({\\mathcal{M}})]^N \\text{ for }|D^c u|\\text{-a.e. }x \\in \\O\\,.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nWe first show (\\ref{aplimM}). By definition of the space\n$BV(\\O;{\\mathcal{M}})$, $u(y)\\in{\\mathcal{M}}$ for a.e. $y\\in \\O$. Therefore for any\n$x\\in\\O\\setminus S_u$, we have $|u(y)-\\tilde u(x)|\\geq\n\\text{dist}(\\tilde u(x),{\\mathcal{M}})$ for a.e. $y\\in\\O$. By definition of\n$S_u$, this yields $\\text{dist}(\\tilde u(x),{\\mathcal{M}})=0$, {\\it i.e.},\n$\\tilde u(x)\\in {\\mathcal{M}}$. Arguing as for the approximate limit points,\none obtains (\\ref{jumpM}).\n\nNow it remains to prove \\eqref{gradM} and \\eqref{cantM}. We introduce the function $\\Phi:{\\mathbb{R}}^d\\to{\\mathbb{R}}$ defined by\n$$\\Phi(s)=\\chi(\\delta^{-1}\\text{dist}(s,{\\mathcal{M}})^2)\\, \\text{dist}(s,{\\mathcal{M}})^2\\,,$$\nwhere $\\chi\\in {\\mathcal{C}}_c^\\infty({\\mathbb{R}};[0,1])$ with $\\chi(t)=1$ for $|t|\\leq\n1$, $\\chi(t)=0$ for $|t|\\geq2$, and $\\delta>0$ is small enough so\nthat $\\Phi \\in {\\mathcal{C}}^1({\\mathbb{R}}^d)$. Note that for every $s \\in {\\mathcal{M}}$,\n$\\Phi(s)=0$ and\n\\begin{equation}\\label{ker}\n\\text{Ker}\\,\\nabla\\Phi(s)=T_s({\\mathcal{M}})\\,.\n\\end{equation}\nBy the Chain Rule formula in $BV$ (see, {\\it e.g., }\n\\cite[Theorem~3.96]{AFP}), $\\Phi\\circ u\\in BV(\\O)$ and\n\\begin{align*}\nD(\\Phi\\circ u)=&\\,\\nabla\\Phi(u)\\nabla u \\,\\mathcal{L}^N\\res \\, \\O+\\nabla \\Phi(\\tilde u) D^cu\n+\\big(\\Phi(u^+)-\\Phi(u^-)\\big)\\otimes\\nu_u\\,\\mathcal{H}^{N-1}\\res \\, J_u\\\\\n=&\\,\\nabla\\Phi(u)\\nabla u \\,\\mathcal{L}^N\\res \\, \\O+\\nabla \\Phi(\\tilde u)A|D^cu|\\,,\n\\end{align*}\nthanks to \\eqref{jumpM}. On the other hand, $\\Phi\\circ u=0$ a.e. in\n$\\O$ since $u(x)\\in{\\mathcal{M}}$ for a.e. $x\\in\\O$. Therefore we have that\n$D(\\Phi\\circ u)\\equiv 0$. Since $\\mathcal{L}^N\\res \\, \\O$ and $|D^c\nu|$ are mutually singular measures, we infer that\n$\\nabla\\Phi(u(x))\\nabla u(x)=0$ for $\\mathcal{L}^N$-a.e. $x\\in\\O$\nand $\\nabla \\Phi(\\tilde u(x))A(x)=0$ for $|D^cu|$-a.e. $x\\in\\O$.\nHence \\eqref{gradM} and \\eqref{cantM} follow from \\eqref{ker}\ntogether with \\eqref{aplimM}.\n\\end{proof}\n\n\nIn \\cite{B,BZ}, density results of smooth functions between\nmanifolds into Sobolev spaces have been established. In the\nfollowing theorem, we summarize these results only in $W^{1,1}$. Let\n$\\mathcal S$ be the family of all finite unions of subsets contained\nin a $(N-2)$-dimensional submanifold of ${\\mathbb{R}}^N$.\n\n\\begin{theorem}\\label{density}Let ${\\mathcal{D}}(\\O;{\\mathcal{M}}) \\subset\nW^{1,1}(\\O;{\\mathcal{M}})$ be defined by\n$${\\mathcal{D}}(\\O;{\\mathcal{M}}):=\\begin{cases}\nW^{1,1}(\\O;{\\mathcal{M}})\\cap{\\mathcal{C}}^\\infty(\\O;{\\mathcal{M}}) & \\text{if $\\,\\pi_1({\\mathcal{M}})=0$}\\,,\\\\[10pt]\n\\big\\{ u \\in W^{1,1}(\\O;{\\mathcal{M}})\\cap {\\mathcal{C}}^\\infty(\\O \\setminus \\Sigma;{\\mathcal{M}})\n\\text{ for some } \\Sigma \\in \\mathcal S \\big\\}\n & \\text{otherwise}\\,.\n\\end{cases}$$\nThen ${\\mathcal{D}}(\\O;{\\mathcal{M}})$ is dense in $W^{1,1}(\\O;{\\mathcal{M}})$ for the strong\n$W^{1,1}(\\O;{\\mathbb{R}}^d)$-topology.\n\\end{theorem}\n\nWe now present a useful projection technique (taken from \\cite{Dem}\nfor ${\\mathcal{M}}={\\mathbb{S}^{d-1}}$). It was first introduced in \\cite{HKL,HL}, and makes\nuse of an averaging device going back to \\cite{FF}. We sketch the\nproof for the convenience of the reader.\n\n\\begin{proposition}\\label{proj}\nLet ${\\mathcal{M}}$ be a compact connected $m$-dimensional smooth submanifold\nof ${\\mathbb{R}}^d$ without boundary, and let $v \\in W^{1,1}(\\O;{\\mathbb{R}}^d) \\cap\n{\\mathcal{C}}^\\infty(\\O\\setminus \\Sigma;{\\mathbb{R}}^d)$ for some $\\Sigma\\in\\mathcal{S}$\nsuch that $v(x) \\in {\\rm co}({\\mathcal{M}})$ for a.e. $x \\in \\O$. Then there\nexists $w \\in W^{1,1}(\\O;{\\mathcal{M}})$ satisfying $w=v$ a.e. in $\\big\\{x \\in\n\\O\\setminus\\Sigma: v(x) \\in {\\mathcal{M}} \\big\\}$ and\n\\begin{equation}\\label{1127}\n\\int_\\O |\\nabla w|\\, dx \\leq C_\\star \\int_\\O |\\nabla\nv|\\,dx\\,,\\end{equation} for some constant $C_\\star >0$ which only\ndepends on $d$ and ${\\mathcal{M}}$.\n\\end{proposition}\n\n\\begin{proof} According to \\cite[Lemma 6.1]{HL} (which holds for\n$p=1$), there exist a compact Lipschitz polyhedral set $X\\subset{\\mathbb{R}}^d$\nof codimension greater or equal to $2$, and a locally Lipschitz map $\\pi:{\\mathbb{R}}^d \\setminus X \\to\n{\\mathcal{M}}$ such that\n\\begin{equation}\\label{gradproj}\n\\int_{B^d(0,R)} |\\nabla \\pi(s)|\\, ds <+\\infty \\quad \\text{ for every\n}R<+\\infty\\,.\n\\end{equation}\nMoreover, in a neighborhood of ${\\mathcal{M}}$ the mapping $\\pi$ is smooth of\nconstant rank equal to $m$.\n\nWe argue as in the proof of \\cite[Theorem 6.2]{HL}. Let $B$ be an\nopen ball in ${\\mathbb{R}}^d$ containing ${\\mathcal{M}} \\cup X$, and let $\\d>0$ small\nenough so that the nearest point projection on ${\\mathcal{M}}$ is a well\ndefined smooth mapping in the $\\d$-neighborhood of ${\\mathcal{M}}$. Fix $\\sigma <\n\\inf\\{\\d,\\text{dist}({\\rm co}({\\mathcal{M}}),\\partial B)\\}$ small enough, and for $a\n\\in B^d(0,\\sigma)$ we define the translates\n$B_a:=a+B$ and $X_a:=a+X$,\nand the projection $\\pi_a:B_a \\setminus X_a \\to {\\mathcal{M}}$ by\n$\\pi_a(s):=\\pi(s-a)$. Since $\\pi$ has full rank and is smooth in a neighborhood of ${\\mathcal{M}}$,\nby the Inverse Function Theorem the number\n\\begin{equation}\\label{gradlambda}\n\\Lambda:=\\sup_{a \\in B^d(0,\\sigma)} {\\rm\nLip}\\big({\\pi_a}_{|{\\mathcal{M}}}\\big)^{-1}\n\\end{equation}\nis finite and only depends on ${\\mathcal{M}}$. Using Sard's lemma, one can show\nthat $\\pi_a \\circ v \\in W^{1,1}(\\O;{\\mathcal{M}})$ for\n${\\mathcal{L}}^d$-a.e. $a \\in B^d(0,\\sigma)$. Then Fubini's theorem together\nwith the Chain Rule formula yields\n\\begin{multline*}\n\\int_{B^d(0,\\sigma)}\n\\int_{\\O} |\\nabla (\\pi_a \\circ v)(x)|\\, d{\\mathcal{L}}^N(x) \\, d{\\mathcal{L}}^d(a)\n\\leq\\,\\\\\n\\leq \\int_\\O |\\nabla v(x)|\n\\left(\\int_{B^d(0,\\sigma)}|\\nabla \\pi(v(x)-a)| \\, d{\\mathcal{L}}^d(a) \\right)\\, d{\\mathcal{L}}^N(x)\n \\leq \\,\\\\\n \\leq \\left(\\int_{B} |\\nabla \\pi(s)|\\, d{\\mathcal{L}}^d(s)\\right)\n\\left(\\int_\\O |\\nabla v(x)|\\,d{\\mathcal{L}}^N(x)\\right)\\,.\n\\end{multline*}\nTherefore we can find $a \\in B^d(0,\\sigma)$ such that\n\\begin{equation}\\label{gradpia}\\int_\\O |\\nabla (\\pi_a \\circ v)|\\, dx\n\\leq C{\\mathcal{L}}^d\\left(B^d(0,\\sigma)\\right)^{-1} \\int_\\O |\\nabla\nv|\\,dx\\,,\\end{equation} where we used (\\ref{gradproj}). To conclude,\nit suffices to set $w:=\\big({\\pi_a}_{|{\\mathcal{M}}}\\big)^{-1} \\circ \\pi_a \\circ v$,\nand (\\ref{1127}) arises as a consequence of (\\ref{gradlambda}) and\n(\\ref{gradpia}).\n\\end{proof}\n\n\n\n\n\\section{Properties of homogenized energy densities}\n\nIn this section we present the main properties of the energy\ndensities $Tf_\\text{hom}$ and $\\vartheta_\\text{hom}$ defined in (\\ref{Tfhom})\nand (\\ref{thetahom}). In particular we will prove that\n$\\vartheta_\\text{hom}$ is well defined in the sense that the limit in\n(\\ref{thetahom}) exists.\n\n\\subsection{The tangentially homogenized bulk energy}\\label{thbe}\n\nWe start by considering the bulk energy density $Tf_{\\rm hom}$ defined in \\eqref{Tfhom}. \nAs in \\cite{BM} we first construct a new energy density\n$g:{\\mathbb{R}}^N\\times{\\mathbb{R}}^d\\times{\\mathbb{R}}^{d\\times N}\\to[0,+\\infty)$ satisfying\n$$g(\\cdot,s,\\xi)=f(\\cdot,\\xi)\\quad \\text{and}\\quad g_{\\rm hom}(s,\\xi)=Tf_{\\rm\nhom}(s,\\xi)\\quad \\text{for $s\\in{\\mathcal{M}}$ and $\\xi\\in[T_s({\\mathcal{M}})]^N$}\\,.$$\nHence upon extending $Tf_\\text{hom}$ by $g_\\text{hom}$ outside the set\n$\\big\\{(s,\\xi) \\in {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N}:\\; s \\in {\\mathcal{M}},\\, \\xi\n\\in [T_s({\\mathcal{M}})]^N\\big\\}$, we will tacitly assume $Tf_\\text{hom}$ to be\ndefined over the whole ${\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N}$. We proceed\nas follow. \\vskip5pt\n\nFor $s \\in {\\mathcal{M}}$ we denote by $P_s:{\\mathbb{R}}^d \\to T_s({\\mathcal{M}})$ the orthogonal\nprojection from ${\\mathbb{R}}^d$ into $T_s({\\mathcal{M}})$, and we set\n$$\\mathbf{P}_s(\\xi):=(P_s(\\xi_1),\\ldots,P_s(\\xi_N)) \\quad \\text{for\n$\\xi=(\\xi_1,\\ldots,\\xi_N)\\in{\\mathbb{R}}^{d\\times N}\\,$.}$$ For $\\delta_0>0$\nfixed, let $\\mathcal U:=\\big\\{s\n\\in{\\mathbb{R}}^d\\,:\\,\\text{dist}(s,{\\mathcal{M}})<\\d_0\\big\\}$ be the $\\d_0$-neighborhood of\n${\\mathcal{M}}$. Choosing $\\delta_0>0$ small enough, we may assume that the\nnearest point projection $\\Pi: \\mathcal U \\to {\\mathcal{M}}$ is a well defined\nLipschitz mapping. Then the map $s \\in \\mathcal U \\mapsto\nP_{\\Pi(s)}$ is Lipschitz. Now we introduce a cut-off function $\\chi\n\\in {\\mathcal{C}}^\\infty_c({\\mathbb{R}}^d;[0,1])$ such that $\\chi(t)=1$ if $\\text{dist}(s,{\\mathcal{M}})\n\\leq \\delta_0\/2$, and $\\chi(s)=0$ if $\\text{dist}(s,{\\mathcal{M}}) \\geq 3\\delta_0\/4$,\nand we define\n$$\\mathbb{P}_s(\\xi):=\\chi(s) \\mathbf{P}_{\\Pi(s)}(\\xi)\\quad \\text{for $(s,\\xi) \\in {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N}\\,$.}$$\nGiven the Carath\\'eodory integrand $f:{\\mathbb{R}}^N \\times {\\mathbb{R}}^{d\\times N}\n\\to [0,+\\infty)$ satisfying assumptions $(H_1)$ to $(H_3)$, we\nconstruct the new integrand $g: {\\mathbb{R}}^N\\times{\\mathbb{R}}^d\\times{\\mathbb{R}}^{d\\times\nN}\\to[0,+\\infty)$ as\n\\begin{equation}\\label{defig}\ng(y,s,\\xi):=f(y,\\mathbb{P}_s(\\xi))+|\\xi-\\mathbb{P}_s(\\xi)|\\,.\n\\end{equation}\n\n\nWe summarize in the following lemma the main properties of $g$.\n\n\\begin{lemma}\\label{defg}\nThe integrand $g$ as defined in (\\ref{defig}) is a Carath\\'eodory function satisfying\n\\begin{equation}\\label{idfg}\ng(y,s,\\xi)=f(y,\\xi)\\quad\\text{and}\\quad g^\\infty(y,s,\\xi)=f^\\infty(y,\\xi)\\quad \\text{for $s\n\\in {\\mathcal{M}}$ and $\\xi \\in [T_s({\\mathcal{M}})]^N\\,$,}\n\\end{equation}\nand\n\\begin{itemize}\n\\item[(i)] $g$ is $1$-periodic in the first variable;\n\\item[(ii)] there exist $0<\\alpha'\\leq \\b'$ such that\n\\begin{equation}\\label{pgrowth}\n\\alpha'|\\xi|\\leq g(y,s,\\xi)\\leq\n\\b'(1+|\\xi|)\\quad\\text{for every $(s,\\xi)\\in {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d\n\\times N}$ and a.e. $y \\in {\\mathbb{R}}^N$}\\,;\\end{equation}\n\\item[(iii)] there exist $C>0$ and $C'>0$ such that\n\\begin{equation}\\label{moduluscont}\n|g(y,s,\\xi)-g(y,s',\\xi)| \\leq C|s-s'|\\; |\\xi|\\,,\\end{equation}\n\\begin{equation}\\label{lipg}\n|g(y,s,\\xi) - g(y,s,\\xi')| \\leq C'|\\xi-\\xi'|\\end{equation} for every\n$s$, $s' \\in {\\mathbb{R}}^d$, every $\\xi \\in {\\mathbb{R}}^{d \\times N}$ and a.e. $y\n\\in {\\mathbb{R}}^N$;\n\\item[(iv)] if in addition $(H_4)$ holds, there exists $00$ such that\n\\begin{equation}\\label{grec}\n|g(y,s,\\xi) - g^\\infty(y,s,\\xi)| \\leq C''(1+|\\xi|^{1-q}) \\end{equation}\nfor every $(s,\\xi) \\in {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N}$ and a.e.\n$y \\in {\\mathbb{R}}^N\\,$.\n\\end{itemize}\n\\end{lemma}\n\\vskip5pt\n\nWe can now state the properties of $Tf_\\text{hom}$ and the relation\nbetween $Tf_\\text{hom}$ and $g_\\text{hom}$ through the homogenization procedure.\n\n\n\\begin{proposition}\\label{properties1}\nLet $f:{\\mathbb{R}}^N \\times {\\mathbb{R}}^{d\\times N} \\to [0,+\\infty)$ be a\nCarath\\'eodory integrand satisfying $(H_1)$ to $(H_3)$.\nThen the following properties hold:\n\\begin{itemize}\n\\item[(i)] for every $s \\in {\\mathcal{M}}$ and $\\xi \\in [T_s({\\mathcal{M}})]^N$,\n\\begin{equation}\\label{identhomform}\nTf_{\\rm hom}(s,\\xi)=g_{\\rm hom}(s,\\xi)\\,,\n\\end{equation}\nwhere\n$$g_{\\rm hom}(s,\\xi):=\\lim_{t\\to+\\infty}\\inf_{\\varphi}\\bigg\\{\n- \\hskip -1em \\int_{(0,t)^N} g(y,s,\\xi+\\nabla \\varphi(y))\\, dy: \\varphi\n\\in W^{1,\\infty}_0((0,t)^N;{\\mathbb{R}}^d) \\bigg\\}$$ is the usual homogenized\nenergy density of $g$ (see, e.g.,\n\\cite[Chapter~14]{BD});\\\\\n\\item[(ii)] the function $Tf_{\\rm hom}$ is tangentially\nquasiconvex, {\\it i.e.}, for all $s \\in {\\mathcal{M}}$ and all $\\xi \\in\n[T_s({\\mathcal{M}})]^N$,\n$$Tf_{\\rm hom}(s,\\xi) \\leq \\int_Q Tf_{\\rm hom}(s,\\xi + \\nabla \\varphi(y))\\, dy$$\nfor every $\\varphi \\in W^{1,\\infty}_0(Q;T_s({\\mathcal{M}}))$. In particular\n$Tf_{\\rm hom}(s,\\cdot)$ is rank one convex;\\\\\n\\item[(iii)] there exists\n$C>0$ such that\n\\begin{equation}\\label{pgT}\n\\a|\\xi|\\leq Tf_{\\rm hom}(s,\\xi) \\leq \\b(1+|\\xi|)\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{plipT}\n|Tf_{\\rm hom}(s,\\xi)-Tf_{\\rm hom}(s,\\xi')| \\leq\nC|\\xi-\\xi'|\n\\end{equation}\nfor every $s \\in {\\mathcal{M}}$ and $\\xi$, $\\xi' \\in [T_s({\\mathcal{M}})]^N$;\n\\item[(iv)] there exists $C_1>0$ such that\n\\begin{equation}\\label{hyp4}\n|Tf_{\\rm hom}(s,\\xi)-Tf_{\\rm hom}(s',\\xi)| \\leq\nC_1|s-s'|(1+|\\xi|)\\,,\n\\end{equation}\nfor every $s$, $s' \\in {\\mathbb{R}}^d$ and $\\xi \\in {\\mathbb{R}}^{d \\times N}$. In\nparticular $Tf_{\\rm hom}$ is continuous;\\\\\n\\item[(v)] if in addition $(H_4)$ holds, there exist $C_2>0$ and $00$,\nwe may find $k \\in {\\mathbb{N}}$ and $\\varphi \\in\nW_0^{1,\\infty}((0,k)^N;{\\mathbb{R}}^d)$ such that\n$$- \\hskip -1em \\int_{(0,k)^N} g(y,s,\\xi+\\nabla \\varphi)\\, dy \\leq g_\\text{hom}(s,\\xi)\n+ \\eta\\,.$$ We infer from \\eqref{pgrowth}\nthat $\\alpha'|\\xi|\\leq g_{\\rm hom}(s,\\xi)\\leq \\beta'(1+|\\xi|)$ and consequently\n\\begin{equation}\\label{varphi}\n- \\hskip -1em \\int_{(0,k)^N}|\\nabla \\varphi|\\, dy \\leq C(1+|\\xi|)\\,,\n\\end{equation}\nfor some constant $C>0$ depending only on $\\a'$ and $\\b'$.\nThen from (\\ref{identhomform}) and \\eqref{moduluscont} it follows\nthat\n\\begin{multline*}\nTf_\\text{hom}(s',\\xi)-Tf_\\text{hom}(s,\\xi)= g_\\text{hom}(s',\\xi)-g_\\text{hom}(s,\\xi)\\leq\\\\\n\\leq - \\hskip -1em \\int_{(0,k)^N} \\big(g(y,s',\\xi+\\nabla \\varphi)-g(y,s,\\xi+\\nabla\n\\varphi)\\big)\\, dy+ \\eta \\leq\\\\ \\leq C |s-s'|- \\hskip -1em \\int_{(0,k)^N}|\\xi\n+\\nabla \\varphi|\\, dy+\\eta\\leq C|s-s'|(1+|\\xi|)+\\eta\\,.\n\\end{multline*}\nWe deduce relation (\\ref{hyp4}) inverting the roles of\n$s$ and $s'$, and sending $\\eta$ to zero. In particular, we obtain\nthat $Tf_\\text{hom}$ is continuous as a consequence of (\\ref{hyp4}) and\n(\\ref{plipT}).\n\nTo show (\\ref{hyp5}), let us consider sequences $t_n \\nearrow +\\infty$, $k_n\n\\in {\\mathbb{N}}$ and $\\varphi_n\\in W^{1,\\infty}_0((0,k_n)^N;T_s({\\mathcal{M}}))$ such\nthat\n\\begin{equation}\\label{comptfhoninf}\nTf_\\text{hom}^{\\, \\infty}(s,\\xi)=\\lim_{n \\to\n+\\infty}\\frac{Tf_\\text{hom}(s,t_n\\xi)}{t_n}\\,,\n\\end{equation}\nand\n$$- \\hskip -1em \\int_{(0,k_n)^N}f(y,t_n \\xi + t_n \\nabla \\varphi_n)\\, dy \\leq\nTf_\\text{hom}(s,t_n\\xi)+\\frac{1}{n}\\,.$$ \nThen $(H_2)$ and\n(\\ref{pgT}) yield\n\\begin{equation}\\label{varphi_n}- \\hskip -1em \\int_{(0,k_n)^N}|\\nabla \\varphi_n|\\, dy \\leq\nC(1+|\\xi|)\\,,\n\\end{equation}\nfor some constant $C>0$ depending only on\n$\\a$ and $\\b$. Using $(H_4)$ and \\eqref{comptfhoninf}, we derive that\n\\begin{align*}\n Tf_\\text{hom}(s,\\xi) - Tf_\\text{hom}^{\\,\\infty}(s,\\xi)\n\\leq & \\liminf_{n \\to +\\infty} \\Bigg\\{- \\hskip -1em \\int_{(0,k_n)^N}\\bigg|\nf(y,\\xi+\\nabla\\varphi_n)- f^{\\,\\infty}(y,\\xi +\\nabla \\varphi_n)\\bigg|\\, dy\\, +\\\\\n&\\,+ - \\hskip -1em \\int_{(0,k_n)^N}\\bigg| f^{\\,\\infty}(y,\\xi+\\nabla\n\\varphi_n)-\\frac{ f(y,t_n \\xi + t_n \\nabla\n\\varphi_n)}{t_n}\\bigg|\\, dy\\Bigg\\}\\\\\n\\leq & \\liminf_{n \\to +\\infty} \\Bigg\\{C\n- \\hskip -1em \\int_{(0,k_n)^N}\\!(1+|\\xi+\\nabla \\varphi_n|^{1-q})\\, dy\\\\\n&+ \\frac{C}{t_n}- \\hskip -1em \\int_{(0,k_n)^N}\\!(1+t_n^{1-q}|\\xi+\\nabla\n\\varphi_n|^{1-q})\\, dy\\Bigg\\}\\,,\n\\end{align*}\nwhere we have also used the fact that $f^{\\,\\infty}(y,\\cdot)$ is\npositively homogeneous of degree one in the last inequality. Then\n(\\ref{varphi_n}) and H\\\"older's inequality lead to\n\\begin{equation}\\label{firstineq}\nTf_\\text{hom}(s,\\xi) - Tf_\\text{hom}^{\\,\\infty}(s,\\xi) \\leq C(1+|\\xi|^{1-q})\\,.\n\\end{equation}\nConversely, given $k\\in {\\mathbb{N}}$ and $\\varphi \\in\nW_0^{1,\\infty}((0,k)^N;T_s({\\mathcal{M}}))$, we deduce from $(H_2)$\nthat\n$$\\frac{f(\\cdot,t(\\xi+\\nabla\\varphi(\\cdot)))}{t} \\leq \\b(1+|\\xi+\\nabla\n\\varphi|) \\in L^1((0,k)^N)$$ whenever $t > 1$. Then Fatou's lemma\nimplies\n\\begin{equation*}\nTf_\\text{hom}^\\infty(s,\\xi)\n\\leq \\limsup_{t \\to +\\infty} - \\hskip -1em \\int_{(0,k)^N}\\frac{f(y,t\\xi+t\n\\nabla \\varphi)}{t}\\, dy \\leq - \\hskip -1em \\int_{(0,k)^N}f^\\infty(y,\\xi+ \\nabla \\varphi)\\, dy\\,.\n\\end{equation*}\nTaking the infimum over all admissible $\\varphi$'s and letting $k\\to+\\infty$, we\ninfer\n\\begin{equation}\\label{remdensinf}\nTf_\\text{hom}^\\infty(s,\\xi) \\leq T(f^\\infty)_\\text{hom}(s,\\xi)\\,.\n\\end{equation}\nFor $\\eta>0$ arbitrary small, consider $k \\in {\\mathbb{N}}$ and $\\varphi \\in\nW^{1,\\infty}_0((0,k)^N;T_s({\\mathcal{M}}))$ such that\n$$- \\hskip -1em \\int_{(0,k)^N}f(y,\\xi + \\nabla \\varphi)\\, dy \\leq\nTf_\\text{hom}(s,\\xi)+\\eta\\,.$$\nIn view of $(H_2)$ and (\\ref{pgT}), it turns\nout that \\eqref{varphi} holds \nwith constant $C>0$ only depending on $\\a$ and $\\b$. Then it\nfollows from \\eqref{remdensinf} that\n\\begin{multline*}\nTf_\\text{hom}^\\infty(s,\\xi) - Tf_\\text{hom}(s,\\xi) \\leq\nT(f^\\infty)_\\text{hom}(s,\\xi) - Tf_\\text{hom}(s,\\xi)\\leq\\\\\n\\leq - \\hskip -1em \\int_{(0,k)^N}|f^\\infty(y,\\xi + \\nabla \\varphi)- f(y,\\xi\n+ \\nabla \\varphi)|\\, dy+\\eta \\leq C- \\hskip -1em \\int_{(0,k)^N}(1+|\\xi + \\nabla \\varphi|^{1-q})\\, dy+\\eta\\,,\n\\end{multline*}\nwhere we have used $(H_4)$ in the last inequality. Using H\\\"older's\ninequality, relation (\\ref{varphi}) together with the arbitrariness of\n$\\eta$ yields\n\\begin{equation}\\label{secineq}Tf_\\text{hom}^\\infty(s,\\xi) -\nTf_\\text{hom}(s,\\xi) \\leq C(1+|\\xi|^{1-q})\\,.\\end{equation} Gathering\n(\\ref{firstineq}) and (\\ref{secineq}) we conclude the proof of\n(\\ref{hyp5}). \\prbox\n\n\\subsection{The homogenized surface energy}\\label{sectsurf}\n\n\\noindent We now present the homogenized surface energy density\n$\\vartheta_\\text{hom}$. We start by introducing some useful notations.\n\nGiven $\\nu=(\\nu_1,\\ldots,\\nu_N)$ an orthonormal basis of ${\\mathbb{R}}^N$ and\n$(a,b) \\in {\\mathcal{M}} \\times {\\mathcal{M}} $, we denote by\n$$Q_\\nu:=\\Big\\{\\alpha_1\\nu_1+\\ldots+\\alpha_N\\nu_N\\;:\\;\\alpha_1,\\ldots,\\alpha_N \\in(-1\/2,1\/2)\\Big\\}\\,,$$\nand for $x \\in {\\mathbb{R}}^N$, we set $\\|x\\|_{\\nu,\\infty}:=\\sup_{i \\in\n\\{1,\\ldots,N\\}}|x\\cdot\\nu_i|$, $x_\\nu:=x\\cdot \\nu_1$ and\n$x':=(x\\cdot\\nu_2)\\nu_2 +\\ldots+(x\\cdot \\nu_N)\\nu_N$ so that $x$ can\nbe identified to the pair $(x',x_\\nu)$. Let $u_{a,b,\\nu}:Q_\\nu \\to\n{\\mathcal{M}}$ be the function defined by\n$$u_{a,b,\\nu}(x):=\\begin{cases}\na & \\text{if } x_\\nu>0\\,,\\\\[5pt]\nb & \\text{if } x_\\nu \\leq 0\\,.\n\\end{cases}$$\nWe introduce the class of functions\n$${\\mathcal{A}}_t(a,b,\\nu) :=\\Big\\{\\varphi \\in W^{1,1}(t Q_\\nu;{\\mathcal{M}}):\n\\varphi=u_{a,b,\\nu} \\text{ on }\\partial(tQ_\\nu)\\Big\\}\\,.$$\nWe have the following result.\n\n\\begin{proposition}\\label{limitsurfenerg}\nFor every $(a,b,\\nu_1)\\in{\\mathcal{M}}\\times{\\mathcal{M}}\\times{\\mathbb{S}^{N-1}}$, there exists\n\\begin{eqnarray*}\n\\vartheta_{\\rm hom}(a,b,\\nu_1) &: = & \\lim_{t\\to+\\infty}\\,\\inf_\\varphi\n\\left\\{\\frac{1}{t^{N-1}} \\int_{t Q_\\nu} f^\\infty(y,\\nabla\n\\varphi(y))\\, dy : \\varphi \\in {\\mathcal{A}}_t(a,b,\\nu) \\right\\}\\,,\n\\end{eqnarray*}\nwhere $\\nu=(\\nu_1,\\ldots,\\nu_N)$ is any orthonormal basis of ${\\mathbb{R}}^N$ with first element equal to $\\nu_1$ (the limit being independent of such a\nchoice).\n\\end{proposition}\n\nThe proof of Proposition \\ref{limitsurfenerg} is quite indirect and\nis based on an analogous result for a similar surface energy density\n$\\tilde \\vartheta_\\text{hom}$ (see \\eqref{surfen2} below). We will prove in Proposition \\ref{limsurf2}\nthat the two densities coincide.\n\\vskip5pt\n\n\nGiven $a$ and $b\\in {\\mathcal{M}}$, we introduce the family of geodesic curves\n between $a$ and $b$ by\n$$\\mathcal{G}(a,b):=\\bigg\\{\\g\\in {\\mathcal{C}}^{\\infty}({\\mathbb{R}};{\\mathcal{M}}):\\;\\g(t)=a\\text{ if } t\\geq 1\/2,\\, \\g(t)=b\\text{ if }\nt\\leq -1\/2\\,,\\;\\int_{\\mathbb{R}}|\\dot \\g|\\, dt=\\mathbf\nd_{\\mathcal{M}}(a,b)\\bigg\\}\\,,$$ where $\\mathbf d_{\\mathcal{M}}$ denotes the geodesic\ndistance on ${\\mathcal{M}}$. We define for $\\e>0$ and\n$\\nu=(\\nu_1,\\ldots,\\nu_N)$ an orthonormal basis of ${\\mathbb{R}}^N$,\n$$\\mathcal{B}_\\e(a,b,\\nu):=\\Big\\{u \\in W^{1,1}(Q_\\nu;{\\mathcal{M}})\\;:\\;u(x)=\\g(x_\\nu\/\\e)\n\\text{ on }\\partial Q_\\nu\\text{ for some }\n\\g\\in\\mathcal{G}(a,b)\\Big\\}\\,.$$\n\n\\begin{proposition}\\label{limsurf2}\nFor every $(a,b)\\in{\\mathcal{M}}\\times{\\mathcal{M}}$ and every orthonormal basis\n$\\nu=(\\nu_1,\\ldots,\\nu_N)$ of ${\\mathbb{R}}^N$, there exists the limit\n\\begin{equation}\\label{surfen2}\n\\tilde\\vartheta_{\\rm hom}(a,b,\\nu) := \\lim_{\\e\\to 0}\\,\\inf_u \\left\\{\n\\int_{Q_\\nu} f^\\infty\\left(\\frac{x}{\\e},\\nabla u\\right) dx : u \\in\n\\mathcal{B}_\\e(a,b,\\nu) \\right\\}\\,.\n\\end{equation}\nMoreover $\\tilde\\vartheta_{\\rm hom}(a,b,\\nu)$ only depends on $a$, $b$ and $\\nu_1$.\n\\end{proposition}\n\n\\begin{proof}\nThe proof follows the scheme of the one in\n\\cite[Proposition~2.2]{BDV}. We fix $a$ and $b\\in {\\mathcal{M}}$. For every\n$\\e>0$ and every orthonormal basis $\\nu=(\\nu_1,\\ldots,\\nu_N)$ of\n${\\mathbb{R}}^N$, we set\n$$ I_\\e(\\nu)= I_\\e(a,b,\\nu):= \\inf \\left\\{ \\int_{Q_\\nu}\nf^\\infty\\left(\\frac{x}{\\e},\\nabla u\\right) dx : u \\in\n\\mathcal{B}_\\e(a,b,\\nu) \\right\\}\\,.\n$$\nWe divide the proof into several steps.\\vskip5pt\n\n{\\bf Step 1.} Let $\\nu$ and $\\nu'$ be two orthonormal bases of\n${\\mathbb{R}}^N$ with equal first vector, {\\it i.e.}, $\\nu_1=\\nu'_1$. Suppose\nthat $\\nu$ is a rational basis, {\\it i.e.}, for all\n$i\\in\\{1,\\ldots,N\\}$ there exists $\\gamma_i\\in{\\mathbb{R}}\\setminus\\{0\\}$\nsuch that $v_i:=\\gamma_i\\nu_i\\in{\\mathbb{Z}}^N$. Similarly to Step 1 of the\nproof of \\cite[Proposition 2.2]{BDV}, we readily obtain that\n\\begin{equation}\\label{complimsupinf}\n\\limsup_{\\e \\to0}\\, I_\\e (\\nu')\\leq \\liminf_{\\e \\to0}\\, I_\\e(\\nu)\\,.\n\\end{equation}\n\n{\\bf Step 2.} Let $\\nu$ and $\\nu'$ be two orthonormal rational bases\nof ${\\mathbb{R}}^N$ with equal first vector. By Step~1 we immediately obtain\nthat the limits $\\displaystyle \\lim_{\\e\\to0}I_\\e(\\nu)$ and\n$\\displaystyle\\lim_{\\e\\to0}I_\\e(\\nu')$ exist and are equal.\n\n\\vskip5pt\n\n{\\bf Step 3.} We claim that for every $\\sigma>0$ there exists\n$\\delta>0$ (independent of $a$ and $b$) such that if $\\nu$ and\n$\\nu'$ are two orthonormal bases of ${\\mathbb{R}}^N$ with\n$|\\nu_i-\\nu'_i|<\\delta$ for every $i=1,\\ldots,N$, then\n$$\\liminf_{\\e\\to0} I_\\e(\\nu)-K\\sigma\\leq \\liminf_{\\e\\to0}I_\\e(\\nu')\\leq\\limsup_{\\e\\to0}I_\\e(\\nu')\\leq\n\\limsup_{\\e\\to0} I_\\e(\\nu)+K\\sigma$$ where $K$ is a positive\nconstant which only depends on ${\\mathcal{M}}$, $\\beta$ and $N$.\n\nWe use the notation $Q_{\\nu,\\eta}:=(1-\\eta)Q_\\nu$ where $0<\\eta<1$.\nLet $\\sigma>0$ be fixed and let $0<\\eta<1$ be such that\n\\begin{equation}\\label{condeta}\n\\eta<\\frac{1}{34}\\quad\\text{and}\\quad\\max\\bigg\\{1-(1-\\eta)^{N-1}\\,,\\;\n\\frac{(1-\\eta)^{N-1}(1-2\\eta)^{N-1}}{(1-3\\eta)^{N-1}}-(1-2\\eta)^{N-1}\\bigg\\}<\\sigma.\n\\end{equation}\nConsider $\\delta_0>0$ (that may be chosen so that $\\d_0 \\leq\n\\eta\/(2\\sqrt{N})$) such that for every $0<\\delta\\leq \\delta_0$ and\nevery pair $\\nu$ and $\\nu'$ of orthonormal basis of ${\\mathbb{R}}^N$\nsatisfying $|\\nu_i-\\nu'_i|\\leq \\delta$ for $i=1,\\ldots,N$, one has\n\\begin{equation}\\label{approxnu}\nQ_{\\nu,3\\eta}\\subset Q_{\\nu',2\\eta}\\subset Q_{\\nu,\\eta}\\,,\n\\end{equation}\nand $\\{x\\cdot\\nu_1'=0\\}\\cap \\partial Q_{\\nu,\\eta} \\subset \\{|x\\cdot\\nu_1|\\leq 1\/8\\}$.\n\nGiven $\\e>0$ small, we consider $u_\\e\\in\\mathcal{B}_{\\e}(a,b,\\nu')$\nsuch that\n$$\\int_{Q_{\\nu'}}f^\\infty\\left(\\frac{x}{\\e},\\nabla u_\\e\\right)dx\\leq I_{\\e}(\\nu')\n+\\sigma\\,,$$\n where $u_\\e(x)=\\g_\\e(x_{\\nu'}\/\\e)$ for $x\\in\\partial\nQ_{\\nu'}$. Now we construct\n$v_\\e\\in\\mathcal{B}_{(1-2\\eta)\\e}(a,b,\\nu)$ satisfying the boundary condition\n$v_\\e(x)=\\g_\\e\\big(x_\\nu\/(1-2\\eta)\\e\\big)$ for $x\\in\\partial\nQ_{\\nu}$. Consider $F_\\eta:{\\mathbb{R}}^N \\to {\\mathbb{R}}$,\n$$F_\\eta(x):=\\bigg(\\frac{1- 2\\|x'\\|_{\\nu,\\infty}}{\\eta}\\bigg)\\frac{x_{\\nu'}}{1-2\\eta}+\n\\bigg(\\frac{\\eta-1+\n2\\|x'\\|_{\\nu,\\infty}}{\\eta}\\bigg)\\frac{x_\\nu}{1-2\\eta}\\,,$$\nand define\n$$v_\\e(x):=\\begin{cases}\n\\displaystyle u_\\e\\bigg(\\frac{x}{1-2\\eta}\\bigg) & \\text{if }x\\in\nQ_{\\nu',2\\eta},\\\\[10pt]\n\\displaystyle \\g_\\e\\bigg(\\frac{x_{\\nu'}}{(1-2\\eta)\\e}\\bigg) &\\text{if }\nx\\in Q_{\\nu,\\eta}\\setminus Q_{\\nu',2\\eta}\\,,\\\\[8pt]\n\\displaystyle a & \\displaystyle\\text{if } x \\in Q_\\nu \\setminus Q_{\\nu,\\eta} \\text{ and }x_\\nu \\geq \\frac{1}{4}\\,,\\\\[8pt]\n\\displaystyle \\gamma_\\e\\bigg(\\frac{F_\\eta(x)}{\\e}\\bigg) & \\text{if }x \\in\nA_\\eta :=\\big\\{x: |x_\\nu|\\leq 1\/4\\big\\}\\cap(Q_\\nu\\setminus\nQ_{\\nu,\\eta})\\,,\\\\[8pt]\n\\displaystyle b & \\displaystyle\\text{if } x \\in Q_\\nu \\setminus Q_{\\nu,\\eta} \\text{ and\n}x_\\nu \\leq -\\frac{1}{4}\\,.\n\\end{cases}$$\nWe can check that $v_\\e$ is well defined for $\\e$ small enough\nand that $v_\\e\\in\\mathcal{B}_{(1-2\\eta)\\e}(a,b,\\nu)$. Therefore\n\\begin{align}\\label{Ebis}\nI_{(1-2\\eta)\\e}(\\nu)\\leq &\n\\int_{Q_\\nu}f^\\infty\\bigg(\\frac{x}{(1-2\\eta)\\e},\\nabla v_\\e\\bigg)\\,\ndx\\nonumber\\\\\n=&\\int_{Q_{\\nu',2\\eta}} f^\\infty\\bigg(\\frac{x}{(1-2\\eta)\\e},\\nabla\nv_\\e\\bigg)\\,dx+\\int_{Q_{\\nu,\\eta}\\setminus\nQ_{\\nu',2\\eta}}f^\\infty\\bigg(\\frac{x}{(1-2\\eta)\\e},\\nabla\nv_\\e\\bigg)\\, dx\\nonumber\\\\\n&+\\int_{A_{\\eta}}f^\\infty\\bigg(\\frac{x}{(1-2\\eta)\\e},\\nabla\nv_\\e\\bigg)\\, dx=:\\,I_1+I_2+I_3\\,.\n\\end{align}\nWe now estimate these three integrals. First, we easily get that\n\\begin{equation}\\label{E1bis}I_1= (1-2\\eta)^{N-1}\\int_{Q_{\\nu'}}f^\\infty\\left(\\frac{y}{\\e},\\nabla u_\\e\\right)\ndy\\leq I_\\e(\\nu')+\\sigma\\,. \\end{equation} In view of\n\\eqref{approxnu} we have $Q_{\\nu,\\eta}\\subset\n(1-\\eta)(1-2\\eta)(1-3\\eta)^{-1}Q_{\\nu'}=:D_{\\eta}$. Then we infer\nfrom the growth condition (\\ref{finfty1gc}) together with Fubini's\ntheorem that\n\\begin{eqnarray}\\label{E2bis}\nI_2 & \\leq & \\beta\\int_{D_\\eta\\setminus Q_{\\nu',2\\eta}}|\\nabla\nv_\\e|\\, dx = \\frac{\\beta}{(1-2\\eta)\\e} \\int_{(D_\\eta\\setminus\nQ_{\\nu',2\\eta})\\cap\\{|x_{\\nu'}|\\leq (1-2\\eta)\\e\/2\\}}\n\\bigg|\\, \\dot \\g_\\e\\bigg(\\frac{x_{\\nu'}}{(1-2\\eta)\\e}\\bigg)\\bigg|\\, dx\\nonumber\\\\\n&= & \\beta\\mathcal{H}^{N-1}\\big((D_\\eta\\setminus Q_{\\nu',2\\eta})\\cap\n\\{x_{\\nu'}=0\\}\\big)\\frac{1}{(1-2\\eta)\\e}\\int_{-(1-2\\eta)\\e\/2}^{(1-2\\eta)\\e\/2}\n\\bigg|\\, \\dot \\g_\\e\\bigg(\\frac{t}{(1-2\\eta)\\e}\\bigg)\\bigg|\\, dt\\nonumber\\\\\n&=& \\beta \\mathbf\nd_{\\mathcal{M}}(a,b)\\bigg(\\frac{(1-\\eta)^{N-1}(1-2\\eta)^{N-1}}{(1-3\\eta)^{N-1}}-(1-2\\eta)^{N-1}\\bigg)\\,.\n\\end{eqnarray}\nNow it remains to estimate $I_3$. To this purpose we first observe\nthat \\eqref{approxnu} yields \\begin{equation}\\label{Feta1} \\|\\nabla\nF_\\eta\\|_{L^\\infty(A_\\eta;{\\mathbb{R}}^N)}\\leq C\\,,\n\\end{equation}\nfor some absolute\nconstant $C>0$, and\n\\begin{equation}\\label{Feta2}|\\nabla\nF_\\eta(x)\\cdot \\nu_1|\\geq 1\\quad\\text{for a.e. $x\\in\nA_\\eta\\,$.}\n\\end{equation}\nHence, thanks the growth condition\n(\\ref{finfty1gc}), (\\ref{Feta1}) and (\\ref{Feta2}), we get that\n\\begin{multline*}\nI_3 \\leq \\beta\\int_{A_\\eta}|\\nabla v_\\e|\\, dx \\leq\n\\frac{C\\beta}{\\e} \\int_{A_\\eta}\\bigg|\\, \\dot\n\\g_\\e\\bigg(\\frac{F_\\eta(x)}{\\e}\\bigg)\\bigg |\\, dx \\leq \\frac{C \\beta}{\\e} \\int_{A_\\eta}\\bigg|\\, \\dot\n\\g_\\e\\bigg(\\frac{F_\\eta(x)}{\\e}\\bigg)\\bigg |\\, |\\nabla\nF_\\eta(x)\\cdot\\nu_1| \\,dx=\\\\\n=C\\beta\\int_{A'_\\eta}\\bigg(\\frac{1}{\\e}\\int_{-1\/4}^{1\/4}\\bigg|\\,\n\\dot \\g_\\e\\bigg(\\frac{F_\\eta(t\\nu_1+x')}{\\e}\\bigg)\\bigg |\\, |\\nabla\nF_\\eta(t\\nu_1+x')\\cdot\\nu_1|\\, dt\\bigg)\\, d\\mathcal{H}^{N-1}(x')\\,,\n\\end{multline*}\nwhere we have set $A'_\\eta:=A_\\eta\\cap\\{x_\\nu=0\\}$, and used Fubini's\ntheorem in the last equality. Changing variables\n$s=(1\/\\e)F_{\\eta}(t\\nu_1+x')$, we obtain that for\n$\\mathcal{H}^{N-1}$-a.e. $x' \\in A'_\\eta$,\n$$\\frac{1}{\\e}\\int_{-1\/4}^{1\/4}\\bigg|\\, \\dot \\g_\\e\\bigg(\\frac{F_\\eta(t\\nu_1+x')}{\\e}\\bigg)\\bigg|\\,\n|\\nabla F_\\eta(t\\nu_1+x')\\cdot\\nu_1|\\, dt\\leq \\int_{\\mathbb{R}} |\\dot\n\\g_\\e(s)|\\,ds=\\mathbf d_{\\mathcal{M}}(a,b)\\,.$$ Consequently,\n\\begin{equation}\\label{E3bis}I_3\\leq C\\beta\\, \\mathcal{H}^{N-1}(A'_\\eta)\\, \\mathbf\nd_{\\mathcal{M}}(a,b)=C\\beta \\big(1-(1-\\eta)^{N-1})\\, \\mathbf\nd_{\\mathcal{M}}(a,b)\\,.\\end{equation} In view of (\\ref{Ebis}), (\\ref{condeta})\nand estimates (\\ref{E1bis}), (\\ref{E2bis}) and (\\ref{E3bis}), we\nconclude that\n$$I_{(1-2\\eta)\\e}(\\nu)\\leq I_\\e(\\nu')+K\\sigma\\,,$$\nwhere $K=1+\\beta \\Delta(1+C)$, $\\Delta$ is the diameter of ${\\mathcal{M}}$ and\n$C$ is the constant given by (\\ref{Feta1}). Finally, letting\n$\\e\\to0$ we derive\n$$\\liminf_{\\e\\to 0}I_\\e(\\nu)\\leq \\liminf_{\\e\\to0}I_\\e(\\nu')+K\\sigma\\,, \\text{ and }\n \\limsup_{\\e\\to 0}I_\\e(\\nu)\\leq \\limsup_{\\e\\to0}I_\\e(\\nu')+K\\sigma\\,. $$\nThe symmetry of the roles of $\\nu$ and $\\nu'$ allows us to invert\nthem, thus concluding the proof of Step~3.\\vskip5pt\n\n{\\bf Step 4.} Let $\\nu$ and $\\nu'$ be two orthonormal bases of\n${\\mathbb{R}}^N$ with equal first vector. Similarly to Step~4 of the proof of\n\\cite[Proposition 2.2]{BDV}, by Steps 2 and 3 we readily obtain that\nthe limits $\\displaystyle\\lim_{\\e\\to0}I_\\e(\\nu)$ and\n$\\displaystyle\\lim_{\\e\\to0}I_\\e(\\nu')$ exist and are equal.\n\\end{proof}\n\n\\noindent{\\bf Proof of Proposition \\ref{limitsurfenerg}.}\nWe use the notation of the previous proof. Given $\\e>0$ and an\northonormal basis $\\nu=(\\nu_1,\\ldots,\\nu_N)$ of ${\\mathbb{R}}^N$, we set\n\\begin{align*}\nJ_\\e(\\nu)=J_\\e(a,b,\\nu) : = & \\inf \\left\\{ \\int_{Q_\\nu}\nf^\\infty\\left(\\frac{x}{\\e},\\nabla u\\right)\\, dx : u \\in\n\\mathcal{A}_1(a,b,\\nu) \\right\\} \\\\\n=&\\inf\\bigg\\{\\e^{N-1}\\int_{\\frac{1}{\\e}Q_\\nu}f^\\infty(y,\\nabla\n\\varphi)\\,dy\\;:\\;\\varphi\\in \\mathcal{A}_{1\/\\e}(a,b,\\nu)\n\\bigg\\}\\,.\\end{align*}\nWe claim that\n\\begin{equation}\\label{idsurfen}\n\\lim_{\\e\\to0} J_{\\e}(\\nu)=\\lim_{\\e\\to 0} I_\\e(\\nu)\\,.\n\\end{equation}\nFor $0<\\e<1$ we set $\\tilde \\e=\\e\/(1-\\e)$, and we consider $u_{\\tilde\n\\e}\\in\\mathcal{B}_{\\tilde \\e}(a,b,\\nu)$ satisfying\n$$\\int_{Q_\\nu}f^{\\infty}\\left(\\frac{x}{\\tilde\\e},\\nabla u_{\\tilde\\e}\\right)dx\\leq I_{\\tilde \\e}(\\nu)+ \\e\\,, $$\nwhere $u_{\\tilde \\e}(x)=\\g_{\\tilde \\e}(x_\\nu\/\\tilde \\e)$ if\n$x\\in\\partial Q_\\nu$, for some $\\g_{\\tilde \\e}\\in\\mathcal{G}(a,b)$.\nWe define for every $x\\in Q_\\nu$,\n$$v_\\e(x):=\\begin{cases}\n\\displaystyle u_{\\tilde \\e}\\left(\\frac{x}{1-\\e}\\right) & \\text{if $x\\in Q_{\\nu,\\e}\\,$,}\\\\[10pt]\n\\displaystyle \\g_{\\tilde\\e}\\bigg(\\frac{x_\\nu}{1-2\\|x'\\|_{\\nu,\\infty}}\\bigg)\n&\\text{otherwise}\\,.\n\\end{cases}$$\nOne may check that $v_\\e\\in\\mathcal{A}_1(a,b,\\nu)$, and hence\n$$J_\\e(\\nu)\\leq \\int_{Q_\\nu}f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right)dx=\\int_{Q_{\\nu,\\e}}f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right)dx\n+\\int_{Q_\\nu\\setminus Q_{\\nu,\\e}}f^\\infty\\left(\\frac{x}{\\e},\\nabla\nv_\\e\\right)dx=I_1+I_2\\,.$$\nWe now estimate these two integrals.\nFirst, we have\n\\begin{equation}\\label{2016}I_1=(1-\\e)^{N-1}\\int_{Q_\\nu}f^\\infty\\left(\\frac{y}{\\tilde\\e},\\nabla\nu_{\\tilde\\e}\\right)dy \\leq (1-\\e)^{N-1}\\big(I_{\\tilde\\e}(\\nu)+\\e\\big)\\,.\\end{equation}\nIn view of the growth condition (\\ref{finfty1gc}),\n\\begin{align*}\nI_2&\\leq \\beta\\int_{Q_\\nu\\setminus Q_{\\nu,\\e}}\\bigg|\\dot \\g_{\\tilde\n\\e}\\bigg(\\frac{x_\\nu}{1-2\\|x'\\|_{\\nu,\\infty}}\\bigg)\\bigg|\n\\bigg(\\frac{1}{1-2\\|x'\\|_{\\nu,\\infty}}+\\frac{|x_\\nu||\\nabla(\\|x'\\|_{\\nu,\\infty})|}{(1-2\\|x'\\|_{\\nu,\\infty})^2}\\bigg)\\,dx\\\\\n&\\leq 2\\beta\\int_{(Q_\\nu\\setminus Q_{\\nu,\\e})\\cap\\{|x_\\nu|\\leq\n(1-2\\|x'\\|_{\\nu,\\infty})\/2\\}}\\bigg|\\dot \\g_{\\tilde\n\\e}\\bigg(\\frac{x_\\nu}{1-2\\|x'\\|_{\\nu,\\infty}}\\bigg)\\bigg|\n\\bigg(\\frac{1}{1-2\\|x'\\|_{\\nu,\\infty}}\\bigg)\\,dx\\,,\n\\end{align*}\nwhere we have used the facts that $\\dot \\g_{\\tilde\n\\e}(x_\\nu\/(1-2\\|x'\\|_{\\nu,\\infty}))=0$ in the set $\\{|x_\\nu|>\n(1-2\\|x'\\|_\\infty)\/2\\}$ and\n$\\|\\nabla(\\|x'\\|_{\\nu,\\infty})\\|_{L^\\infty(Q_\\nu;{\\mathbb{R}}^N)}\\leq 1$.\nSetting $Q'_\\nu=Q_\\nu\\cap\\{x_\\nu=0\\}$ and\n$Q'_{\\nu,\\e}=Q_{\\nu,\\e}\\cap\\{x_\\nu=0\\}$, we infer from Fubini's\ntheorem that\n\\begin{multline}\\label{2017}\nI_2\\leq 2\\beta\\int_{Q'_\\nu\\setminus\nQ'_{\\nu,\\e}}\\bigg(\\int_{-(1-2\\|x'\\|_{\\nu,\\infty})\/2}^{(1-2\\|x'\\|_{\\nu,\\infty})\/2}\n\\bigg|\\dot \\g_{\\tilde\n\\e}\\bigg(\\frac{t}{1-2\\|x'\\|_{\\nu,\\infty}}\\bigg)\\bigg|\n\\bigg(\\frac{1}{1-2\\|x'\\|_{\\nu,\\infty}}\\bigg)\\, dt\\bigg)\\, d\\mathcal{H}^{N-1}(x')\\leq\\\\\n\\leq 2\\beta\\, \\mathcal{H}^{N-1}(Q'_\\nu\\setminus Q'_{\\nu,\\e})\\,\n\\mathbf d_{\\mathcal{M}}(a,b)\\leq 2\\beta \\mathbf d_{\\mathcal{M}}(a,b)\n\\big(1-(1-\\e)^{N-1}\\big)\\,.\n\\end{multline}\nIn view of the estimates (\\ref{2016}) and (\\ref{2017}) obtained for\n$I_1$ and $I_2$, we derive that\n\\begin{equation}\\label{2019}\n\\limsup_{\\e\\to0}J_\\e(\\nu)\\leq \\lim_{\\e\\to0}\nI_\\e(\\nu)\\,.\\end{equation}\n\nConversely, given $0<\\e<1$, we consider $\\tilde\nu_\\e\\in\\mathcal{A}_1(a,b,\\nu)$ such that\n$$\\int_{Q_\\nu}f^\\infty\\left(\\frac{x}{\\e},\\nabla\\tilde u_\\e\\right)dx\\leq J_\\e(\\nu)+\\e\\,,$$\nand $\\g\\in\\mathcal{G}(a,b)$ fixed. We define for $x\\in Q_\\nu$,\n$$w_\\e(x):=\\begin{cases}\n\\displaystyle \\tilde u_\\e\\left(\\frac{x}{1-\\e}\\right) &\\text{if $x\\in Q_{\\nu,\\e}$,}\\\\[10pt]\n\\displaystyle\n\\g\\bigg(\\frac{x_\\nu}{(1-\\e)(2\\|x'\\|_{\\nu,\\infty}-1+\\e)}\\bigg)&\\text{otherwise.}\n\\end{cases}$$\nWe can check that $w_\\e\\in\\mathcal{B}_{(1-\\e)\\e}(a,b,\\nu)$, and arguing as previously we infer that\n\\begin{eqnarray*}I_{(1-\\e)\\e}(\\nu)\n& \\leq & \\int_{Q_{\\nu,\\e}}\\!f^\\infty\\left(\\frac{x}{(1-\\e)\\e}\\,,\\nabla\nw_\\e\\right)\\, dx+ \\int_{Q_\\nu\\setminus\nQ_{\\nu,\\e}}f^\\infty\\left(\\frac{x}{(1-\\e)\\e}\\,,\\nabla\nw_\\e\\right)dx\\\\\n&\\leq & (1-\\e)^{N-1}\\big(J_\\e(\\nu)+\\e\\big) +2\\beta \\mathbf d_{\\mathcal{M}}(a,b)\\big(1-(1-\\e)^{N-1}\\big)\n\\,.\n\\end{eqnarray*}\nConsequently, $\\displaystyle\\lim_{\\e\\to0}I_\\e(\\nu)\\leq \\liminf_{\\e\\to0}J_\\e(\\nu)$, which, together with\n(\\ref{2019}), completes the proof of Proposition\n\\ref{limitsurfenerg}.\n\\prbox\n\\vskip5pt\n\nWe now state the\nfollowing properties of the surface energy density.\n\n\\begin{proposition}\\label{contsurfenerg}\nThe function $\\vartheta_\\text{hom}$ is continuous on ${\\mathcal{M}} \\times {\\mathcal{M}} \\times\n{\\mathbb{S}^{N-1}}$ and there exist constants $C_1>0$ and $C_2>0$ such that\n\\begin{equation}\\label{propsurf}\n|\\vartheta_{\\rm hom}(a_1,b_1,\\nu_1) - \\vartheta_{\\rm hom}(a_2,b_2,\\nu_1)| \\leq C_1\n(|a_1-a_2|+ |b_1-b_2|)\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{propsurf2}\n\\vartheta_{\\rm hom}(a_1,b_1,\\nu_1)\\leq C_2|a_1-b_1|\n\\end{equation}\nfor every $a_1,b_1,a_2,b_2\\in {\\mathcal{M}}$ and $\\nu_1 \\in {\\mathbb{S}^{N-1}}$.\n\\end{proposition}\n\n\\begin{proof}\nWe use the notation of the previous proof. By Proposition\n\\ref{limitsurfenerg} together with steps 3 and 4 of the proof of\nProposition \\ref{limsurf2}, we get that $\\vartheta_\\text{hom}(a,b,\\cdot)$\nis continuous on ${\\mathbb{S}^{N-1}}$ uniformly with respect to $a$ and $b$. Hence\nit is enough to show that (\\ref{propsurf}) holds to get the\ncontinuity of $\\vartheta_\\text{hom}$.\n\\vskip5pt\n\n{\\bf Step 1.} We start with the proof of \\eqref{propsurf}. Fix\n$\\nu_1 \\in {\\mathbb{S}^{N-1}}$ and let $\\nu=(\\nu_1,\\nu_2,\\ldots,\\nu_N)$ be any\northonormal basis of ${\\mathbb{R}}^N$.\nFor every $\\e>0$, let $\\tilde\n\\e:= \\e\/(1-\\e)$ and consider $\\g_{\\tilde \\e} \\in \\mathcal\nG(a_1,b_1)$ and $u_{\\tilde \\e} \\in \\mathcal B_{\\tilde\n\\e}(a_1,b_1,\\nu)$ such that $u_{\\tilde \\e}(x)=\\g_{\\tilde \\e}(x_\\nu\/\\tilde \\e)$ for\n$x\\in\\partial Q_\\nu$ and\n$$\\int_{Q_\\nu} f^\\infty\\left(\\frac{x}{\\tilde \\e},\\nabla u_{\\tilde\n\\e}\\right) dx \\leq I_{\\tilde \\e}(a_1,b_1,\\nu) + \\e\\,.$$\nWe shall now\ncarefully modify $u_{\\tilde \\e}$ in order to\nget another function $v_\\e \\in \\mathcal A_1(a_2,b_2,\\nu)$. We will\nproceed as in the proofs of Propositions \\ref{limitsurfenerg} and\n\\ref{limsurf2}. Let $\\g_a \\in \\mathcal G(a_2,a_1)$ and $\\g_b \\in\n\\mathcal G(b_2,b_1)$, and define\n$$v_\\e(x):=\\left\\{\n\\begin{array}{lll}\n\\displaystyle u_{\\tilde\\e}\\left(\\frac{x}{1-\\e}\\right) & \\text{ if } & x \\in\nQ_{\\nu,\\e}\\,,\\\\[0.3cm]\n\\displaystyle \\g_{\\tilde\\e}\\left(\\frac{x_\\nu}{1-2\\|x'\\|_{\\nu,\\infty}} \\right)\n& \\text{ if } & \\displaystyle x \\in A_1\\,,\\\\[0.3cm]\n\\displaystyle \\g_a\\left(\\frac{2\\|x\\|_{\\nu,\\infty}-1}{\\e}+\\frac{1}{2} \\right) &\n\\text{ if } & \\displaystyle x \\in A_2:=(Q_\\nu\\setminus\nQ_{\\nu,\\e})\\cap\\{x_\\nu\\geq \\e\/2\\}\\,,\\\\[0.3cm]\n\\displaystyle \\g_b\\left(\\frac{2\\|x\\|_{\\nu,\\infty}-1}{\\e}+\\frac{1}{2} \\right) &\n\\text{ if } & \\displaystyle x \\in A_3:=(Q_\\nu\\setminus Q_{\\nu,\\e})\\cap\\{x_\\nu\\leq -\\e\/2\\}\\,,\\\\[0.3cm]\n\\displaystyle \\g_a\\left(\\frac{2\\|x'\\|_{\\nu,\\infty}-1}{2x_\\nu}+\\frac{1}{2}\n\\right) & \\text{ if } & \\displaystyle x \\in A_4:=\\left\\{ 0 < x_\\nu \\leq\n\\frac{\\e}{2}\\,,\\;\n\\frac{1}{2}-x_\\nu \\leq \\|x'\\|_{\\nu,\\infty} < \\frac{1}{2}\\right\\}\\,,\\\\[0.3cm]\n\\displaystyle \\g_b\\left(\\frac{1-2\\|x'\\|_{\\nu,\\infty}}{2x_\\nu}+\\frac{1}{2}\n\\right) & \\text{ if } & \\displaystyle x \\in A_5:=\\left\\{ -\\frac{\\e}{2} < x_\\nu\n\\leq 0\\,,\\; \\frac{1}{2}+x_\\nu \\leq \\|x'\\|_{\\nu,\\infty} <\n\\frac{1}{2}\\right\\}\\,,\n\\end{array}\\right.$$\nwith\n$$ A_1:=\\left\\{ \\frac{1-\\e}{2} \\leq\n\\|x'\\|_{\\nu,\\infty} < \\frac{1}{2}\n\\text{ and } |x_\\nu|\\leq -\\|x'\\|_{\\nu,\\infty}+\\frac{1}{2}\\right\\}\\,.$$\nOne may check that the function $v_\\e$ has been constructed in such\na way that $v_\\e \\in {\\mathcal{A}}_1(a_2,b_2,\\nu)$, and thus\n\\begin{equation}\\label{Je}J_\\e(a_2,b_2,\\nu) \\leq\n\\int_{Q_\\nu}f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx\\,.\n\\end{equation}\nArguing exactly as in the proof of Proposition \\ref{limsurf2}, one\ncan show that\n\\begin{equation}\\label{cunu}\\int_{Q_{\\nu,\\e}}\nf^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx \\leq I_{\\tilde\n\\e}(a_1,b_1,\\nu) + \\e\\,,\\end{equation}\nand\n\\begin{eqnarray}\\label{A1}\\int_{A_1}\nf^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx \\leq C\\mathbf\nd_{\\mathcal{M}}(a_1,b_1)(1-(1-\\e)^{N-1}))\\,.\n\\end{eqnarray}\nNow we only estimate the integrals over $A_2$ and $A_4$, the ones\nover $A_3$ and $A_5$ being very similar. Define the Lipschitz\nfunction $F_\\e:{\\mathbb{R}}^{N} \\to {\\mathbb{R}}$ by\n$$F_\\e(x):=\\frac{2\\|x\\|_{\\nu,\\infty}-1}{\\e}+\\frac{1}{2}\\,.$$\nUsing the growth condition (\\ref{finfty1gc}) together with Fubini's\ntheorem, and the fact that $A_2 \\subset\nF_\\e^{-1}\\big([-1\/2,1\/2)\\big)$, we derive\n\\begin{multline*}\n\\int_{A_2} f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx \\leq\n\\b \\int_{A_2} |\\dot\\g_a (F_\\e(x))|\\, |\\nabla F_\\e(x)|\\, dx\n \\leq\\\\\n \\leq \\b \\int_{F_\\e^{-1}([-1\/2,1\/2))} |\\dot\\g_a\n(F_\\e(x))|\\, |\\nabla F_\\e(x)|\\, dx\n \\leq \\b \\int_{-1\/2}^{1\/2}|\\,\\dot \\g_a(t)| \\,\n{\\mathcal{H}}^{N-1}(F_\\e^{-1}\\{t\\})\\, dt\n\\,,\n\\end{multline*}\nwhere we used the Coarea\nformula in the last inequality.\nWe observe that for every $t\\in(-1\/2,1\/2)$, $F^{-1}_\\e\\{t\\}=\\partial Q_{\\nu,\\frac{\\e(1-2t)}{2}}$ so that\n${\\mathcal{H}}^{N-1}(F_\\e^{-1}\\{t\\})\\leq {\\mathcal{H}}^{N-1}(\\partial Q)$. Therefore\n\\begin{eqnarray}\\label{A2}\\int_{A_2}\nf^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx \\leq \\b\n{\\mathcal{H}}^{N-1}(\\partial Q)\\mathbf d_{\\mathcal{M}}(a_1,a_2)\\,.\\end{eqnarray} Define\nnow $G : {\\mathbb{R}}^N\\setminus\\{x_\\nu=0\\} \\to {\\mathbb{R}}$ by\n$$G(x):=\\frac{2\\|x'\\|_{\\nu,\\infty}-1}{2x_\\nu}+\\frac{1}{2}\\,.$$\nThe growth condition (\\ref{finfty1gc}) and Fubini's theorem yield\n\\begin{multline*}\n\\int_{A_4} f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx \\,\\leq \\\\\n\\leq \\b\n\\int_{0}^{\\e\/2} \\left(\\int_{G(\\cdot,x_\\nu)^{-1}([-1\/2,1\/2))}\n|\\dot\\g_a (G(x',x_\\nu))|\\, |\\nabla G(x',x_\\nu)|\\, d{\\mathcal{H}}^{N-1}(x')\n\\right)dx_\\nu\\,.\n\\end{multline*}\nAs $|\\nabla_{x'} G(x)| = 1\/x_\\nu$ and\n$|\\nabla_{x_\\nu} G(x)| \\leq 1\/x_\\nu$ for a.e. $x \\in A_4$, it\nfollows that $|\\nabla G(x)| \\leq 2 |\\nabla_{x'} G(x)|$ for a.e. $x\n\\in A_4$. Hence\n\\begin{multline*}\n\\int_{A_4} f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx \\,\\leq \\\\\n\\leq 2\\b\n\\int_{0}^{\\e\/2} \\left(\\int_{G(\\cdot,x_\\nu)^{-1}([-1\/2,1\/2))}\n|\\dot\\g_a (G(x',x_\\nu))|\\, |\\nabla_{x'} G(x',x_\\nu)|\\,\nd{\\mathcal{H}}^{N-1}(x') \\right)dx_\\nu\\,.\n\\end{multline*}\nFor every $x_\\nu \\in (0,\\e\/2)$ the\nfunction $G(\\cdot,x_\\nu):{\\mathbb{R}}^{N-1} \\to {\\mathbb{R}}$ is Lipschitz, and thus the\nCoarea formula implies\n\\begin{eqnarray}\\label{A4}\n\\int_{A_4} f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx & \\leq &\n2\\b \\int_0^{\\e\/2} \\left( \\int_{-1\/2}^{1\/2}|\\, \\dot\\g_a(t)|\\,\n{\\mathcal{H}}^{N-2}(\\{x': G(x',x_\\nu)=t\\})\\, dt \\right)dx_\\nu\\nonumber\\\\\n&\\leq & C\\e\\, \\mathbf d_{\\mathcal{M}}(a_1,a_2)\\,,\n\\end{eqnarray}\nwhere we used as previously the estimate ${\\mathcal{H}}^{N-2}(\\{x': G(x',x_\\nu)=t\\})\\leq {\\mathcal{H}}^{N-2}\\big(\\partial (\\frac{-1}{2},\\frac{1}{2})^{N-1}\\big)$.\nGathering (\\ref{Je}) to (\\ref{A4}) and considering the analogous estimates\nfor the integrals over $A_3$ and $A_5$ (with $b_1$ and $b_2$ instead\nof $a_1$ and $a_2$), we infer that\n$$J_\\e(a_2,b_2,\\nu)\n\\leq \\int_{Q_\\nu}f^\\infty\\left(\\frac{x}{\\e},\\nabla v_\\e\\right) dx\n\\leq I_{\\tilde\\e}(a_1,b_1,\\nu) + C\\big(\\e + \\mathbf d_{\\mathcal{M}}(a_1,a_2) +\n\\mathbf d_{\\mathcal{M}}(b_1,b_2)\\big)\\,.$$ Taking the limit as $\\e \\to 0$, we\nget in light of Propositions \\ref{limitsurfenerg} and \\ref{limsurf2}\nthat\n$$\\vartheta_\\text{hom}(a_2,b_2,\\nu) \\leq \\vartheta_\\text{hom}(a_1,b_1,\\nu) + C \\big( \\mathbf\nd_{\\mathcal{M}}(b_1,b_2) + \\mathbf d_{\\mathcal{M}}(a_1,a_2) \\big)\\,.$$ Since the geodesic\ndistance on ${\\mathcal{M}}$ is equivalent to the Euclidian distance, we\nconclude, possibly exchanging the roles of $(a_1,b_1)$ and\n$(a_2,b_2)$, that (\\ref{propsurf}) holds. \\vskip5pt\n\n{\\bf Step 2.} We now prove \\eqref{propsurf2}. Given an arbitrary orthonormal basis $\\nu=(\\nu_1,\\ldots,\\nu_N)$ of ${\\mathbb{R}}^N$, let $\\g \\in\n\\mathcal G(a_1,b_1)$ and define $u_\\e(x):=\\g(x_\\nu\/\\e)$. Obviously\n$u_\\e \\in \\mathcal B_\\e(a_1,b_1,\\nu)$. Using (\\ref{idsurfen})\ntogether with the growth condition (\\ref{finfty1gc}) satisfied by\n$f^\\infty$, we derive that\n$$\\vartheta_\\text{hom}(a_1,b_1,\\nu_1) \\leq \\liminf_{\\e \\to\n0}\\int_{Q_\\nu}f^\\infty\\left(\\frac{x}{\\e},\\nabla u_\\e\\right)dx \\leq\n\\liminf_{\\e \\to 0} \\frac{\\b}{\\e}\\int_{Q_\\nu}\n\\left|\\dot\\g\\left(\\frac{x\\cdot\\nu_1}{\\e}\\right)\\right|\\,\ndx=\\b\\mathbf d_{{\\mathcal{M}}}(a_1,b_1)\\,.$$ Then (\\ref{propsurf2}) follows\nfrom the equivalence between $\\mathbf d_{{\\mathcal{M}}}$ and the Euclidian\ndistance.\n\\end{proof}\n\n\n\\section{Localization and integral repersentation on partitions}\\label{BV}\n\nIn this section we first show that the $\\Gamma$-limit defines a measure. Then we prove an abstract representation on\npartitions in sets of finite perimeter. This two facts will allow us to obtain the upper bound on the $\\Gamma$-limit in the next section.\n\n\\subsection{Localization}\n\nWe consider an arbitrary given sequence\n$\\{\\e_n\\} \\searrow 0^+$ and we localize the functionals\n$\\{{\\mathcal{F}}_{\\e_n}\\}_{n\\in{\\mathbb{N}}}$ on the family ${\\mathcal{A}}(\\O)$, {\\it i.e.}, for\nevery $u \\in L^1(\\O;{\\mathbb{R}}^d)$ and every $A \\in {\\mathcal{A}}(\\O)$, we set\n$${\\mathcal{F}}_{\\e_n}(u,A):= \\begin{cases}\n\\displaystyle \\int_A\nf\\left(\\frac{x}{\\e_n},\\nabla u\\right) dx & \\text{if }u \\in\nW^{1,1}(A;{\\mathcal{M}})\\,,\\\\[8pt]\n+\\infty & \\text{otherwise}\\,.\n\\end{cases}$$\nNext we define for $u\\in L^1(\\O;{\\mathbb{R}}^d)$ and $A \\in {\\mathcal{A}}(\\O)$,\n\\begin{equation*}\n{\\mathcal{F}}(u,A):= \\inf_{\\{u_n\\}} \\bigg\\{ \\liminf_{n \\to +\\infty}\\, {\\mathcal{F}}_{\\e_n}\n(u_n,A)\\, :\\, u_n \\to u \\text{ in }L^1(A;{\\mathbb{R}}^d) \\bigg\\}\\,.\n\\end{equation*}\nNote that ${\\mathcal{F}}(u,\\cdot)$ is an increasing set function for every\n$u\\in L^1(\\O;{\\mathbb{R}}^d)$ and that ${\\mathcal{F}}(\\cdot,A)$ is lower semicontinuous\nwith respect to the strong $L^1(A;{\\mathbb{R}}^d)$-convergence for every\n$A\\in {\\mathcal{A}}(\\O)$.\n\nSince $L^1(A;{\\mathbb{R}}^d)$ is separable, \\cite[Theorem~8.5]{DM} and a\ndiagonalization argument bring the existence of a subsequence (still\ndenoted $\\{\\e_n\\}$) such that ${\\mathcal{F}}(\\cdot,A)$ is the $\\G$-limit of\n${\\mathcal{F}}_{\\e_n}(\\cdot,A)$ for the strong $L^1(A;{\\mathbb{R}}^d)$-topology for\nevery $A \\in {\\mathcal{R}}(\\O)$ (or $A=\\O$). \\vskip5pt\n\nWe have the following locality property of the\n$\\G$-limit which, in the $BV$ setting, parallels \\cite[Lemma 3.1]{BM}.\n\n\\begin{lemma}\\label{measbis}\nFor every $u \\in BV(\\O;{\\mathcal{M}})$, the set function ${\\mathcal{F}}(u,\\cdot)$ is the\nrestriction to ${\\mathcal{A}}(\\O)$ of a Radon measure absolutely continuous\nwith respect to ${\\mathcal{L}}^N+|Du|$.\n\\end{lemma}\n\n\\begin{proof}\nLet $u \\in BV(\\O;{\\mathcal{M}})$ and $A \\in {\\mathcal{A}}(\\O)$. By Theorem 3.9 in\n\\cite{AFP}, there exists a sequence $\\{u_n\\} \\subset\nW^{1,1}(A;{\\mathbb{R}}^d) \\cap {\\mathcal{C}}^\\infty(A;{\\mathbb{R}}^d)$ such that $u_n \\to u$ in\n$L^1(A;{\\mathbb{R}}^d)$ and $\\int_A |\\nabla u_n|\\, dx \\to |Du|(A)$. Moreover,\n$u_n(x)\\in {\\rm co}({\\mathcal{M}})$ for a.e. $x \\in A$ and every $n \\in {\\mathbb{N}}$.\nApplying Proposition \\ref{proj} to $u_n$, we obtain a new sequence\n$\\{w_n\\} \\subset W^{1,1}(A;{\\mathcal{M}})$ satisfying\n$$\\int_A |\\nabla w_n|\\, dx \\leq C_\\star \\int_A |\\nabla u_n|\\, dx,$$ for\nsome constant $C_\\star>0$ depending only on ${\\mathcal{M}}$ and $d$. From\nconstruction of $w_n$, we have that $w_n \\to u$ in $L^1(A;{\\mathbb{R}}^d)$.\nTaking $\\{w_n\\}$ as admissible sequence, we deduce in light of the\ngrowth condition $(H_2)$ that\n\\begin{equation*}\n{\\mathcal{F}}(u,A) \\leq \\beta\\big({\\mathcal{L}}^N(A)+C_\\star|Du|(A)\\big)\\,.\n\\end{equation*}\n\nWe now prove that\n\\begin{equation*}\n{\\mathcal{F}}(u,A) \\leq {\\mathcal{F}}(u,B) + {\\mathcal{F}}(u,A \\setminus \\overline C)\n\\end{equation*}\nfor every $A$, $B$ and $C \\in {\\mathcal{A}}(\\O)$ satisfying $\\overline C\n\\subset B \\subset A$. Then the measure property of\n${\\mathcal{F}}(u,\\cdot)$ can be obtained as in the proof of \\cite[Lemma 3.1]{BM} with\nminor modifications. For this reason, we shall omit it.\n\nLet $R \\in {\\mathcal{R}}(\\O)$ such that $C \\subset\\subset R \\subset\\subset B$\nand consider $\\{u_n\\} \\subset W^{1,1}(R;{\\mathcal{M}})$ satisfying $u_n \\to u$\nin $L^1(R;{\\mathbb{R}}^d)$ and\n\\begin{equation}\\label{u_nbis}\n\\lim_{n \\to +\\infty} {\\mathcal{F}}_{\\e_n}(u_n,R) = {\\mathcal{F}}(u,R)\\,.\n\\end{equation}\nGiven $\\eta>0$ arbitrary, there exists a sequence $\\{v_n\\} \\subset\nW^{1,1}(A \\setminus \\overline C;{\\mathcal{M}})$ such that $v_n \\to u$ in\n$L^1(A\\setminus \\overline C;{\\mathbb{R}}^d)$ and\n\\begin{equation}\\label{v_nbis}\n\\liminf_{n \\to +\\infty}\\, {\\mathcal{F}}_{\\e_n}(v_n,A \\setminus \\overline C) \\leq\n{\\mathcal{F}}(u,A \\setminus \\overline C) + \\eta\\,.\n\\end{equation}\nBy Theorem \\ref{density}, we can assume without loss of generality\nthat $u_n \\in {\\mathcal{D}}(R;{\\mathcal{M}})$ and $v_n \\in {\\mathcal{D}}(A \\setminus \\overline\nC;{\\mathcal{M}})$. Let $L:=\\text{dist}(C,\\partial R)$ and define for every $i \\in\n\\{0,\\ldots,n\\}$,\n$$R_i:=\\bigg\\{x \\in R:\\, \\text{dist}(x,\\partial R)\n>\\frac{iL}{n}\\bigg\\}\\,.$$\nGiven $i \\in \\{0,\\ldots,n-1\\}$, let $S_i:=R_i \\setminus\n\\overline{R_{i+1}}$ and consider a cut-off function $\\zeta_i \\in\n{\\mathcal{C}}^\\infty_c(\\O;[0,1])$ satisfying $\\zeta_i(x) = 1$ for $x\\in\nR_{i+1}$, $\\zeta_i(x) = 0$ for $x\\in \\O\\setminus R_{i}$ and $|\\nabla\n\\zeta_i|\\leq 2n\/L$. Define\n$$z_{n,i}:=\\zeta_i u_n + (1-\\zeta_i)v_n \\in W^{1,1}(A;{\\mathbb{R}}^d)\\,.$$\nIf $\\pi_1({\\mathcal{M}}) \\neq 0$, $z_{n,i}$ is smooth in $A\\setminus\n\\Sigma_{n,i}$ with $\\Sigma_{n,i} \\in \\mathcal S$, while $z_{n,i}$ is\nsmooth in $A$ if $\\pi_1({\\mathcal{M}}) = 0$. Observe that $z_{n,i}(x) \\in {\\rm\nco}({\\mathcal{M}})$ for a.e. $x \\in A$ and actually, $z_{n,i}$ fails to be\n${\\mathcal{M}}$-valued exactly in the set $S_i$. To get an admissible sequence,\nwe project $z_{n,i}$ on ${\\mathcal{M}}$ using Proposition \\ref{proj}. It yields\na sequence $\\{w_{n,i}\\} \\subset W^{1,1}(A;{\\mathcal{M}})$ satisfying $w_{n,i}\n=z_{n,i}$ a.e. in $A \\setminus S_i$,\n\\begin{equation}\\label{L1}\n\\int_A|w_{n,i} -u |\\, dx \\leq \\int_A |z_{n,i} - u|\\, dx+C\n{\\mathcal{L}}^N(S_i)\\,,\n\\end{equation}\nfor some constant $C>0$ depending only on the diameter of ${\\rm\nco}({\\mathcal{M}})$, and\n$$\\int_{S_i} |\\nabla w_{n,i}|\\, dx \\leq C_\\star \\int_{S_i} |\\nabla z_{n,i}|\\, dx\n \\leq C_\\star \\int_{S_i}\\left(|\\nabla u_n| + |\\nabla v_n| +\n\\frac{n}{2L} |u_n - v_n|\\right)\\, dx\\,.$$ Arguing exactly as in the\nproof of \\cite[Lemma 3.1]{BM}, we now find an index $i_n \\in\n\\{0,\\ldots,n-1\\}$ such that\n\\begin{multline}\\label{nin}\n{\\mathcal{F}}_{\\e_n}(w_{n,i_n},A) \\leq\n{\\mathcal{F}}_{\\e_n}(u_n,R)+{\\mathcal{F}}_{\\e_n}(v_n,A \\setminus \\overline C)\\,+\\\\\n+ C_0 \\int_{R \\setminus \\overline C}|u_n - v_n|\\, dx +\n\\frac{C_0}{n}\\sup_{k \\in {\\mathbb{N}}}\\int_{R \\setminus \\overline C}(1+|\\nabla\nu_k|+ |\\nabla v_k|)\\, dx\\,,\n\\end{multline}\nfor some constant $C_0$ independent of $n$.\n\nA well\nknown consequence of the Coarea formula yields (see, {\\it e.g.},\n\\cite[Lemma 3.2.34]{Federer}),\n\\begin{equation}\\label{vanslice}\n{\\mathcal{L}}^N(S_{i_n}) = \\int_{i_n\nL\/n}^{(i_n+1)L\/n} {\\mathcal{H}}^{N-1}(\\{x \\in R: \\text{dist}(x,\\partial R)=t\\})\\, dt \\to 0 \\quad\\text{as $n \\to\n+\\infty$\\,.}\n\\end{equation}\nAs a consequence of (\\ref{L1}) and \\eqref{vanslice}, $w_{n,i_n}\n\\to u$ in $L^1(A;{\\mathbb{R}}^d)$. Taking the $\\liminf$ in\n(\\ref{nin}) and using (\\ref{u_nbis}) together with (\\ref{v_nbis}), we derive\n$${\\mathcal{F}}(u,A) \\leq {\\mathcal{F}}(u,R) + {\\mathcal{F}}(u,A \\setminus \\overline C)+\\eta \\leq {\\mathcal{F}}(u,B) + {\\mathcal{F}}(u,A \\setminus \\overline C)+\\eta\\,.$$\nThe conclusion follows from the arbitrariness of $\\eta$.\n\\end{proof}\n\n\\begin{remark}\\label{measconstr}\nIn view of Lemma \\ref{measbis}, for every $u\\in BV(\\O;{\\mathcal{M}})$, the set\nfunction ${\\mathcal{F}}(u,\\cdot)$ can be uniquely extended to a Radon measure\non $\\O$. Such a measure is given by\n$${\\mathcal{F}}(u,B):= \\inf \\big\\{{\\mathcal{F}}(u,A)\\,:\\,A\\in{\\mathcal{A}}(\\O),\\,B\\subset A\\big\\}\\,, $$\nfor every $B\\in\\mathcal{B}(\\O)$ (see, \\emph{e.g.}, \\cite[Theorem\n1.53]{AFP}).\n\\end{remark}\n\n\n\n\\subsection{Integral representation on partitions}\n\n\n\nBesides the locality of ${\\mathcal{F}}(u,\\cdot)$, another key point of the\nanalysis is to prove an abstract integral representation on\npartitions. Similarly to {\\it e.g.} \\cite[Lemma 3.7]{BDV}, using\n$(H_1)$ we easily obtain the translation invariance property of the\n$\\G$-limit, the proof of which is omitted.\n\n\\begin{lemma}\\label{invtrans}\nFor every $u \\in BV(\\O;{\\mathcal{M}})$, every $A \\in {\\mathcal{A}}(\\O)$ and every\n$y\\in{\\mathbb{R}}^N$ such that $y+A \\subset \\O$, we have\n$${\\mathcal{F}}(\\tau_yu,y+A)={\\mathcal{F}}(u,A) \\,,$$\nwhere $(\\tau_yu)(x):=u(x-y)$.\n\\end{lemma}\n\nWe are now in position to prove the integral representation of\nthe $\\G$-limit on partitions. \n\n\\begin{proposition}\\label{reppart}\nThere exists a unique function $K:{\\mathcal{M}}\\times{\\mathcal{M}}\\times{\\mathbb{S}^{N-1}}\\to[0,+\\infty)$\ncontinuous in the last variable and such that\n\\newline\n\\emph{(i)} $K(a,b,\\nu)=K(b,a,-\\nu)$ for every\n$(a,b,\\nu)\\in{\\mathcal{M}}\\times{\\mathcal{M}}\\times{\\mathbb{S}^{N-1}}$,\n\\newline\n\\emph{(ii)} for every finite subset $T$ of ${\\mathcal{M}}$,\n\\begin{equation}\\label{intsurfbor}\n{\\mathcal{F}}(u,S)=\\int_{S} K(u^+,u^-,\\nu_u)\\, d{\\mathcal{H}}^{N-1}\\,,\n\\end{equation}\nfor every $u\\in BV(\\O;T)$ and every Borel subset $S$ of $\\O\\cap S_u\\,$.\n\\end{proposition}\n\n\\begin{proof}\nIt follows the argument of \\cite[Proposition 4.2]{BDV} that is based\non the general result \\cite[Theorem 3.1]{AB}, on account to Lemmas\n\\ref{measbis}, \\ref{invtrans} and Remark \\ref{measconstr}. We omit\nany further details.\n\\end{proof}\n\n\n\\section{The upper bound}\n\n\n\n\\noindent We now adress the $\\G$-$\\limsup$ inequality. The upper\nbound on the diffuse part will be obtained using an extension of the\nrelaxation result of \\cite{AEL} (see Theorem \\ref{relax} in the\nAppendix) together with the partial representation of the $\\G$-limit\nalready established in $W^{1,1}$ (see Theorem~\\ref{babmilp=1}). The\nestimate of the jump part relies on the integral representation on\npartitions in sets of finite perimeter stated in Proposition\n\\ref{reppart}. \\vskip5pt\n\nIn view of the measure property of the $\\G$-limit, we may write for\nevery $u\\in BV(\\O;{\\mathcal{M}})$,\n\\begin{equation}\\label{decompupbd}\n{\\mathcal{F}}(u,\\O)={\\mathcal{F}}(u,\\O\\setminus S_u)+{\\mathcal{F}}(u,\\O\\cap S_u)\\,.\n\\end{equation}\nHence the desired upper bound ${\\mathcal{F}}(u,\\O)\\leq {\\mathcal{F}}_{\\rm hom}(u)$ will follow estimating separately the two terms in the right handside of \\eqref{decompupbd}.\n\n\\begin{lemma}\\label{upperboundBV}\nFor every $u \\in BV(\\O;{\\mathcal{M}})$, we have\n$${\\mathcal{F}}(u,\\O\\setminus S_u) \\leq \\int_\\O Tf_{\\rm hom}(u,\\nabla u)\\,\ndx+ \\int_\\O Tf^\\infty_{\\rm hom}\\left(\\tilde\nu,\\frac{dD^cu}{d|D^cu|}\\right)\\, d|D^cu|\\,.$$\n\\end{lemma}\n\n\\begin{proof}\nLet $A \\in {\\mathcal{A}}(\\O)$ and $\\{u_n\\} \\subset W^{1,1}(A;{\\mathcal{M}})$ be such that\n$u_n \\to u$ in $L^1(A;{\\mathbb{R}}^d)$. Since ${\\mathcal{F}}(\\cdot,A)$ is sequentially\nlower semicontinuous for the strong $L^1(A;{\\mathbb{R}}^d)$ convergence, it\nfollows from Theorem~\\ref{babmilp=1} that\n$${\\mathcal{F}}(u,A) \\leq \\liminf_{n \\to +\\infty}\\, {\\mathcal{F}}(u_n,A)=\\liminf_{n \\to +\\infty}\\, \\int_A Tf_\\text{hom}(u_n,\\nabla u_n)\\, dx\\,.$$\nSince the sequence $\\{u_n\\}$ is arbitrary, we deduce\n$${\\mathcal{F}}(u,A) \\leq \\inf\\left\\{\\liminf_{n \\to +\\infty} \\int_A Tf_\\text{hom}(u_n,\\nabla u_n)\\, dx : \\{u_n\\} \\subset W^{1,1}(A;{\\mathcal{M}}),\n\\, u_n \\to u\\text{ in }L^1(A;{\\mathbb{R}}^d)\\right\\}.$$ According to\nProposition \\ref{properties1}, the energy density $Tf_\\text{hom}$ is a\ncontinuous and tangentially quasiconvex function which fulfills the\nassumptions of Theorem \\ref{relax}. Hence\n\\begin{equation}\\label{FUA}\n{\\mathcal{F}}(u,A) \\leq \\int_A Tf_\\text{hom}(u,\\nabla u)\\,\ndx+ \\int_A Tf^\\infty_\\text{hom}\\left(\\tilde u,\\frac{dD^cu}{d|D^cu|}\\right)\nd|D^cu| + \\int_{S_u \\cap A} H(u^+,u^-,\\nu_u)\\, d{\\mathcal{H}}^{N-1}\n\\end{equation}\nfor some function $H:{\\mathcal{M}} \\times {\\mathcal{M}} \\times {\\mathbb{S}^{N-1}} \\to [0,+\\infty)$. By\nouter regularity, (\\ref{FUA}) holds for every $A \\in \\mathcal\nB(\\O)$. Taking $A=\\O \\setminus S_u$, we obtain\n$${\\mathcal{F}}(u,\\O \\setminus S_u) \\leq \\int_\\O Tf_\\text{hom}(u,\\nabla u)\\,\ndx+ \\int_\\O Tf^\\infty_\\text{hom}\\left(\\tilde\nu,\\frac{dD^cu}{d|D^cu|}\\right)\\, d|D^cu|\\,,\n$$ and the proof is complete.\n\\end{proof}\n\nTo prove the upper bound of the jump part, we first need to compare the energy density $K$ obtained in Proposition \\ref{reppart}\nwith the expected density $\\vartheta_\\text{hom}$.\n\n\\begin{lemma}\\label{upbdsurf}\nWe have $K(a,b,\\nu_1)\\leq \\vartheta_{\\rm hom}(a,b,\\nu_1)$ for every\n$(a,b,\\nu_1)\\in {\\mathcal{M}}\\times{\\mathcal{M}}\\times\\mathbb{S}^{N-1}$.\n\\end{lemma}\n\n\\begin{proof}\nWe will partially proceed as in the proof of Proposition\n\\ref{limsurf2} and we refer to it for the notation. Consider\n$\\nu=(\\nu_1,\\ldots,\\nu_N)$ an orthonormal basis of ${\\mathbb{R}}^N$. We shall\nprove that $K(a,b,\\nu_1)\\leq \\vartheta_{\\rm hom}(a,b,\\nu_1)$. Since\n$K$ and $\\vartheta_{\\rm hom}$ are continuous in the last variable,\nwe may assume that $\\nu$ is a rational basis, {\\it i.e.}, for all $i\n\\in \\{1,\\ldots,N\\}$, there exists $\\g_i \\in {\\mathbb{R}}\\setminus \\{0\\}$ such\nthat $v_i:= \\g_i \\nu_i \\in {\\mathbb{Z}}^N$, and the general case follows by\ndensity. \\vskip5pt\n\nGiven $0<\\eta<1$ arbitrary, by Proposition \\ref{limitsurfenerg} and\n\\eqref{idsurfen} we can find $\\e_0>0$,\n$u_0\\in\\mathcal{B}_{\\e_0}(a,b,\\nu)$ and\n$\\gamma_{\\e_0}\\in\\mathcal{G}(a,b)$ such that\n$u_0(x)=\\gamma_{\\e_0}(x\\cdot\\nu_1\/\\e_0)$ and\n$$\\int_{Q_{\\nu}} f^{\\infty}\\bigg(\\frac{x}{\\e_0},\\nabla u_0\\bigg)\\, dx\\leq \\vartheta_{\\rm hom}(a,b,\\nu_1)+\\eta\\,.$$\nFor every $\\lambda=(\\lambda_2,\\ldots,\\lambda_N)\\in{\\mathbb{Z}}^{N-1}$, we set\n$x_n^{(\\lambda)}:=\\e_n\\sum_{i=2}^N\\lambda_iv_i$ and\n$Q_{\\nu,n}^{(\\lambda)}:=x^{(\\lambda)}_n+(\\e_n\/\\e_0)Q_\\nu$. We define\nthe set $\\Lambda_n$ by\n\\begin{multline*}\n\\Lambda_n\n:=\\Bigg\\{\\lambda\\in{\\mathbb{Z}}^{N-1}\\;:\\;Q_{\\nu,n}^{(\\lambda)}\\subset\nQ_{\\nu} \\text{ and } x_n^{(\\lambda)}\\in \\sum_{i=2}^N\nl_i\\left(\\frac{\\e_n}{\\e_0}+\\e_n \\gamma_i\\right)\\nu_i+\\e_n P\\\\\n\\text{ for some }(l_2,\\ldots,l_N)\\in{\\mathbb{Z}}^{N-1}\\Bigg\\}\\,,\n\\end{multline*}\nwhere $$P:=\\Big\\{\\a_2v_2 + \\ldots + \\a_N v_N : \\, \\a_2,\\ldots,\\a_N\n\\in [-1\/2,1\/2)\\Big\\}.$$ Next consider\n$$u_n(x)=\\begin{cases}\n\\displaystyle u_0\\bigg(\\frac{\\e_0(x-x_n^{(\\lambda)})\\cdot\\nu_1}{\\e_n}\\bigg) & \\text{if $x\\in Q_{\\nu,n}^{(\\lambda)}$ for some $\\lambda\\in\\Lambda_n$}\\,,\\\\[10pt]\n\\displaystyle \\gamma_{\\e_0}\\bigg(\\frac{x\\cdot\\nu_1}{\\e_n}\\bigg) & \\text{otherwise}\\,.\n\\end{cases}$$\nNote that $u_n\\in W^{1,1}(Q_\\nu;{\\mathcal{M}})$, $\\{\\nabla u_n\\}$ is bounded in\n$L^1(Q_\\nu;{\\mathbb{R}}^{d \\times N})$, and $u_n\\to u^{a,b}_{\\nu_1}$ in\n$L^1(Q_\\nu;{\\mathbb{R}}^d)$ as $n\\to+\\infty$ with $u^{a,b}_{\\nu_1}$ given\nby \\begin{equation*}\nu_{\\nu_1}^{a,b}(x):=\\begin{cases}\na & \\text{if } x\\cdot\\nu_1 \\geq 0\\,, \\\\\nb & \\text{if } x\\cdot\\nu_1 <0\\,,\n\\end{cases}\n\\quad \\Pi_{\\nu_1}:=\\big\\{x\\in{\\mathbb{R}}^N : x\\cdot\\nu_1=0\\big\\}\\,.\n\\end{equation*}\nArguing as in Step 1 of the proof of \\cite[Proposition 2.2]{BDV}, we\nobtain that \n\\begin{equation}\\label{pasidee1}\n\\limsup_{n\\to+\\infty}\\,\n\\int_{Q_\\nu}f^\\infty\\bigg(\\frac{x}{\\e_n},\\nabla u_n\\bigg)\\, dx\\leq\n\\int_{Q_\\nu} f^\\infty\\bigg(\\frac{x}{\\e_0},\\nabla u_0\\bigg)\\, dx \\leq\n\\vartheta_{\\rm hom}(a,b,\\nu_1)+\\eta\\,.\n\\end{equation}\nFor $\\rho>0$ define $A_\\rho:=Q_\\nu\\cap\\{|x\\cdot\\nu_1|<\\rho\\}$. By\nconstruction the sequence $\\{u_n\\}$ is admissible for\n${\\mathcal{F}}(u^{a,b}_{\\nu_1},A_\\eta)$ so that\n\\begin{multline}\\label{pasidee1b}\n{\\mathcal{F}}\\big(u^{a,b}_{\\nu_1},A_\\eta\\cap\\Pi_{\\nu_1}\\big)\\leq{\\mathcal{F}}(u^{a,b}_{\\nu_1},A_\\eta)\\leq\n\\liminf_{n\\to+\\infty} \\,\\int_{A_\\eta}f\\bigg(\\frac{x}{\\e_n},\\nabla u_n\\bigg)\\, dx\\leq \\\\\n\\leq\\beta\n\\mathcal{L}^N(A_\\eta)+\\liminf_{n\\to+\\infty}\\,\\int_{A_{\\e_n}}f\\bigg(\\frac{x}{\\e_n},\\nabla\nu_n\\bigg)\\, dx\\leq\n\\liminf_{n\\to+\\infty}\\,\\int_{A_{\\e_n}}f\\bigg(\\frac{x}{\\e_n},\\nabla\nu_n\\bigg)\\, dx +\\beta\\eta\\,,\n\\end{multline}\nwhere we have used $(H_2)$ and the fact that $\\nabla u_n=0$ outside\n$A_{\\e_n}$. On the other hand, Proposition~\\ref{reppart} yields\n\\begin{equation}\\label{pasidee2}\n{\\mathcal{F}}\\big(u^{a,b}_{\\nu_1},A_\\eta\\cap\\Pi_{\\nu_1}\\big)=\\int_{A_\\eta\\cap\\Pi_{\\nu_1}}K(a,b,\\nu_1)\\,\nd{\\mathcal{H}}^{N-1}= K(a,b,\\nu_1)\\,.\n\\end{equation}\nUsing $(H_4)$, the boundedness of $\\{\\nabla u_n\\}$ in\n$L^1(Q_\\nu;{\\mathbb{R}}^{d \\times N})$, the fact that\n$f^\\infty(\\cdot,0)\\equiv 0$, and H\\\"older's inequality, we derive\n\\begin{eqnarray}\\label{pasidee3}\n\\bigg|\\int_{A_{\\e_n}}f\\bigg(\\frac{x}{\\e_n},\\nabla u_n\\bigg) \\,dx-\n\\int_{Q_\\nu}f^\\infty\\bigg(\\frac{x}{\\e_n},\\nabla u_n\\bigg)\\,dx\\bigg|& \\leq & C\\int_{A_{\\e_n}}(1+|\\nabla u_n|^{1-q})\\,dx\\nonumber\\\\\n& \\leq & C\\big(\\e_n+\\e_n^q\\|\\nabla u_n\\|^{1-q}_{L^1(Q_\\nu;{\\mathbb{R}}^{d\n\\times N})}\\big)\\to 0\\,\n\\end{eqnarray}\nas $n \\to \\infty$. Gathering \\eqref{pasidee1}, \\eqref{pasidee1b}, \\eqref{pasidee2} and\n\\eqref{pasidee3}, we obtain $K(a,b,\\nu_1)\\leq \\vartheta_{\\rm\nhom}(a,b,\\nu_1)+(\\beta+1)\\eta$ and the conclusion follows from the\narbitrariness of $\\eta$.\n\\end{proof}\n\nWe are now in position to prove the upper bound on the jump part of the energy. The argument is based on\nLemma \\ref{upbdsurf} together with an approximation procedure of \\cite{AMT}. In view of Lemma \\ref{upperboundBV}\nand \\eqref{decompupbd}, this will complete the proof of the upper bound ${\\mathcal{F}}(u,\\O)\\leq {\\mathcal{F}}_{\\rm hom}(u)$.\n\n\\begin{corollary}\\label{upbdjp}\nFor every $u\\in BV(\\O;{\\mathcal{M}})$, we have\n$${\\mathcal{F}}(u,\\O\\cap S_u)\\leq \\int_{\\O\\cap S_u} \\vartheta_{\\rm hom}(u^+,u^-,\\nu_u)\\, d{\\mathcal{H}}^{N-1}\\,.$$\n\\end{corollary}\n\n\\begin{proof} First assume that $u$ takes a finite number of values, {\\it i.e.}, $u\\in BV(\\O;T)$ for some finite subset $T\\subset {\\mathcal{M}}$.\nThen the conclusion directly follows from Proposition \\ref{reppart} together with Lemma~\\ref{upbdsurf}.\n\nFix an arbitrary function $u\\in BV(\\O;{\\mathcal{M}})$ and an open set\n$A\\in{\\mathcal{A}}(\\O)$. For $\\delta_0>0$ small enough, let $\\mathcal\nU:=\\big\\{s \\in{\\mathbb{R}}^d\\,:\\,\\text{dist}(s,{\\mathcal{M}})<\\d_0\\big\\}$ be the\n$\\d_0$-neighborhood of ${\\mathcal{M}}$ on which the nearest point projection\n$\\Pi: \\mathcal U \\to {\\mathcal{M}}$ is a well defined Lipschitz mapping. We\nextend $\\vartheta_{\\rm hom}$ to a function $\\hat \\vartheta_{\\rm\nhom}$ defined in ${\\mathbb{R}}^d \\times {\\mathbb{R}}^d\n\\times\\mathbb{S}^{N-1}$ by setting\n$$\\hat \\vartheta_{\\rm hom}(a,b,\\nu):=\\chi(a)\\chi(b)\\vartheta_{\\rm hom}\\bigg(\\Pi(a),\\Pi(b), \\nu\\bigg)\\,, $$\nfor a cut-off function $\\chi\n\\in {\\mathcal{C}}^\\infty_c({\\mathbb{R}}^d;[0,1])$ satisfying $\\chi(t)=1$ if $\\text{dist}(s,{\\mathcal{M}})\n\\leq \\delta_0\/2$, and $\\chi(s)=0$ if $\\text{dist}(s,{\\mathcal{M}}) \\geq 3\\delta_0\/4$.\nIn view of Proposition \\ref{contsurfenerg}, we infer that\n$\\hat\\vartheta_{\\rm hom}$ is continuous and satisfies\n$$|\\hat\\vartheta_{\\rm hom}(a_1,b_1,\\nu)-\\hat\\vartheta_{\\rm hom}(a_2,b_2,\\nu)|\\leq C\\big(|a_1-a_2|+|b_1-b_2|\\big)\\,,$$\nand\n$$\\hat\\vartheta_{\\rm hom}(a_1,b_1,\\nu)\\leq C|a_1-b_1|\\,,$$\nfor every $a_1$, $b_1$, $a_2$, $b_2\\in{\\mathbb{R}}^d$, $\\nu\\in{\\mathbb{S}^{N-1}}$, and\nsome constant $C>0$. Therefore we can apply Step~2 in the proof of\n\\cite[Proposition 4.8]{AMT} to obtain a sequence $\\{v_n\\}\\subset\nBV(\\O;{\\mathbb{R}}^d)$ such that, for every $n\\in {\\mathbb{N}}$, $v_n\\in BV(\\O;T_n)$\nfor some finite set $T_n\\subset {\\mathbb{R}}^d$, $v_n\\to u$ in\n$L^\\infty(\\O;{\\mathbb{R}}^d)$ and\n\\begin{eqnarray*}\n\\limsup_{n\\to+\\infty}\\,\\int_{A\\cap S_{v_n}}\\hat\\vartheta_{\\rm\nhom}(v_n^+,v_n^-,\\nu_{v_n})\\, d{\\mathcal{H}}^{N-1} & \\leq & C|Du|(A\\setminus\nS_u)+\n\\int_{A\\cap S_{u}}\\hat\\vartheta_{\\rm hom}(u^+,u^-,\\nu_{u})\\, d{\\mathcal{H}}^{N-1}\\\\\n& = & C|Du|(A\\setminus S_u)+ \\int_{A\\cap S_{u}}\\vartheta_{\\rm\nhom}(u^+,u^-,\\nu_{u})\\, d{\\mathcal{H}}^{N-1}\\,.\n\\end{eqnarray*}\nHence we may assume without loss of generality\nthat for each $n\\in {\\mathbb{N}}$, $\\|v_n - u\\|_{L^\\infty(\\O;{\\mathbb{R}}^d)} < \\d_0\/2$,\nand thus $\\text{dist}(v_n^\\pm(x),{\\mathcal{M}}) \\leq |v_n^\\pm(x)-u^\\pm(x)|< \\d_0\/2$ for\n${\\mathcal{H}}^{N-1}$-a.e. $x \\in S_{v_n}$. In particular, we can define\n$$u_n:=\\Pi(v_n)\\,, $$\nand then $u_n\\in BV(\\O;{\\mathcal{M}})$, $u_n\\to u$ in $L^1(\\O;{\\mathbb{R}}^d)$.\nMoreover, one may check that for each $n\\in\\mathbb{N}$,\n$S_{u_n}\\subset S_{v_n}$ so that\n${\\mathcal{H}}^{N-1}\\big(S_{u_n}\\setminus(J_{u_n}\\cap J_{v_n})\\big)\\leq\n{\\mathcal{H}}^{N-1}(S_{u_n}\\setminus J_{u_n})+ {\\mathcal{H}}^{N-1}(S_{v_n}\\setminus\nJ_{v_n})=0$, and\n$$u_n^\\pm(x)=\\Pi(v_n^\\pm(x)) \\quad \\text{ and } \\quad \\nu_{u_n}(x)=\\nu_{v_n}(x)\n\\quad\\text{ for every $x\\in J_{u_n}\\cap J_{v_n}$}\\,.$$\nConsequently,\n\\begin{multline}\\label{estilimsurf}\n\\limsup_{n\\to+\\infty}\\, \\int_{A\\cap S_{u_n}}\\vartheta_{\\rm\nhom}(u_n^+,u_n^-,\\nu_{u_n})\\, d{\\mathcal{H}}^{N-1} \\,\\leq \\\\\n\\leq \\limsup_{n\\to+\\infty}\\int_{A\\cap S_{v_n}}\n\\hat\\vartheta_{\\rm hom}(v_n^+,v_n^-,\\nu_{v_n})\\, d{\\mathcal{H}}^{N-1}\\,\\leq\\\\\n\\leq C|Du|(A\\setminus S_u)+ \\int_{A\\cap S_{u}}\\vartheta_{\\rm\nhom}(u^+,u^-,\\nu_{u})\\, d{\\mathcal{H}}^{N-1}\\,.\n\\end{multline}\nSince $u_n$ takes a finite number of values, Proposition\n\\ref{reppart} and Lemma \\ref{upbdsurf} yield \n\\begin{equation}\\label{estisurfn}\n{\\mathcal{F}}(u_n,A\\cap S_{u_n})\\leq \\int_{A\\cap S_{u_n}}\\vartheta_{\\rm\nhom}(u_n^+,u_n^-,\\nu_{u_n})\\, d{\\mathcal{H}}^{N-1}\\,,\n\\end{equation}\nand, in view of Lemma \\ref{measbis},\n\\begin{equation}\\label{estidiffn}\n{\\mathcal{F}}(u_n,A\\setminus S_{u_n}) \\leq C{\\mathcal{L}}^N(A)\\,.\n\\end{equation}\nCombining \\eqref{estilimsurf}, \\eqref{estisurfn} and \\eqref{estidiffn}, we deduce\n\\begin{eqnarray*}\n\\limsup_{n\\to+\\infty} \\,{\\mathcal{F}}(u_n,A) & = & \\limsup_{n\\to+\\infty}\\big({\\mathcal{F}}(u_n,A\\setminus S_{u_n})+{\\mathcal{F}}(u_n,A\\cap S_{u_n})\\big)\\\\\n& \\leq & \\int_{A\\cap S_{u}}\\vartheta_{\\rm hom}(u^+,u^-,\\nu_{u})\\,\nd{\\mathcal{H}}^{N-1}+ C\\big({\\mathcal{L}}^N(A)+|Du|(A\\setminus S_u)\\big)\\,.\n\\end{eqnarray*}\nOn the other hand, ${\\mathcal{F}}(\\cdot,A)$ is lower semicontinuous with\nrespect to the strong $L^1(A;{\\mathbb{R}}^d)$-convergence, and thus\n$\\displaystyle{\\mathcal{F}}(u,A)\\leq \\liminf_{n\\to+\\infty}\\,{\\mathcal{F}}(u_n,A)$ which leads to\n$${\\mathcal{F}}(u,A)\\leq \\int_{A\\cap S_{u}}\\vartheta_{\\rm hom}(u^+,u^-,\\nu_{u})\\, d{\\mathcal{H}}^{N-1}+\nC\\big({\\mathcal{L}}^N(A)+|Du|(A\\setminus S_u)\\big)\\,.$$ Since $A$ is\narbitrary, the above inequality holds for any open set $A\\in{\\mathcal{A}}(\\O)$\nand, by Remark \\ref{measconstr}, it also holds if $A$ is any Borel\nsubset of $\\O$. Then taking $A=\\O\\cap S_u$ yields the desired\ninequality.\n\\end{proof}\n\n\n\n\n\\section{The lower bound}\n\n\n\n\n\\noindent We adress in this section with the $\\G$-$\\liminf$\ninequality. Using the blow-up method, we follow the approach of\n\\cite{FM2}, estimating separately the Cantor part and the jump part,\nwhile the bulk part is obtained exactly as in the $W^{1,1}$ analysis, see\n\\cite[Lemma 5.2]{BM}.\n\n\\begin{lemma}\\label{lowerboundBV}\nFor every $u \\in BV(\\O;{\\mathcal{M}})$, we have ${\\mathcal{F}}(u,\\O)\\geq {\\mathcal{F}}_{\\rm hom}\n(u)$.\n\\end{lemma}\n\n\\begin{proof}\nLet $u \\in BV(\\O;{\\mathcal{M}})$ and $\\{u_n\\} \\subset W^{1,1}(\\O;{\\mathcal{M}})$ be such\nthat $${\\mathcal{F}}(u,\\O)= \\lim_{n \\to +\\infty}\\int_\\O f\n\\left(\\frac{x}{\\e_n},\\nabla u_n \\right)dx\\,.$$ Define the sequence\nof nonnegative Radon measures\n$$\\mu_n:=f \\left(\\frac{\\cdot}{\\e_n},\\nabla u_n \\right){\\mathcal{L}}^N\n\\res\\, \\O\\,.$$\nUp to the extraction of a subsequence, we can assume that\nthere exists a nonnegative Radon measure $\\mu \\in {\\mathcal{M}}(\\O)$\nsuch that $\\mu_n \\xrightharpoonup[]{*} \\mu$ in ${\\mathcal{M}}(\\O)$. By the\nBesicovitch Differentiation Theorem,\nwe\ncan split $\\mu$ into the sum of four mutually singular nonnegative measures\n$\\mu=\\mu^a + \\mu^j+\\mu^c+\\mu^s$ where $\\mu^a \\ll \\mathcal L^N$,\n$\\mu^j \\ll {\\mathcal{H}}^{N-1}\\res\\, S_u$ and $\\mu^c \\ll |D^c u|$. Since we\nhave $\\mu(\\O) \\leq {\\mathcal{F}}(u,\\O)$, it is enough to check that\n\\begin{equation}\\label{lambda^a}\n\\frac{d\\mu}{d{\\mathcal{L}}^N}(x_0) \\geq Tf_\\text{hom}(u(x_0),\\nabla u(x_0))\\quad\n\\text{ for }{\\mathcal{L}}^N\\text{-a.e. }x_0 \\in \\O\\,,\n\\end{equation}\n\\begin{equation}\\label{lambda^c}\n\\frac{d\\mu}{d|D^c u|}(x_0) \\geq Tf_\\text{hom}^\\infty\\left(\\tilde\nu(x_0),\\frac{dD^c u}{d|D^c u|}(x_0)\\right)\\quad \\text{ for }|D^c\nu|\\text{-a.e. }x_0 \\in \\O\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{lambda^j}\n\\frac{d\\mu}{d{\\mathcal{H}}^{N-1}\\res\\, S_u}(x_0) \\geq\n\\vartheta_\\text{hom}(u^+(x_0),u^-(x_0),\\nu_u(x_0))\\quad \\text{ for\n}{\\mathcal{H}}^{N-1}\\text{-a.e. }x_0 \\in S_u\\,.\n\\end{equation}\nThe proof of (\\ref{lambda^a}) follows the one in \\cite[Lemma\n5.2]{BM} and we shall omit it. The proofs of (\\ref{lambda^c}) and\n(\\ref{lambda^j}) are postponed to the remaining of this subsection.\n\\end{proof}\n\n\n\\noindent {\\bf Proof of \\eqref{lambda^c}.} The lower bound on the\ndensity of the Cantor part will be achieved in three steps. We shall\nuse the blow-up method to reduce the study to constant limits, and\nthen a truncation argument as in the proof of \\cite[Lemma 5.2]{BM},\nto replace the starting sequence by a uniformly converging one.\n\\vskip5pt\n\n{\\bf Step 1.} Choose a point $x_0 \\in \\O$ such that\n\\begin{equation}\\label{cantor2}\n\\lim_{\\rho \\to 0^+}- \\hskip -1em \\int_{Q(x_0,\\rho)}|u(x)-\\tilde u(x_0)|\\, dx=0\\,,\n\\end{equation}\n\\begin{equation}\\label{cantor3}\nA(x_0):=\\lim_{\\rho \\to\n0^+}\\frac{Du(Q(x_0,\\rho))}{|Du|(Q(x_0,\\rho))}\\in [T_{\\tilde\nu(x_0)}({\\mathcal{M}})]^N\\;\\text{ is a rank one matrix with }\\;|A(x_0)|=1\\,,\n\\end{equation}\n\\begin{equation}\\label{cantor1}\n\\frac{d\\mu}{d|D^cu|}(x_0) \\quad \\text{exists and is finite and}\\quad\n\\frac{d|Du|}{d|D^cu|}(x_0)=1\\,,\n\\end{equation}\n\\begin{equation}\\label{cantor4}\n\\lim_{\\rho \\to 0^+}\\frac{|Du|(Q(x_0,\\rho))}{\\rho^{N-1}}=0 \\quad\\text{and}\\quad\n\\lim_{\\rho \\to 0^+}\\frac{|Du|(Q(x_0,\\rho))}{\\rho^N}=+\\infty\\,,\n\\end{equation}\n\\begin{equation}\\label{cantor5}\n\\liminf_{\\rho \\to 0^+}\\,\\frac{|Du|(Q(x_0,\\rho) \\setminus\nQ(x_0,\\tau\\rho))}{|Du|(Q(x_0,\\rho))}\\leq 1-\\tau^N\\quad\\text{for every $0<\\tau<1$}\\,.\n\\end{equation}\nIt turns out that $|D^c u|$-a.e. $x_0\\in\\O$ satisfy these\nproperties. Indeed (\\ref{cantor1}) is immediate while\n(\\ref{cantor2}) is a consequence of the fact that $S_u$ is $|D^c\nu|$-negligible. Property (\\ref{cantor3}) comes from Alberti Rank One\nTheorem together with Lemma \\ref{manifold}, (\\ref{cantor4}) from\n\\cite[Proposition~3.92~(a),~(c)]{AFP} and (\\ref{cantor5}) from\n\\cite[Lemma~2.13]{FM2}. Write $A(x_0)=a \\otimes \\nu$ for some $a \\in\n{\\mathcal{M}}$ and $\\nu \\in {\\mathbb{S}^{N-1}}$. Upon rotating the coordinate axis, one may\nassume without loss of generality that $\\nu=e_N$. To simplify the\nnotations, we set $s_0:=\\tilde u(x_0)$ and $A_0:=A(x_0)$. \\vskip5pt\n\nFix $t\\in(0,1)$ arbitrarily close to $1$, and in view of\n\\eqref{cantor5}, find a sequence $\\rho_k \\searrow 0^+$ such\nthat\n\\begin{equation}\\label{cantor5bis}\n\\limsup_{k\\to+\\infty}\\,\\frac{|Du|(Q(x_0,\\rho_k) \\setminus\nQ(x_0,t\\rho_k))}{|Du|(Q(x_0,\\rho_k))}\\leq 1-t^N\\,.\n\\end{equation}\nNow fix $t<\\gamma<1$ and set $\\gamma':=(1+\\gamma)\/2$. Using (\\ref{cantor1}), we derive\n\\begin{multline}\\label{infmu}\n\\frac{d\\mu}{d|D^cu|}(x_0) =\\lim_{k \\to +\\infty}\\frac{\\mu(Q(x_0,\\rho_k))}{|Du|(Q(x_0,\\rho_k))}\\geq\n\\limsup_{k \\to +\\infty}\\,\\frac{\\mu(\\overline{Q(x_0,\\gamma'\\rho_k)})}{|Du|(Q(x_0,\\rho_k))}\\geq\\\\\n\\geq \\limsup_{k\\to+\\infty}\\,\\limsup_{n\\to+\\infty}\\,\n\\frac{1}{|Du|(Q(x_0,\\rho_k))}\\int_{Q(x_0,\\gamma'\n\\rho_k)}f\\bigg(\\frac{x}{\\e_n},\\nabla u_n\\bigg)dx\\,.\n\\end{multline}\nArguing as in the proof of \\cite[Lemma 5.2]{BM} with minor\nmodifications, we construct a sequence $\\{\\bar v_{n}\\} \\subset\nW^{1,\\infty}(Q(0,\\rho_k);{\\mathbb{R}}^d)$ satisfying $\\bar v_{n} \\to\nu(x_0+\\cdot)$ in $L^1(Q(0,\\rho_k);{\\mathbb{R}}^d)$ and\n\\begin{equation}\\label{passuv}\n\\limsup_{n\\to+\\infty}\\, \\int_{Q(x_0,\\gamma'\n\\rho_k)}f\\bigg(\\frac{x}{\\e_n},\\nabla u_n\\bigg)\\, dx\\geq\n\\limsup_{n\\to+\\infty}\\, \\int_{Q(0,\\gamma\n\\rho_k)}g\\bigg(\\frac{x}{\\e_n}, \\bar v_n, \\nabla \\bar v_n\\bigg)\\,\ndx\\,,\n\\end{equation}\nwhere $g$ is given by \\eqref{defg}. Setting $w_{n,k}(x):=\\bar\nv_{n}(\\rho_k\\, x)$, a change of variable together with \\eqref{infmu}\nand \\eqref{passuv} yields\n\\begin{equation}\\label{c1}\n\\frac{d\\mu}{d|D^cu|}(x_0) \\geq \\limsup_{k \\to +\\infty} \\limsup_{n\n\\to +\\infty}\\, \\frac{\\rho_k^N}{|Du|(Q(x_0,\\rho_k))} \\int_{\\gamma Q}\ng\\left(\\frac{\\rho_k\\, x}{\\e_n},w_{n,k}, \\frac{1}{\\rho_k} \\nabla\nw_{n,k}\\right)dx\\,.\n\\end{equation}\nThen we infer from (\\ref{cantor2}) that\n\\begin{equation}\\label{c2}\n\\lim_{k \\to +\\infty} \\lim_{n \\to +\\infty}\\int_Q|w_{n,k}-s_0|\\, dx=0\\,,\n\\end{equation}\nand\n\\begin{multline}\\label{c3}\n\\lim_{k \\to +\\infty} \\lim_{n \\to\n+\\infty}\\frac{\\rho_k^{N-1}}{|Du|(Q(x_0,\\rho_k))}\\int_Q\n\\bigg|w_{n,k}(x)-u(x_0+\\rho_k x)\\\\\n- \\int_Q \\big(w_{n,k}(y)-u(x_0+\\rho_k y)\\big)\\, dy\\bigg|\\, dx =0\\,.\n\\end{multline}\nBy \\eqref{c1}, \\eqref{c2} and (\\ref{c3}), we can extract a diagonal sequence $n_k\n\\to+\\infty$ such that $\\d_k:=\\e_{n_k}\/\\rho_k\\to 0$,\n$w_k:=w_{n_k,k}\\to s_0$ in\n$L^1(Q;{\\mathbb{R}}^d)$,\n$$\\frac{d\\mu}{d|D^cu|}(x_0) \\geq \\limsup_{k \\to +\\infty}\\,\n\\frac{\\rho_k^N}{|Du|(Q(x_0,\\rho_k))} \\int_{\\gamma Q}\ng\\left(\\frac{x}{\\d_k},w_k, \\frac{1}{\\rho_k} \\nabla w_k\\right)dx\\,,$$\nand\n\\begin{equation}\\label{c5}\n\\lim_{k \\to +\\infty}\\frac{\\rho_k^{N-1}}{|Du|(Q(x_0,\\rho_k))}\\int_Q\n\\bigg|w_k(x)-u(x_0+\\rho_k\\, x) - \\int_Q \\big(w_k(y)-u(x_0+\\rho_k\\, y)\\big)\\,\ndy\\bigg|\\,dx =0\\,.\n\\end{equation}\n\\vskip5pt\n\n{\\bf Step 2.} Now we reproduce the truncation argument used in Step\n2 of the proof of \\cite[Lemma~5.2]{BM} with minor modifications\n(make use of (\\ref{cantor4}) and \\cite[Lemma~2.12]{FM2} instead of\n\\cite[Lemma~2.6]{FM}, see \\cite{FM2} for details). Setting\n$a_k:=\\int_Q w_k(y)\\, dy$, it yields a sequence of cut-off functions\n$\\{\\zeta_k\\} \\subset {\\mathcal{C}}^\\infty_c({\\mathbb{R}};[0,1])$ such that\n$\\zeta_k(\\tau)=1$ if $|\\tau| \\leq s_k$, $\\zeta_k(\\tau)=0$ is\n$|\\tau|\\geq t_k$ for some\n$$\\|w_k-a_k\\|^{1\/2}_{L^1(Q;{\\mathbb{R}}^d)} s_k\\}}|w_k(x)-\\overline w_k(x)|\\, dx\\leq \\\\\n \\leq \\frac{\\rho_k^{N-1}}{|Du|(Q(x_0,\\rho_k))} \\int_{\\{|w_k-a_k|> s_k\\}}|w_k(x)-a_k|\\, dx= \\int_{\\{|w_k-a_k|> s_k\\}} |z_k(x)|\\, dx\\,.\n\\end{multline}\nBy Chebyshev inequality, we have\n\\begin{equation}\\label{ln}{\\mathcal{L}}^N(\\{|w_k-a_k|> s_k\\}) \\leq \\frac{1}{s_k}\\int_Q|w_k(x)-a_k|\\, dx\n\\leq \\|w_k-a_k\\|^{1\/2}_{L^1(Q;{\\mathbb{R}}^d)}\\to 0\\,,\n\\end{equation}\nand thus (\\ref{ei}), (\\ref{ln}) and the equi-integrability of $\\{z_k\\}$ imply $\\|\\overline z_k -\nz_k\\|_{L^1(Q;{\\mathbb{R}}^d)} \\to 0$. Therefore $\\overline z_k\n\\to v$ in $L^1(Q;{\\mathbb{R}}^d)$, and setting\n$\\a_k:=|Du|(Q(x_0,\\rho_k))\/\\rho_k^N \\to +\\infty$,\n\\begin{equation}\\label{tilde}\n\\frac{d\\mu}{d|D^cu|}(x_0) \\geq \\limsup_{k \\to +\\infty}\\,\n\\frac{1}{\\a_k} \\int_{\\gamma Q} g\\left(\\frac{x}{\\d_k},s_0,\\a_k \\nabla\n\\overline z_k\\right)dx\\,.\n\\end{equation}\nUsing (\\ref{grec}) and the positive $1$-homogeneity of the recession function\n$g^\\infty(y,s,\\cdot)$, we infer that\n\\begin{align*}\n \\int_{\\gamma Q} \\left| \\frac{1}{\\a_k}\\, g\\left(\\frac{x}{\\d_k},s_0,\\a_k\n\\nabla \\overline z_k\\right) -\ng^\\infty\\left(\\frac{x}{\\d_k},s_0, \\nabla \\overline z_k\\right)\\right| dx &\\leq \\frac{C}{\\a_k} \\int_{\\gamma Q} (1+ \\a_k^{1-q} |\\nabla \\overline z_k|^{1-q})\\, dx\\\\\n&\\leq C\\big(\\a_k^{-1} + \\a_k^{-q} \\|\\nabla \\overline\nz_k\\|^{1-q}_{L^1(\\gamma Q;{\\mathbb{R}}^{d \\times N})}\\big) \\to 0\\,,\n\\end{align*}\nwhere we have used H\\\"older's inequality and the boundedness of\n$\\{\\nabla \\overline z_k\\}$ in $L^1(\\gamma Q;{\\mathbb{R}}^{d\n\\times N})$ (which follows from (\\ref{pgrowth}) and \\eqref{tilde}).\nConsequently,\n$$\\frac{d\\mu}{d|D^cu|}(x_0) \\geq \\limsup_{k \\to +\\infty}\\,\n\\int_{\\gamma Q} g^\\infty \\left(\\frac{x}{\\d_k},s_0,\\nabla \\overline\nz_k\\right)dx\\,.$$\n\\vskip5pt\n\n{\\bf Step 3.} Extend $\\theta$ continuously to ${\\mathbb{R}}$ by the values of\nits traces at $\\pm 1\/2$. Define $v_k(x)=v_k(x_N):=a \\theta * \\varrho_k (x_N)$ where\n$\\varrho_k$ is a sequence of (one dimensional) mollifiers. Then $v_k\n\\to v$ in $L^1(Q;{\\mathbb{R}}^d)$ and thus, since $\\overline u_k - v_k \\to\n0$ in $L^1(Q;{\\mathbb{R}}^d)$, it follows that (up to a subsequence)\n\\begin{equation}\\label{D}\nD\\overline u_k(\\tau Q) - Dv_k(\\tau Q) \\to 0\n\\end{equation}\nfor ${\\mathcal{L}}^1$-a.e. $\\tau \\in (0,1)$. Fix $\\tau\\in (t,\\gamma)$\nfor which \\eqref{D} holds. Since $\\|\\bar z_k -v_k\\|_{L^1(Q;{\\mathbb{R}}^d)}\\to 0$, one can use a standard cut-off\nfunction argument (see \\cite[p. 29--30]{FM2}) to modify the sequence $\\{\\overline z_k\\}$\nand produce a new sequence\n$\\{\\overline\\varphi_k\\} \\subset W^{1,\\infty}(\\tau Q;{\\mathbb{R}}^d)$ satisfying\n$\\overline \\varphi_k \\to v$ in $L^1(\\tau Q;{\\mathbb{R}}^d)$, $\\overline\n\\varphi_k=v_k$ on a neighborhood of $\\partial (\\tau Q)$ and\n\\begin{equation}\\label{may}\n\\frac{d\\mu}{d|D^cu|}(x_0) \\geq \\limsup_{k \\to +\\infty} \\int_{\\tau Q}\ng^\\infty \\left(\\frac{x}{\\d_k},s_0,\\nabla \\overline\n\\varphi_k\\right)dx\\,.\n\\end{equation}\nA simple computation shows that\n\\begin{equation}\\label{D2}\nD\\overline u_k (\\tau Q)=\\frac{Du(Q(x_0,\\tau\n\\rho_k))}{|Du|(Q(x_0,\\rho_k))}\\quad \\text{ and }\\quad Dv_k(\\tau\nQ)=\\tau^N \\,A_k\\,,\n\\end{equation}\nwhere $A_k \\in {\\mathbb{R}}^{d \\times N}$ is the matrix given by\n$$A_k:= a \\otimes e_N \\frac{\\theta *\\varrho_k (\\tau\/2)- \\theta *\\varrho_k (-\\tau\/2)}{\\tau}\\,.$$\nWe observe that $A_k$ is bounded in $k$ since $\\theta$ has bounded\nvariation.\n\nLet $m_k:=[\\tau\/\\delta_k]+1\\in {\\mathbb{N}}$, and define for $x=(x',x_N)\\in \\delta_km_k Q$,\n$$\\varphi_k(x):=\\begin{cases}\n\\overline \\varphi_k(x)-A_kx & \\text{if $x\\in \\tau Q$}\\,,\\\\\nv_k(x_N)-A_k\\, x & \\text{if $|x_N|\\leq \\tau\/2$ and $|x'|\\geq \\tau\/2$}\\, ,\\\\\nv_k(\\tau\/2)-A_k(x',\\tau\/2) & \\text{if $x_N\\geq \\tau\/2$}\\,,\\\\\nv_k(-\\tau\/2)-A_k(x',-\\tau\/2) & \\text{if $x_N\\leq -\\tau\/2$}\\,.\n\\end{cases}$$\nOne may check that $\\varphi_k\\in W^{1,\\infty}(\\delta_km_kQ;{\\mathbb{R}}^d)$, $\\varphi_k$ is $\\delta_km_k$-periodic, and that\n\\begin{equation}\\label{periodization}\n \\limsup_{k \\to +\\infty} \\int_{\\tau Q}\ng^\\infty \\left(\\frac{x}{\\d_k},s_0,\\nabla \\overline\n\\varphi_k\\right)dx= \\limsup_{k \\to +\\infty} \\int_{\\delta_km_k Q}\ng^\\infty \\left(\\frac{x}{\\d_k},s_0,A_k+\\nabla\n\\varphi_k\\right)dx\\,.\n\\end{equation}\nSetting $\\phi_k(y):=\\tau^{N}\\delta_k^{-1}\\varphi_k(\\delta_k y)$ for $y\\in m_k Q$, we have $\\phi_k\\in W^{1,\\infty}_{\\#}(m_kQ;{\\mathbb{R}}^d)$, and a change of variables yields\n\\begin{align}\n\\nonumber\\int_{\\delta_km_k Q} g^\\infty\n\\left(\\frac{x}{\\d_k},s_0,A_k+\\nabla varphi_k\\right)dx&=\n\\tau^{-N}\\delta_k^Nm_k^N- \\hskip -1em \\int_{m_k Q} g^\\infty\n\\left(y,s_0,\\tau^{N}A_k+\\nabla\n\\phi_k\\right)dy \\\\\n\\label{compper} &\\geq \\tau^{-N}\\delta_k^Nm_k^N (g^\\infty)_{\\rm hom}(s_0,\\tau^N A_k)\\,,\n\\end{align}\nsince $(g^\\infty)_\\text{hom}$ can be computed as follows\n(see Remark \\ref{reminfty} and {\\it e.g., } \\cite[Remark~14.6]{BD}),\n\\begin{eqnarray*}\n(g^\\infty)_\\text{hom}(s,\\xi)\n= \\inf \\left\\{- \\hskip -1em \\int_{(0,m)^N} g^\\infty(y,s,\\xi +\\nabla \\phi(y))\\, dy : m\\in {\\mathbb{N}},\\,\n\\phi \\in W^{1,\\infty}_\\#((0,m)^N;{\\mathbb{R}}^d) \\right\\}\\,.\n\\end{eqnarray*}\nGathering \\eqref{may}, \\eqref{periodization} and \\eqref{compper}, we derive\n\\begin{equation*}\n\\frac{d\\mu}{d|D^cu|}(x_0) \\geq\\limsup_{k \\to +\\infty}\n\\,(g^\\infty)_\\text{hom}(s_0,\\tau^N A_k)\\,.\n\\end{equation*}\nIn view (\\ref{D}), (\\ref{D2}), (\\ref{cantor5bis}) and (\\ref{cantor3}), we have\n\\begin{multline*}\n\\limsup_{k \\to +\\infty}|\\tau^N A_k -A_0|=\\limsup_{k \\to +\\infty}|Dv_k(\\tau Q) - A_0|=\\limsup_{k \\to +\\infty}|D\\overline u_k(\\tau Q)-A_0|=\\\\\n=\\limsup_{k \\to +\\infty}\\left| \\frac{Du(Q(x_0,\\tau \\rho_k))}{|Du|(Q(x_0,\\rho_k))}-A_0\\right|\n= \\limsup_{k \\to +\\infty}\n\\frac{|Du|(Q(x_0,\\rho_k)\\setminus Q(x_0,\\tau\\rho_k))}\n{|Du|(Q(x_0,\\rho_k))}\\leq 1- t^{N}\\,.\n\\end{multline*}\nBy Remark \\ref{reminfty}, $(g^\\infty)_\\text{hom}(s_0,\\cdot)$ is Lipschitz continuous, and consequently\n$$\\frac{d\\mu}{d|D^cu|}(x_0) \\geq (g^\\infty)_\\text{hom}(s_0,A_0)-C(1-t^{N})\\,.$$\nFrom the arbitrariness of $t$, we finally infer that\n$$\\frac{d\\mu}{d|D^cu|}(x_0) \\geq (g^\\infty)_\\text{hom}(s_0,A_0)\\,. $$\nSince $s_0 \\in {\\mathcal{M}}$ and $A_0 \\in [T_{s_0}({\\mathcal{M}})]^N$, Remark\n\\ref{reminfty} and \\eqref{remdensinf} yield\n$(g^\\infty)_\\text{hom}(s_0,A_0)= T(f^\\infty)_\\text{hom}(s_0,A_0)\\geq\nTf_\\text{hom}^\\infty(s_0,A_0)$, and the proof is complete. \\prbox\n\\vskip10pt\n\n\\noindent{\\bf Proof of \\eqref{lambda^j}.} The strategy used in that\npart follows the one already used for the bulk and Cantor parts. It\nstill rests on the blow up method together with the projection\nargument in Proposition~\\ref{proj}. \\vskip5pt\n\n{\\bf Step 1.} Let $x_0 \\in S_u$ be such that\n\\begin{equation}\\label{jump1}\n\\lim_{\\rho \\to 0^+}- \\hskip -1em \\int_{Q_{\\nu_u(x_0)}^\\pm(x_0,\\rho)} |u(x)-u^\\pm(x_0)|\\, dx=0\\,,\n\\end{equation}\nwhere $u^\\pm(x_0) \\in {\\mathcal{M}}$,\n\\begin{equation}\\label{jump2}\n\\lim_{\\rho \\to 0^+}\\frac{{\\mathcal{H}}^{N-1}(S_u \\cap Q_{\\nu_u(x_0)}(x_0,\\rho))}{\\rho^{N-1}}=1\\,,\n\\end{equation}\nand such that the Radon-Nikod\\'ym derivative of $\\mu$ with respect\nto ${\\mathcal{H}}^{N-1}\\res \\, S_u$ exists and is finite. By\nLemma~\\ref{manifold}, Theorem 3.78 and Theorem 2.83 (i) in\n\\cite{AFP} (with cubes instead of balls), it turns out that\n${\\mathcal{H}}^{N-1}$-a.e. $x_0\\in S_u$ satisfy these properties. Set\n$s_0^\\pm:=u^\\pm(x_0)$, $\\nu_0:=\\nu_u(x_0)$.\n\nUp to a further subsequence, we may assume that $(1+|\\nabla u_n|) {\\mathcal{L}}^N\n\\res\\, \\O \\xrightharpoonup[]{*} \\lambda$ in ${\\mathcal{M}}(\\O)$ for some nonnegative\nRadon measure $\\lambda \\in {\\mathcal{M}}(\\O)$. Consider a sequence $\\rho_k \\searrow 0^+$ satisfying $\\mu(\\partial\nQ_{\\nu_0}(x_0,\\rho_k))=\\lambda(\\partial\nQ_{\\nu_0}(x_0,\\rho_k))=0$ for each $k \\in {\\mathbb{N}}$. Using\n(\\ref{jump2}) we derive\n\\begin{multline*}\n\\frac{d\\mu}{d{\\mathcal{H}}^{N-1}\\res \\, S_u}(x_0)\\lim_{k \\to +\\infty}\\frac{\\mu(Q_{\\nu_0}(x_0,\\rho_k))}{{\\mathcal{H}}^{N-1}(S_u \\cap Q_{\\nu_0}(x_0,\\rho_k))}\n=\\lim_{k \\to +\\infty}\\frac{\\mu(Q_{\\nu_0}(x_0,\\rho_k))}{\\rho_k^{N-1}}=\\\\\n= \\lim_{k \\to +\\infty}\\lim_{n \\to\n+\\infty}\\frac{1}{\\rho_k^{N-1}}\\int_{Q_{\\nu_0}(x_0,\\rho_k)}\nf\\left(\\frac{x}{\\e_n},\\nabla u_n \\right)dx\\,.\n\\end{multline*}\nThanks to Theorem \\ref{density}, one can assume without loss of\ngenerality that $u_n \\in \\mathcal D(\\O;{\\mathcal{M}})$ for each $n \\in {\\mathbb{N}}$.\nArguing exactly as in Step 1 of the proof of \\cite[Lemma 5.2]{BM}\n(with $Q_{\\nu_0}(x_0,\\rho_k)$ instead of $Q(x_0,\\rho_k)$) we obtain\na sequence $\\{v_n\\} \\subset \\mathcal D(Q_{\\nu_0}(0,\\rho_k);{\\mathcal{M}})$ such\nthat $v_n \\to u(x_0+\\cdot)$ in $L^1(Q_{\\nu_0}(0,\\rho_k);{\\mathbb{R}}^d)$ as\n$n \\to +\\infty$, and\n$$\\frac{d\\mu}{d{\\mathcal{H}}^{N-1}\\res \\, S_u}(x_0) \\geq \\limsup_{k \\to +\\infty}\\,\\limsup_{n \\to +\\infty}\\,\n\\frac{1}{\\rho_k^{N-1}}\\int_{Q_{\\nu_0}(0,\\rho_k)}\nf\\left(\\frac{x}{\\e_n},\\nabla v_n \\right)dx$$\n(note that the\nconstruction process to obtain $v_n$ from $u_n$ does not affect the\nmanifold constraint). Changing variables and setting\n$w_{n,k}(x)=v_n(\\rho_k\\, x)$ lead to\n$$\\frac{d\\mu}{d{\\mathcal{H}}^{N-1}\\res \\, S_u}(x_0) \\geq \\limsup_{k \\to\n+\\infty}\\,\\limsup_{n \\to +\\infty} \\rho_k\\int_{Q_{\\nu_0}}\nf\\left(\\frac{\\rho_k \\, x}{\\e_n},\\frac{1}{\\rho_k}\\nabla w_{n,k}\n\\right)dx\\,.$$\nDefining\n$$u_0(x):=\\begin{cases}\ns_0^+ & \\text{ if } x\\cdot \\nu_0 > 0\\,,\\\\\ns_0^- & \\text{ if } x\\cdot \\nu_0 \\leq 0\\,,\n\\end{cases}$$\nwe infer from (\\ref{jump1}) that\n$$\\lim_{k \\to +\\infty}\\lim_{n \\to +\\infty}\n\\int_{Q_{\\nu_0}}|w_{n,k}-u_0|\\, dx=0\\,.$$ By a standard diagonal\nargument, we find a sequence $n_k \\nearrow +\\infty$ such that\n $\\d_k:=\\e_{n_k}\/\\rho_k\\to 0$, $w_k:=w_{n_k,k} \\in \\mathcal\nD(Q_{\\nu_0};{\\mathcal{M}})$ converges to $u_0$ in $L^1(Q_{\\nu_0};{\\mathbb{R}}^d)$, and\n\\begin{equation}\\label{dhnwj}\n\\frac{d\\mu}{d{\\mathcal{H}}^{N-1}\\res \\, S_u}(x_0) \\geq \\limsup_{k \\to +\\infty}\\,\n\\rho_k\\int_{Q_{\\nu_0}}f\\left(\\frac{x}{\\d_k},\\frac{1}{\\rho_k}\\nabla\nw_k \\right)dx\\,.\n\\end{equation}\nAccording to $(H_4)$ and the positive $1$-homogeneity of $f^\\infty(y,\\cdot)$, we have\n\\begin{align}\\label{dhnwj2}\n\\nonumber\\int_{Q_{\\nu_0}}\\left|\\rho_k\\,\nf\\left(\\frac{x}{\\d_k},\\frac{1}{\\rho_k}\\nabla w_k \\right) -\nf^\\infty\\left(\\frac{x}{\\d_k},\\nabla w_k \\right)\\right| dx & \\leq C\\rho_k\\int_{Q_{\\nu_0}} (1 + \\rho_k^{q-1}\n|\\nabla w_k|^{1-q})\\, dx\\\\\n&\\leq C\\left(\\rho_k +\\rho_k^q \\|\\nabla\nw_k\\|^{1-q}_{L^1(Q_{\\nu_0};{\\mathbb{R}}^{d \\times N})}\\right)\\,,\n\\end{align}\nwhere we have used H\\\"older's inequality and $00$ and $\\b>0$ such that\n$$\\a |\\xi| \\leq f(x,s,\\xi) \\leq \\b(1+|\\xi|) \\quad \\text{ for every\n}(x,s,\\xi) \\in \\O \\times {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N}\\,;$$\n\n\\item[$(H_3')$] for every compact set $K \\subset \\O$, there exists a\ncontinuous function $\\omega : [0,+\\infty) \\to [0,+\\infty)$\nsatisfying $\\omega(0)=0$ and\n$$|f(x,s,\\xi) - f(x',s',\\xi)| \\leq \\omega(|x-x'| + |s-s'|)\n(1+|\\xi|)$$\nfor every $x$, $x' \\in \\O$, $s$, $s' \\in {\\mathbb{R}}^d$ and\n$\\xi \\in {\\mathbb{R}}^{d \\times N}$;\n\n\\item[$(H_4')$] there exist $C>0$ and $q \\in (0,1)$ such that\n$$|f(x,s,\\xi) - f^\\infty(x,s,\\xi)| \\leq C(1+|\\xi|^{1-q}), \\quad \\text{ for every\n}(x,s,\\xi) \\in \\O \\times {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N}\\,,$$ where\n$f^\\infty : \\O \\times {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N} \\to [0,+\\infty)$\nis the recession function of $f$ defined by\n$$f^\\infty(x,s,\\xi):=\\limsup_{t \\to +\\infty} \\frac{f(x,s,t\\xi)}{t}\\,\n.$$\n\\end{itemize}\n\nConsider the functional $F:L^1(\\O;{\\mathbb{R}}^d) \\to [0,+\\infty]$ given by\n$$F(u):=\\left\\{\n\\begin{array}{ll}\n\\displaystyle \\int_\\O f(x,u,\\nabla u)\\, dx & \\text{ if }u \\in\nW^{1,1}(\\O;{\\mathcal{M}}),\\\\[0.4cm]\n+\\infty & \\text{ otherwise}, \\end{array}\\right.$$ and its relaxation\nfor the strong $L^1(\\O;{\\mathbb{R}}^d)$-topology $\\overline F:L^1(\\O;{\\mathbb{R}}^d)\n\\to [0,+\\infty]$ defined by\n$$\\overline F(u):=\\inf_{\\{u_n\\}} \\left\\{ \\liminf_{n \\to +\\infty}\nF(u_n) : u_n \\to u \\text{ in }L^1(\\O;{\\mathbb{R}}^d)\\right\\}\\,.$$ Then the\nfollowing integral representation result holds:\n\n\\begin{theorem} \\label{relax}\nLet ${\\mathcal{M}}$ be a smooth compact and connected submanifold of ${\\mathbb{R}}^d$\nwithout boundary, and let $f:{\\mathbb{R}}^N \\times {\\mathbb{R}}^{d \\times N} \\to\n[0,+\\infty)$ be a continuous function satisfying $(H_1')$ to\n$(H_4')$. Then for every $u \\in L^1(\\O;{\\mathbb{R}}^d)$,\n\\begin{equation}\\label{reprel}\n\\overline F(u)= \\begin{cases} \\displaystyle\n\\begin{multlined}[8.5cm]\n\\,\\int_\\O f(x,u,\\nabla u)dx +\n\\int_{\\O\\cap S_u}K(x,u^+,u^-,\\nu_u)d{\\mathcal{H}}^{N-1}\\,+ \\\\[-15pt]\n+ \\int_\\O f^\\infty\\bigg(x,\\tilde u,\\frac{dD^cu}{d|D^cu|}\\bigg)\\,\nd|D^cu|\n\\end{multlined}\n& \\text{\\it if }\\,u \\in BV(\\O;{\\mathcal{M}})\\,,\\\\\n& \\\\\n\\,+\\infty & \\text{\\it otherwise}\\,,\n\\end{cases}\n\\end{equation}\nwhere for every $(x,a,b,\\nu) \\in \\O \\times {\\mathcal{M}} \\times {\\mathcal{M}} \\times {\\mathbb{S}^{N-1}}$,\n\\begin{multline*}\nK(x,a,b,\\nu) := \\inf_\\varphi \\bigg\\{\\int_{Q_\\nu}\nf^\\infty(x,\\varphi(y),\\nabla \\varphi(y))\\, dy : \\varphi \\in\nW^{1,1}(Q_\\nu;{\\mathcal{M}}),\\; \\varphi=a \\text{ on } \\{x\\cdot \\nu=1\/2\\},\\\\\n\\varphi=b \\text{ on }\\{x\\cdot \\nu=-1\/2\\} \\text{ {\\it and} } \\varphi \\text{\n\\it is $1$-periodic in the }\\nu_2,\\ldots,\\nu_{N} \\text{ directions}\n\\bigg\\}\\,,\n\\end{multline*}\n$\\{\\nu,\\nu_2,\\ldots,\\nu_N\\}$ forms any orthonormal basis of ${\\mathbb{R}}^N$,\nand $Q_\\nu$ stands for the open unit cube in ${\\mathbb{R}}^N$ centered at the\norigin associated to this basis.\n\\end{theorem}\n\n\n\\noindent{\\bf Sketch of the Proof.} The proof of the lower bound\n``$\\geq$\" in \\eqref{reprel} can be obtained as in \\cite[Lemma~5.2]{BM} and Lemma \\ref{lowerboundBV} using standard techniques to\nhandle with the dependence on the space variable. The lower bounds\nfor the bulk and Cantor parts rely on the construction of a suitable\nfunction $\\tilde f: \\O \\times {\\mathbb{R}}^d \\times {\\mathbb{R}}^{d \\times N} \\to\n[0,+\\infty)$ replacing $f$ as we already pursued in Section\n\\ref{thbe}. On the other hand, the jump part rests on the projection\non ${\\mathcal{M}}$ of \\cite{HL} as in Proposition \\ref{proj} instead of the\nstandard projection on the sphere used in\n\\cite[Proposition~5.2]{AEL}. \\vskip5pt\n\n\\noindent\nTo obtain the upper bound, we localize as usual the functionals setting for every $u \\in\nL^1(\\O;{\\mathbb{R}}^d)$ and $A \\in {\\mathcal{A}}(\\O)$,\n$$F(u,A):=\\begin{cases}\n\\displaystyle \\int_A f(x,u,\\nabla u)\\, dx & \\text{ if }u \\in\nW^{1,1}(A;{\\mathcal{M}})\\,,\\\\\n+\\infty & \\text{ otherwise}\\,,\n\\end{cases}$$\n$$\\overline F(u,A):=\\inf_{\\{u_n\\}} \\left\\{ \\liminf_{n \\to +\\infty}\nF(u_n,A) : u_n \\to u \\text{ in }L^1(A;{\\mathbb{R}}^d)\\right\\}\\,.$$\nArguing as in the proof of Lemma \\ref{measbis}, we obtain that for every $u \\in BV(\\O;{\\mathcal{M}})$, the set function\n$\\overline F(u,\\cdot)$ is the restriction to ${\\mathcal{A}}(\\O)$ of a Radon\nmeasure absolutely continuous with respect to ${\\mathcal{L}}^N+|Du|$. Hence it uniquely extends into a Radon measure on $\\Omega$ (see Remark \\ref{measconstr}),\nand it suffices to prove that for any $u\\in BV(\\O;{\\mathcal{M}})$,\n\\begin{equation}\\label{jppartrel}\n\\overline F(u,\\Omega\\cap S_u)\\leq \\int_{\\Omega \\cap S_u}K(x,u^+,u^-,\\nu_u)\\, d{\\mathcal{H}}^{N-1}\\,,\n\\end{equation}\n\\begin{equation}\\label{contpartrel}\n\\frac{d\\overline F(u,\\cdot)}{d {\\mathcal{L}}^N}(x_0)\\leq f(x_0,u(x_0),\\nabla u(x_0))\\quad \\text{for ${\\mathcal{L}}^N$-a.e. $x_0\\in \\Omega$}\\,,\n\\end{equation}\n\\begin{equation}\\label{cantpartrel}\n\\frac{d\\overline F(u,\\cdot)}{d |D^cu|}(x_0)\\leq f^\\infty\\bigg(x_0,\\tilde u(x_0),\\frac{dD^c u}{d|D^cu|}(x_0)\\bigg)\\quad \\text{for $|D^cu|$-a.e. $x_0\\in \\Omega$}\\,,\n\\end{equation}\n\\vskip5pt\n\n\\noindent{\\it Proof of \\eqref{jppartrel}.} Concerning the jump part, one can proceed as in \\cite[Lemma~6.5]{AEL}.\nA slight difference lies in the third step of its proof where one needs to approximate in energy\nan arbitrary $u\\in BV(\\O;{\\mathcal{M}})$ by a sequence $\\{u_n\\}\\subset BV(\\O;{\\mathcal{M}})$ such that for each $n$, $u_n$ assumes a finite number of values.\nThis can be performed as in the proof of Corollary \\ref{upbdjp} using the regularity properties of $K$ stated in \\cite[Lemma~4.1]{AEL} for ${\\mathcal{M}}=\\mathbb{S}^{d-1}$.\n\\vskip5pt\n\n\n\\noindent{\\it Proof of \\eqref{contpartrel}.} Let $x_0 \\in \\O$ be a Lebesgue\npoint for $u$ and $\\nabla u$ such that $u(x_0) \\in {\\mathcal{M}}$, $\\nabla u(x_0)\n\\in [T_{u(x_0)}({\\mathcal{M}})]^N$,\n$$\\lim_{\\rho \\to 0^+} - \\hskip -1em \\int_{Q(x_0,\\rho)} |u(x) - u(x_0)|(1+|\\nabla\nu(x)|)\\, dx=0\\,,\\quad \\lim_{\\rho \\to 0^+}\\frac{|D^s\nu|(Q(x_0,\\rho))}{\\rho^N}=0\\,,$$ and\n$$\\frac{d |Du|}{d{\\mathcal{L}}^N}(x_0) \\quad \\text{ and }\\quad \\frac{d\\overline\nF(u,\\cdot)}{d{\\mathcal{L}}^N}(x_0)$$ exist and are finite. Note that\n${\\mathcal{L}}^N$-a.e. $x_0 \\in \\O$ satisfy these properties. We select a sequence $\\rho_k\n\\searrow 0^+$ such that $Q(x_0,2\\rho_k)\\subset \\O$ and $|Du|(\\partial Q(x_0,\\rho_k)) =0$ for\neach $k \\in {\\mathbb{N}}$. Next consider a sequence of standard mollifiers\n$\\{\\varrho_n\\}$, and define $u_n :=\\varrho_n * u \\in\nW^{1,1}(Q(x_0,\\rho_k);{\\mathbb{R}}^d) \\cap {\\mathcal{C}}^\\infty(Q(x_0,\\rho_k);{\\mathbb{R}}^d)$. In the sequel, we shall argue\nas in the proof of Proposition \\ref{proj} and we refer to it for the notation. Fix $\\d>0$ small\nenough such that $\\pi:{\\mathbb{R}}^d\\setminus X\\to {\\mathcal{M}}$ is smooth\nin the $\\d$-neighborhood of ${\\mathcal{M}}$. \nSince $u_n$ takes its values in ${\\rm co}({\\mathcal{M}})$, we can reproduce the proof of Proposition \\ref{proj} to find $a_n^k \\in{\\mathbb{R}}^d$ with $|a_n^k|<\\delta\/4$\nsuch that setting $p_n^k:=(\\pi_{a_n^k}|_{{\\mathcal{M}}})^{-1}\n\\circ \\pi_{a_n^k}$, $w_n^k:=p_n^k\\circ u_n \\in W^{1,1}(Q(x_0,\\rho_k);{\\mathcal{M}})$ and\n\\begin{equation}\\label{Ank}\n\\int_{A_n^k} |\\nabla w_n^k|\\, dx \\leq C_* \\int_{A_n^k}|\\nabla u_n|\\,\ndx\\,,\n\\end{equation}\nwhere $A_n^k$ denotes the open set $A_n^k:=\\big\\{x \\in Q(x_0,\\rho_k) : {\\rm dist}(u_n(x),{\\mathcal{M}}) >\\d \/2\\big\\}$. \nFurthermore, since $ \\pi$ is smooth in the $\\d$-neighborhood\nof ${\\mathcal{M}}$ and $|a_n^k|<\\d\/4$,\nthere exists a constant $C_\\d>0$ independent\nof $n$ and $k$ such that\n\\begin{equation}\\label{dn}\n|\\nabla^2 p_n^k(s)|+|\\nabla p_n^k(s)| \\leq C_\\d \\text{ for every $s \\in {\\mathbb{R}}^d$ satisfying\n$\\text{dist}(s,{\\mathcal{M}})\\leq \\d\/2$}\\,,\n\\end{equation}\nand consequently,\n\\begin{equation}\\label{cAnk}\n|\\nabla w_n^k| \\leq C_\\d |\\nabla u_n| \\quad \\text{${\\mathcal{L}}^N$-a.e. in\n}Q(x_0,\\rho_k) \\setminus A_n^k\\,.\n\\end{equation}\nSince $u(x) \\in {\\mathcal{M}}$ for ${\\mathcal{L}}^N$-a.e. $x \\in \\O$, it follows that\n$${\\mathcal{L}}^N(A_n^k) \\leq\n\\frac{2}{\\d} \\int_{Q(x_0,\\rho_k)} \\text{dist}(u_n,{\\mathcal{M}})\\, dx \\leq\n\\frac{2}{\\d} \\int_{Q(x_0,\\rho_k)} |u_n-u|\\, dx \\xrightarrow[n \\to\n+\\infty]{} 0\\,,$$\nand then (\\ref{dn}) yields\n\\begin{multline*}\n\\int_{Q(x_0,\\rho_k)}|w_n^k - u|\\, dx = \\int_{\nA_n^k}|w_n^k - u|\\, dx + \\int_{Q(x_0,\\rho_k)\n\\setminus A_n^k}|p_n^k(u_n) - p_n^k(u)|\\, dx\\leq\\\\\n \\leq {\\rm diam}({\\mathcal{M}}) {\\mathcal{L}}^N(A_n^k) + C_\\d\n\\int_{Q(x_0,\\rho_k)}|u_n-u|\\, dx \\xrightarrow[n \\to +\\infty]{} 0\\,.\n\\end{multline*}\nHence $w_n^k \\to u$ in $L^1(Q(x_0,\\rho_k);{\\mathbb{R}}^d)$ as $n \\to +\\infty$\nso that we are allowed to take $w_n^k$ as competitor, {\\it i.e.},\n$$\\overline F(u,Q(x_0,\\rho_k)) \\leq \\liminf_{n \\to +\\infty}\n\\int_{Q(x_0,\\rho_k)}f(x,w_n^k,\\nabla w_n^k)\\, dx\\,.$$\nAt this stage we can argue exactly\nas in \\cite[Lemma~6.4]{AEL} to prove that for any $\\eta>0$\nthere exists $\\lambda=\\lambda(\\eta)>0$ such that\n\\begin{multline}\\label{1numb}\n\\overline F(u,Q(x_0,\\rho_k)) \\leq \\liminf_{n \\to\n+\\infty}\\bigg\\{\\int_{Q(x_0,\\rho_k)}f(x_0,u(x_0),\\nabla u_n)\\,\ndx+C\\int_{Q(x_0,\\rho_k)}|\\nabla u_n - \\nabla w_n^k|\\, dx\\, +\\\\\n+ C(\\eta +\\lambda \\rho_k) \\int_{Q(x_0,\\rho_k)}(1+|\\nabla u_n|)\\, dx\n+C\\lambda\\int_{Q(x_0,\\rho_k)}|w_n^k-u(x_0)|(1+|\\nabla\nw_n^k|)\\, dx \\bigg\\}\\,.\n\\end{multline}\nThe first and third term in the right handside of \\eqref{1numb} can be treated as in the proof of \\cite[Theorem~2.16]{FM2}. \nConcerning the remaining terms, we proceed as follows. \nUsing\n(\\ref{Ank}), (\\ref{dn}) and (\\ref{cAnk}), we get that\n\\begin{multline}\\label{2numb}\n\\int_{Q(x_0,\\rho_k)}|w_n^k-u(x_0)||\\nabla w_n^k|\\, dx\n\\leq{\\rm diam}({\\mathcal{M}}) \\int_{A_n^k}|\\nabla w_n^k|\\, dx\\,+\\\\\n+ \\int_{Q(x_0,\\rho_k)\n\\setminus A_n^k}|p_n^k(u_n) - p_n^k(u(x_0))| |\\nabla w_n^k| \\, dx\n \\leq C\\int_{A_n^k}|\\nabla u_n|\\, dx\\,+\\\\\n+ C_\\d \\int_{Q(x_0,\\rho_k) \\setminus A_n^k}|u_n - u(x_0)| |\\nabla\nu_n|\n\\,dx \n\\leq C_\\d \\int_{Q(x_0,\\rho_k)}|u_n - u(x_0)| |\\nabla u_n|\\, dx\\,,\n\\end{multline}\nwhere $C_\\d>0$ still denotes some constant depending on $\\d$ but independent of $k$ and $n$. Arguing\nin a similar way, we also derive\n\\begin{equation}\\label{3}\n\\int_{Q(x_0,\\rho_k)}|\\nabla u_n - \\nabla w_n^k|\\, dx \\leq C_\\delta\n\\int_{Q(x_0,\\rho_k)} |u_n -u(x_0)| |\\nabla u_n|\\, dx+\n\\int_{Q(x_0,\\rho_k) \\setminus A_n^k} |L_n^k \\nabla u_n|\\, dx\\,,\n\\end{equation}\nwhere $L_n^k:={\\rm Id} - \\nabla p_n^k(u(x_0)) \\in {\\rm\nLin}({\\mathbb{R}}^{d\\times d},{\\mathbb{R}}^{d \\times d})$. \nGathering \\eqref{1numb}, \\eqref{2numb} and (\\ref{3}) we finally\nobtain that\n\\begin{multline}\\label{4}\n\\overline F(u,Q(x_0,\\rho_k)) \\leq \\liminf_{n \\to\n+\\infty}\\bigg\\{\\int_{Q(x_0,\\rho_k)}f(x_0,u(x_0),\\nabla u_n)\\,\ndx+C\\int_{Q(x_0,\\rho_k) \\setminus A_n^k} |L_n^k \\nabla u_n|\\, dx\\,+\\\\\n+C(\\eta +\\lambda \\rho_k) \\int_{Q(x_0,\\rho_k)}(1+|\\nabla u_n|)\\, dx\n+C_\\d\\lambda\\int_{Q(x_0,\\rho_k)}|u_n-u(x_0)|(1+|\\nabla u_n|)\\,\ndx \\bigg\\}\\,.\n\\end{multline}\nNow we can follow the argument in \\cite[Lemma~6.4]{AEL} to conclude that\n$$\\frac{d\\overline F(u,\\cdot)}{d{\\mathcal{L}}^N}(x_0) \\leq f(x_0,u(x_0),\\nabla\nu(x_0))\\,,$$\nwhich completes the proof of \\eqref{contpartrel}.\n\\vskip5pt\n\n\\noindent {\\it Proof of \\eqref{cantpartrel}.} Once again the proof\nparallels the one in \\cite[Lemma~6.4]{AEL}. We first proceed as in\nthe previous reasoning leading to \\eqref{4}. Then we can exactly\nfollow the argument of \\cite[Lemma~6.4]{AEL} to obtain\n\\eqref{cantpartrel}. \\prbox\n\n\n\\vskip15pt\n\n\\noindent{\\bf Acknowledgement. }The authors wish to thank Roberto\nAlicandro, Pierre Bousquet, Giovanni Leoni and Domenico Mucci for\nseveral interesting discussions on the subject. This work was initiated while\nV. Millot was visiting the department of {\\it Functional Analysis and Applications} at S.I.S.S.A.,\nhe thanks G. Dal Maso and the whole department for\nthe warm hospitality. The research of\nJ.-F. Babadjian was partially supported by the Marie Curie Research\nTraining Network MRTN-CT-2004-505226 ``Multi-scale modelling and\ncharacterisation for phase transformations in advanced materials''\n(MULTIMAT). V. Millot was partially supported by the Center for\nNonlinear Analysis (CNA) under the National Science Fundation Grant\nNo. 0405343.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:introduction}}\n\\else\n\\section{Introduction}\n\\label{sec:introduction}\n\\fi\n\n\\IEEEPARstart{O}{ne-class} novelty detection refers to the problem of determining if a test data sample is normal (known class) or anomalous (novel class). In real-world applications, novel data is difficult to collect since they are often rare or unsafe. Hence, one-class novelty detection considers training data from only a single known class. Most recent advances in one-class novelty detection are based on the deep Auto-Encoder (AE) style architectures, such as Denoising Auto-Encoder (DAE) \\cite{salehi2020arae,vincent2008extracting}, Variational Auto-Encoder (VAE) \\cite{kingma2013auto}, Adversarial Auto-Encoder (AAE) \\cite{makhzani2015adversarial, pidhorskyi2018generative}, Generative Adversarial Network (GAN) \\cite{goodfellow2014generative,perera2019ocgan,sabokrou2018adversarially,zhangp}, etc. Given an AE that learns the distribution of the known class, normal data are expected to be reconstructed accurately, while anomalous data are not. The reconstruction error of the AE is then used as a score for a test example to perform novelty detection. Although deep novelty detection methods achieve impressive performance, their robustness against adversarial attacks \\cite{goodfellow2015explaining,Szegedy2014Intriguing} lacks exploration.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.46\\textwidth]{cover_pic.pdf}\n\t\\caption{Overview of the proposed adversarially robust one-class novelty detection idea (PLS). The vanilla Auto-Encoder (AE) and AE+PLS are trained with the known class defined as digit 8. AE+PLS reconstructs every adversarial data into the known class (digit 8) and thus produces preferred reconstruction errors for novelty detection, even under attacks.}\n\t\\label{fig:cover_pic}\n\\end{figure}\n\nAdversarial examples pose serious security threats to deep networks as they can fool them with carefully crafted perturbations. Over the past few years, many adversarial attack and defense approaches have been proposed for tasks such as image classification \\cite{guo2017countering,raff2019barrage,Xie_2019_CVPR,xu2017feature}, video recognition \\cite{lo2020defending,wei2019sparse}, optical flow estimation \\cite{ranjan2019attacking} and open-set recognition \\cite{shao2020open}. However, adversarial attacks or defenses have not been thoroughly investigated in the context of one-class novelty detection. We first show that present novelty detectors are vulnerable to adversarial attacks. Subsequently, we demonstrate that many state-of-the-art defenses \\cite{hendrycks2019selfsupervised,shi2021online,xie2020smooth,Xie_2019_CVPR} prove to be sub-optimal to properly defend novelty detectors against adversarial examples. This motivates us to design an effective defense strategy specifically for one-class novelty detection.\n\nTo this end, we propose to leverage task-specific knowledge to protect novelty detectors. These novelty detectors are only required to retain information about normal data, thereby resulting in poor reconstructions for anomalous data. This is favorable to the novelty detection problem. This can be achieved by constraining the latent space to make the features closer to a prior distribution \\cite{perera2019ocgan,park2020learning}. Also, it has been shown that adversarial perturbations can be removed in the feature space \\cite{Xie_2019_CVPR}. Therefore, one can largely manipulate the latent space of novelty detectors to devoid them of feature corruption introduced by adversaries, while maintaining the performance on clean input data. This property is unique to the novelty detection task, as most deep learning applications (e.g., image classification) require a model containing sophisticated semantic information, and a large manipulation on the latent space may limit the model capability, resulting in performance degradation.\n\nIn this paper, we propose a defense strategy, referred to as Principal Latent Space (PLS), to defend novelty detectors against adversarial examples. Specifically, PLS learns the incrementally-trained \\cite{ross2008incremental} cascade principal components in the latent space. This contains a cascade principal component analysis (PCA), which consists of a PCA operating on the vector dimension (i.e., channel) of a latent space \\cite{van2017neural} and the other PCA operating on the spatial dimension. We name these two PCAs as \\textit{Vector-PCA} and \\textit{Spatial-PCA}, respectively. First, Vector-PCA uses a learned \\textit{principal latent vector} to represent a latent space as the Vector-PCA space of a single-channel map. Since the principal latent vector is a pre-trained component that would not be affected by adversarial perturbations, most adversaries are removed at this step, and the remaining adversaries are enclosed within the small \\textit{Vector-PCA space}. Subsequently, Spatial-PCA uses learned \\textit{principal Vector-PCA maps} to represent the Vector-PCA space as the \\textit{Spatial-PCA space} and expel the remaining adversaries. Finally, the corresponding cascade inverse PCA transforms the Spatial-PCA space back to the original dimensionality, resulting in the \\textit{principal latent space}.\n\nWith PLS, the decoder could compute preferred reconstruction errors as novelty scores, even under adversarial attacks (see Fig.~\\ref{fig:cover_pic}). Additionally, we incorporate adversarial training (AT) \\cite{madry2018towards} with PLS to further exert PLS's ability. In contrast to typical defenses which often sacrifice their performance on clean data \\cite{tsipras2018robustness,xie2020adversarial}, the proposed defense strategy does not hurt the performance but rather improves it. The PLS module can be attached to any AE-style architectures (VAE, GAN, etc), so it is applicable to a wide variety of the existing novelty detection approaches, such as \\cite{kingma2013auto,makhzani2015adversarial,sabokrou2018adversarially,pidhorskyi2018generative,salehi2020arae} etc. We extensively evaluate PLS on eight adversarial attacks, three datasets and six different novelty detectors. We further compare PLS with commonly-used defense methods and show that it consistently enhances the adversarial robustness of novelty detectors by significant margins. To the best of our knowledge, this is one of the first adversarially robust novelty detection methods.\n\n\n\\section{Related work} \\label{sec2}\n\\noindent \\textbf{One-class novelty detection.}\nOne-class novelty detection is of great interest to the computer vision community. Earlier algorithms mainly rely on Support Vector Machines (SVM) formulation \\cite{scholkopf1999support,tax2004support}. With the advent of deep learning, AE-based approaches are dominating this area and achieve state-of-the-art performance \\cite{gong2019memorizing,park2020learning,perera2019ocgan,pidhorskyi2018generative,sabokrou2018adversarially,sakurada2014anomaly,salehi2020arae,xia2015learning,zhou2017anomaly}. ALOCC \\cite{sabokrou2018adversarially} considers a DAE \\cite{vincent2008extracting} as a generator and appends a discriminator to train the entire network by the generative adversarial framework \\cite{goodfellow2014generative}. GPND \\cite{pidhorskyi2018generative} is based on AAE \\cite{makhzani2015adversarial}, and it applies a discriminator to the latent space and the other discriminator to the output. OCGAN \\cite{perera2019ocgan} includes two discriminators and a classifier to train a DAE by the generative adversarial framework. ARAE \\cite{salehi2020arae} crafts adversarial examples from the latent space to adversarially train a DAE. Different from our work, ARAE's adversarial examples aim to pursue performance, and its adversarial robustness is not thoroughly evaluated (see Supplementary).\n\n\\noindent \\textbf{Adversarial attacks.}\nSzegedy et al. \\cite{Szegedy2014Intriguing} showed that carefully crafted perturbations can fool deep networks. Goodfellow et al. \\cite{goodfellow2015explaining} introduced the Fast Gradient Sign Method (FGSM), which leverages the sign of gradients to produce adversarial examples. Projected Gradient Descent (PGD) \\cite{madry2018towards} extends FGSM from single iteration gradient descent to an iterative version. MI-FGSM \\cite{dong2018boosting} generates more transferable adversarial attacks by a momentum mechanism. MultAdv \\cite{lo2020multav} produces adversarial examples via the multiplicative operation instead of the additive operation. Physically realizable attacks, which can be implemented in the physical scenarios, is also developed \\cite{Sharif16AdvML,zajac2019adversarial}. For example, Adversarial Framing (AF) \\cite{zajac2019adversarial} adds perturbations on the border of an image, while the remaining pixels are unchanged.\n\n\\noindent \\textbf{Adversarial defenses.}\nAt earlier time, a few studies aim to detect adversarial examples \\cite{hendrycks2016early,jere2020principal,li2017adversarial}. However, it is well-known that detection is inherently weaker than defense in terms of resisting adversarial attacks. Although several defense approaches based on image transformation are proposed afterward \\cite{guo2017countering,xu2017feature,bhagoji2017dimensionality}, they fail to defend against white-box attacks \\cite{carlini2017adversarial,obfuscated}. Recently, Adversarial Training (AT) has been considered one of the most effective defenses, especially in the white-box setting. Madry et al. \\cite{madry2018towards} formulated AT in a min-max optimization framework (PGD-AT), and this has been widely used as a benchmark. Xie et al. \\cite{Xie_2019_CVPR} includes the feature denoising block (FD) in networks to remove adversarial perturbations in the feature domain. SAT \\cite{xie2020smooth} uses smooth approximations of ReLU activation to enhance PGD-AT. Hendrycks et al. \\cite{hendrycks2019selfsupervised} added an auxiliary rotation prediction task \\cite{gidaris2018unsupervised} to improve PGD-AT (RotNet-AT). SOAP \\cite{shi2021online} takes self-supervised signals to purify adversarial examples during inference.\n\nTo the best of our knowledge, APAE \\cite{goodge2020robustness} might be the only present defense designed for anomaly detection. It uses approximate projection and feature weighting to reduce adversarial effects. However, its robustness is not fully tested and only anomalous data are perturbed in its evaluation (see Supplementary). Instead, we provide a generic framework for evaluating the adversarial robustness of novelty detectors and our proposed defense method.\n\n\n\\section{Attacking novelty detection models} \\label{sec3}\n\nWe consider several popular adversarial attacks \\cite{dong2018boosting,goodfellow2015explaining,lo2020multav,madry2018towards,papernot2017practical,zajac2019adversarial} and modify their loss objectives to suit the novelty detection problem setup. Here, we take PGD \\cite{madry2018towards} as an example to illustrate our attack formulation. The other gradient-based attacks can be extended by a similar formulation (see Supplementary).\n\nConsider an AE-based target model with an encoder $Enc$ and a decoder $Dec$, and an input image $\\mathbf{X}$ with the ground-truth label $y \\in \\{-1, 1\\}$, where ``$1$\" denotes the known class and ``$-1$\" denotes the novel classes. We generate the adversarial example $\\mathbf{X}_{adv}$ as follows:\n\\begin{equation}\n\\label{pgd_attack}\n\\mathbf{X}^{t+1} = Proj^{L_\\infty}_{\\mathbf{X}, \\ \\epsilon} \\big\\{ \\mathbf{X}^{t} + \\alpha \\cdot sign(\\bigtriangledown_{\\mathbf{X}^t} \\mathcal{L}(\\hat{\\mathbf{X}}^t, \\mathbf{X}^t, y)) \\big\\},\n\\end{equation}\nwhere, $\\hat{\\mathbf{X}}^t = Dec(Enc(\\mathbf{X}^t))$, $\\alpha>0$ denotes a step size, and $t \\in [0,t_{max}-1]$ is the number of attacking iterations, $\\mathbf{X} = \\mathbf{X}^0$ and $\\mathbf{X}_{adv} = \\mathbf{X}^{t_{max}}$. $Proj^{L_\\infty}_{\\mathbf{X},\\epsilon}\\{\\cdot\\}$ projects its element into an $L_\\infty$-norm bound with perturbation size $\\epsilon$ such that $\\parallel \\mathbf{X}^{t+1} - \\mathbf{X} \\parallel_{\\infty} \\leq \\epsilon$. $\\mathcal{L}$ corresponds to the mean square error (MSE) loss defined as follows:\n\\begin{equation}\n\\label{mse_loss}\n\\mathcal{L}(\\hat{\\mathbf{X}}^t, \\mathbf{X}^t, y) = y \\parallel \\hat{\\mathbf{X}}^t - \\mathbf{X}^t \\parallel_2.\n\\end{equation}\nGiven a test example, if it belongs to the known class, we maximize its reconstruction error (i.e., novelty score) by gradient ascent; while if it belongs to novel classes, we minimize its reconstruction error by gradient descent.\n\nPresent novelty detection methods are vulnerable to this attack (see Sec.~\\ref{robustness}); that is, normal data would be misclassified into novel classes, and anomalous data would be misclassified into the known class. Moreover, this attacking strategy is much stronger than the attacks introduced by \\cite{salehi2020arae}, which perturbs only normal data, and by \\cite{goodge2020robustness}, which perturbs only anomalous data (see Supplementary).\n\n\n\\section{Adversarially robust novelty detection} \\label{method}\nThe proposed defense strategy exploits the task-specific knowledge of one-class novelty detection. Specifically, we leverage the fact that a novelty detector's latent space can be manipulated to a larger extent as long as it retains the known class information. This property is especially useful to remove more adversarial perturbations in the latent space. Therefore, we propose to train a novelty detector by manipulating its latent space such that it can improve adversarial robustness while maintaining the performance on clean data. Note that these characteristics are specific to the novelty detection problem. The majority of visual recognition problems, such as image classification, require a model retaining multiple category information. Hence, a large manipulation on the latent space may hinder the model capability and thus degrade the performance.\n\nIn the following subsections, we first briefly review PCA to define the notations used in this paper, then discuss the proposed PLS in detail.\n\n\\subsection{Preliminary}\nPCA computes the principal components of a collection of data and uses them to conduct a change of basis on the data through a linear transformation. Consider a data matrix $\\mathbf{X} \\in \\mathbb{R}^{n \\times d}$, its mean $\\bm{\\mu} \\in \\mathbb{R}^{1 \\times d}$ and its covariance $\\mathbf{C} = (\\mathbf{X}-\\bm{\\mu})^\\top (\\mathbf{X}-\\bm{\\mu})$. $\\mathbf{C}$ can be written as $\\mathbf{C} = \\mathbf{U} \\bm{\\Lambda} \\mathbf{V}^\\top$ via Singular Vector Decomposition (SVD), where $\\mathbf{U} \\in \\mathbb{R}^{d \\times d}$ is an orthogonal matrix containing the principal components of $\\mathbf{X}$. Here, we define a mapping $h$ which computes the mean vector and the first $k$ principal components of the given $\\mathbf{X}$:\n\\begin{equation}\n\\label{hhh}\nh(\\mathbf{X}, k): \\mathbf{X} \\to \\{\\bm{\\mu}, \\tilde{\\mathbf{U}}\\},\n\\end{equation}\nwhere $\\tilde{\\mathbf{U}} \\in \\mathbb{R}^{d \\times k}$ keeps only the first $k$ columns of $\\mathbf{U}$. Now we define the forward and the inverse PCA transformation as a pair of mapping $(f: \\mathbb{R}^{n \\times d} \\to \\mathbb{R}^{n \\times k}$, $g: \\mathbb{R}^{n \\times k} \\to \\mathbb{R}^{n \\times d})$; $f$ performs the forward PCA: \n\\begin{equation}\n\\label{fff}\nf(\\mathbf{X}; \\bm{\\mu}, \\tilde{\\mathbf{U}}) = (\\mathbf{X}-\\bm{\\mu}) \\tilde{\\mathbf{U}},\n\\end{equation}\nand $g$ performs the inverse PCA:\n\\begin{equation}\n\\label{ggg}\ng(\\mathbf{X}_{PCA}; \\bm{\\mu}, \\tilde{\\mathbf{U}}) = \\mathbf{X}_{PCA} \\tilde{\\mathbf{U}}^\\top + \\bm{\\mu},\n\\end{equation}\nwhere $\\mathbf{X}_{PCA} = f(\\mathbf{X}; \\bm{\\mu}, \\tilde{\\mathbf{U}})$. Finally, we can write the PCA reconstruction of $\\mathbf{X}$ as $\\hat{\\mathbf{X}} = g(f(\\mathbf{X}; \\bm{\\mu}, \\tilde{\\mathbf{U}}); \\bm{\\mu}, \\tilde{\\mathbf{U}})$.\n\n\\subsection{Principal Latent Space (PLS)}\nThe proposed PLS contains two major components: (1) Vector-PCA and (2) Spatial-PCA. In Vector-PCA, we perform $(h, f, g)$ on the vector dimension as $(h_V, f_V, g_V)$, and in Spatial-PCA, we perform $(h, f, g)$ on the spatial dimension as $(h_S, f_S, g_S)$. Let $Enc$ be the encoder and $Dec$ be the decoder of a novelty detection model. Let us denote an adversarial image as $\\mathbf{X}_{adv}$, we have its latent space $\\mathbf{Z}_{adv} = Enc(\\mathbf{X}_{adv}) \\in \\mathbb{R}^{s \\times v}$, where $s = h \\times w$ is the spatial dimensionality obtained by the product of height and width, and $v$ is the vector dimensionality (i.e., the number of channels). Under adversarial attacks, $\\mathbf{Z}_{adv}$ would be corrupted by adversarial perturbations such that the decoder cannot compute reconstruction errors favorable to novelty detection. We define the proposed PLS as a transformation $PLS: \\mathbf{Z}_{adv} \\to \\mathbf{Z}_{PLS}$, which removes adversaries from $\\mathbf{Z}_{adv}$, where $\\mathbf{Z}_{PLS}$ is referred to as principal latent space. $PLS$ is implemented by our incrementally-trained cascade PCA. In the beginning, a sigmoid function replaces the encoder's last activation function to bound $\\mathbf{Z}_{adv}$ values between 0 and 1. The following procedures are described below.\n\nFirst, Vector-PCA computes the mean latent vector and the principal latent vector of $\\mathbf{Z}_{adv}$:\n\\begin{equation}\n\\label{hv}\n\\{\\bm{\\mu}_V, \\tilde{\\mathbf{U}}_V\\} = h_V(\\mathbf{Z}_{adv}, k_V=1),\n\\end{equation}\nwhere, we always set $k_V$ to 1, so $\\tilde{\\mathbf{U}}_V$ is the first principal latent vector of $\\mathbf{Z}_{adv}$. Second, Vector-PCA transforms $\\mathbf{Z}_{adv}$ to its Vector-PCA space $\\mathbf{Z}_V \\in \\mathbb{R}^{s \\times 1}$:\n\\begin{equation}\n\\label{fv}\n\\mathbf{Z}_V = f_V(\\mathbf{Z}_{adv}; \\bm{\\mu}_V, \\tilde{\\mathbf{U}}_V).\n\\end{equation}\nNext, Spatial-PCA computes the mean Vector-PCA map\\footnote{We use the word ``map\" to indicate they are on the spatial dimension.} and the principal Vector-PCA maps of $\\mathbf{Z}_V$:\n\\begin{equation}\n\\label{hs}\n\\{\\bm{\\mu}_S, \\tilde{\\mathbf{U}}_S\\} = h_S(\\mathbf{Z}_V^\\top, k_S),\n\\end{equation}\nwhere, $k_S$ is a hyperparameter. Then, Spatial-PCA transforms $\\mathbf{Z}_V$ to its Spatial-PCA space $\\mathbf{Z}_S \\in \\mathbb{R}^{k_S \\times 1}$:\n\\begin{equation}\n\\label{fs}\n\\mathbf{Z}_S^\\top = f_S(\\mathbf{Z}_V^\\top; \\bm{\\mu}_S, \\tilde{\\mathbf{U}}_S).\n\\end{equation}\nFinally, the inverse Spatial-PCA and the inverse Vector-PCA transform $\\mathbf{Z}_S$ back to its original dimensionality:\n\\begin{equation}\n\\label{gs}\n\\hat{\\mathbf{Z}}_V^\\top = g_S(\\mathbf{Z}_S^\\top; \\bm{\\mu}_S, \\tilde{\\mathbf{U}}_S),\n\\end{equation}\n\\begin{equation}\n\\label{gv}\n\\mathbf{Z}_{PLS} = g_V(\\hat{\\mathbf{Z}}_V; \\bm{\\mu}_V, \\tilde{\\mathbf{U}}_V),\n\\end{equation}\nwhere, $\\hat{\\mathbf{Z}}_V$ is the Spatial-PCA reconstruction of $\\mathbf{Z}_V$, and $\\mathbf{Z}_{PLS}$ is the resulting principal latent space. Fig.~\\ref{fig:big_pic} gives an overview of this procedure. The decoder then uses $\\mathbf{Z}_{PLS}$ to reconstruct the input adversarial example as $\\hat{\\mathbf{X}}_{adv} = Dec(\\mathbf{Z}_{PLS})$ for computing the novelty score.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.46\\textwidth]{big_pic.pdf}\n\t\\caption{Overview of the proposed PLS. $f_V$: forward Vector-PCA, $f_S$: forward Spatial-PCA, $g_S$: inverse Spatial-PCA, $g_V$: inverse Vector-PCA, $h_V$ and $h_S$ are the mappings for computing principal components.}\n\t\\label{fig:big_pic}\n\\end{figure}\n\n\\subsection{Incremental training}\nThe \\textit{principal latent components} $\\{\\bm{\\mu}_V, \\tilde{\\mathbf{U}}_V, \\bm{\\mu}_S, \\tilde{\\mathbf{U}}_S\\}$ are incrementally-trained along with the network weights by exponential moving average (EMA) during training, so we call this process incrementally-trained cascade PCA. Specifically, at training iteration $t$, these components are updated with following equations:\n\\begin{equation}\n\\label{train_V}\n\\{\\bm{\\mu}_V^t, \\tilde{\\mathbf{U}}_V^t\\} = (1-\\eta_V) \\{\\bm{\\mu}_V^{t-1}, \\tilde{\\mathbf{U}}_V^{t-1}\\} + \\eta_V \\cdot h_V(\\mathbf{Z}_{adv}^t),\n\\end{equation}\n\\begin{equation}\n\\label{train_S}\n\\{\\bm{\\mu}_S^t, \\tilde{\\mathbf{U}}_S^t\\} = (1-\\eta_S) \\{\\bm{\\mu}_S^{t-1}, \\tilde{\\mathbf{U}}_S^{t-1}\\} + \\eta_S \\cdot h_S(\\mathbf{Z}_V^{t \\top}),\n\\end{equation}\nwhere $\\eta_V$ and $\\eta_S$ are the EMA learning rates.\n\nConsider the model weights are trained by the mini-batch gradient descent with a batch size $b$, the latent dimensionality is shaped to $\\mathbf{Z}_{adv} \\in \\mathbb{R}^{bs \\times v}$, the resulting $\\mathbf{Z}_V \\in \\mathbb{R}^{bs \\times 1}$ is reshaped to $\\mathbf{Z}_V \\in \\mathbb{R}^{s \\times b}$ after the Vector-PCA $f_V$, and $\\hat{\\mathbf{Z}}_V \\in \\mathbb{R}^{s \\times b}$ is reshaped back to $\\hat{\\mathbf{Z}}_V \\in \\mathbb{R}^{bs \\times 1}$ after the inverse Spatial-PCA $g_S$. Hence, in a mini-batch, both $h_V$ and $h_S$ have $b$ times more data points to acquire better principal latent components at each training iteration. At iteration $t$, $(f_V, g_V)$ performs with the components $\\{\\bm{\\mu}_V^t, \\tilde{\\mathbf{U}}_V^t\\}$, and $(f_S, g_S)$ performs with the components $\\{\\bm{\\mu}_S^t, \\tilde{\\mathbf{U}}_S^t\\}$. When the training process ends, the well-trained components are denoted as $\\{\\bm{\\mu}_V^*, \\tilde{\\mathbf{U}}_V^*, \\bm{\\mu}_S^*, \\tilde{\\mathbf{U}}_S^*\\}$. During infernce, $(f_V, g_V)$ performs with $\\{\\bm{\\mu}_V^*, \\tilde{\\mathbf{U}}_V^*\\}$, and $(f_S, g_S)$ performs with $\\{\\bm{\\mu}_S^*, \\tilde{\\mathbf{U}}_S^*\\}$, while $h_V$ and $h_S$ do not operate (see Fig.~\\ref{fig:big_pic}). The entire process is differentiable during inference and thus does not cause obfuscated gradients \\cite{obfuscated}. This incremental training helps make sure the cascade PCA is aware of the network weight updates at each training step, encouraging mutual learning between the network weights and the principal latent components. The entire model and thus can be trained end-to-end.\n\n\\subsection{Defense mechanism}\nWe further elaborate on how the proposed PLS defends against adversarial attacks. Given an adversarial example $\\mathbf{X}_{adv}$, its latent space $\\mathbf{Z}_{adv}$ is adversarially perturbed. After Vector-PCA, each latent vector of $\\mathbf{Z}_{adv}$ is represented by a scaling factor of the learned principal latent vector $\\tilde{\\mathbf{U}}_V^*$ (with a bias term $\\bm{\\mu}_V^*$). The Vector-PCA space $\\mathbf{Z}_V$ stores these scaling factors on a single-channel map (i.e., on the spatial domain only). Since all the principal latent components are pre-trained parameters, they would not be affected by adversarial perturbations. Replacing the perturbed latent vectors by $\\tilde{\\mathbf{U}}_V^*$ removes the majority of the adversaries. The only place where the remaining adversaries can appear is the scaling factors of $\\tilde{\\mathbf{U}}_V^*$ on the single-channel map. In other words, these adversaries are enclosed within a small subspace, making them easier to expel.\n\nSubsequently, Spatial-PCA reconstructs this small subspace by a set of principal Vector-PCA maps $\\tilde{\\mathbf{U}}_S^*$ (with a bias term $\\bm{\\mu}_S^*$). Since $\\tilde{\\mathbf{U}}_S^*$ and $\\bm{\\mu}_S^*$ are adversary-free, the remaining adversaries are further removed. From another perspective, this step can be viewed as PCA-based denoising performing in the spatial domain of features. With the robust principal latent space $\\mathbf{Z}_{PLS}$, the decoder can obtain a preferred reconstruction error for novelty detection, even in the presence of an adversarial example. Additionally, we perform AT \\cite{madry2018towards} to train the model, further improving the robustness.\n\n\n\n\\section{Experiments}\n\nWe evaluate PLS on eight adversarial attacks, three datasets and six existing novelty detection methods. We further compare PLS with state-of-the-art defense approaches. An extensive ablation study is also presented. \n\n\\renewcommand{\\arraystretch}{0.6}\n\\setlength{\\tabcolsep}{4.5pt}\n\\begin{table*}[htp!]\n\t\\begin{center}\n\t\t\\caption{The mAUROC of models under various adversarial attacks.}\n\t\t\\label{table:main_results_1}\n\t\t\\begin{tabular}{r | r | c | ccccc | c}\n\t\t\t\\hline \\noalign{\\smallskip} \\noalign{\\smallskip}\n\t\t\tDataset & Defense & Clean & FGSM \\cite{goodfellow2015explaining} & PGD \\cite{madry2018towards} & MI-FGSM \\cite{dong2018boosting} & MultAdv \\cite{lo2020multav} & AF \\cite{zajac2019adversarial} & Black-box \\cite{papernot2017practical} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\t\t\t& No Defense & 0.964 & 0.350 & 0.051 & 0.022 & 0.170 & 0.014 & 0.790 \\\\\n\t\t\t\\noalign{\\smallskip} \\cline{2-9} \\noalign{\\smallskip}\n\t\t\t& PGD-AT \\cite{madry2018towards} & 0.961 & 0.604 & 0.357 & 0.369 & 0.444 & 0.155 & 0.691 \\\\\n\t\t\t& FD \\cite{Xie_2019_CVPR} & 0.963 & 0.612 & 0.366 & 0.379 & 0.453 & 0.142 & 0.700 \\\\\n\t\t\tMNIST & SAT \\cite{xie2020smooth} & 0.947 & 0.527 & 0.295 & 0.306 & 0.370 & 0.142 & 0.652 \\\\\n\t\t\t\\cite{lecun2010mnist} & RotNet-AT \\cite{hendrycks2019selfsupervised} & \\textbf{0.967} & 0.598 & 0.333 & 0.333 & 0.424 & 0.101 & 0.695 \\\\\n\t\t\t& SOAP \\cite{shi2021online} & 0.940 & 0.686 & 0.504 & 0.506 & 0.433 & 0.088 & \\textbf{0.863} \\\\\n\t\t\t& APAE \\cite{goodge2020robustness} & 0.925 & 0.428 & 0.104 & 0.105 & 0.251 & 0.022 & 0.730 \\\\\n\t\t\t& PLS (ours) & \\textbf{0.967} & \\textbf{0.786} & \\textbf{0.678} & \\textbf{0.679} & \\textbf{0.701} & \\textbf{0.599} & 0.840 \\\\\n\t\t\t\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\t\t\t& No Defense & 0.892 & 0.469 & 0.088 & 0.047 & 0.148 & 0.112 & 0.562 \\\\\n\t\t\t\\noalign{\\smallskip} \\cline{2-9} \\noalign{\\smallskip}\n\t\t\t& PGD-AT \\cite{madry2018towards} & 0.890 & 0.518 & 0.368 & 0.348 & 0.327 & 0.253 & 0.540 \\\\\n\t\t\t& FD \\cite{Xie_2019_CVPR} & 0.886 & 0.524 & 0.379 & 0.359 & 0.335 & 0.252 & 0.535 \\\\\n\t\t\tF-MNIST & SAT \\cite{xie2020smooth} & 0.878 & 0.444 & 0.306 & 0.285 & 0.273 & 0.231 & 0.492 \\\\\n\t\t\t\\cite{xiao2017fashion} & RotNet-AT \\cite{hendrycks2019selfsupervised} & 0.891 & 0.527 & 0.375 & 0.351 & 0.312 & 0.240 & 0.541 \\\\\n\t\t\t& SOAP \\cite{shi2021online} & 0.876 & 0.639 & 0.475 & 0.475 & 0.327 & 0.274 & 0.611 \\\\\n\t\t\t& APAE \\cite{goodge2020robustness} & 0.861 & 0.510 & 0.174 & 0.174 & 0.220 & 0.135 & 0.513 \\\\\n\t\t\t& PLS (ours) & \\textbf{0.909} & \\textbf{0.677} & \\textbf{0.600} & \\textbf{0.585} & \\textbf{0.573} & \\textbf{0.591} & \\textbf{0.696} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\t\t\t& No Defense & 0.550 & 0.186 & 0.034 & 0.018 & 0.025 & 0.035 & 0.227 \\\\\n\t\t\t\\noalign{\\smallskip} \\cline{2-9} \\noalign{\\smallskip}\n\t\t\t& PGD-AT \\cite{madry2018towards} & 0.546 & 0.236 & 0.145 & 0.139 & 0.107 & 0.096 & 0.223 \\\\\n\t\t\t& FD \\cite{Xie_2019_CVPR} & 0.546 & 0.237 & 0.147 & 0.141 & 0.109 & 0.103 & 0.222 \\\\\n\t\t\tCIFAR-10 & SAT \\cite{xie2020smooth} & 0.537 & 0.223 & 0.141 & 0.135 & 0.101 & 0.079 & 0.219 \\\\\n\t\t\t\\cite{krizhevsky2009learning} & RotNet-AT \\cite{hendrycks2019selfsupervised} & 0.547 & 0.236 & 0.139 & 0.107 & 0.075 & 0.092 & 0.224 \\\\\n\t\t\t& SOAP \\cite{shi2021online} & 0.546 & 0.270 & 0.131 & 0.141 & 0.096 & 0.070 & 0.231 \\\\\n\t\t\t& APAE \\cite{goodge2020robustness} & 0.552 & 0.259 & 0.097 & 0.097 & 0.077 & 0.112 & 0.255 \\\\\n\t\t\t& PLS (ours) & \\textbf{0.578} & \\textbf{0.320} & \\textbf{0.245} & \\textbf{0.242} & \\textbf{0.201} & \\textbf{0.243} & \\textbf{0.331} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table*}\n\n\n\\subsection{Experimental setup} \\label{sec51}\n\n\\noindent \\textbf{Datasets.}\nWe use three datasets for evaluation: MNIST \\cite{lecun2010mnist}, Fashion-MNIST (F-MNIST) \\cite{xiao2017fashion} and CIFAR-10 \\cite{krizhevsky2009learning}. MNIST consists of 28 $\\times$ 28 grayscale handwritten digits from 0 to 9. It contains 60,000 training data and 10,000 test data. F-MNIST is composed of 28 $\\times$ 28 grayscale images from 10 fashion product categories. It comprises of 60,000 training data and 10,000 test data. CIFAR-10 consists of 32 $\\times$ 32 color images from 10 different classes. There are 50,000 training and 10,000 test images in this dataset.\n\n\\noindent \\textbf{Evaluation protocol.}\nWe simulate a one-class novelty detection scenario by the following protocol. Given a dataset, each class is defined as the known class at a time, and a model is trained with the training data of this known class. During inference, the test data of the known class are considered normal, and the test data of the other classes (i.e., novel classes) are considered anomalous. We select the anomalous data from each novel class equally to constitute half of the test set, where the anomalous data within a novel class are selected randomly. Hence, our test set contains 50\\% anomalous data, where each novel class accounts for the same proportion. The area under the Receiver Operating Characteristic curve (AUROC) value is used as the evaluation metric, where the ROC curve is obtained by varying the threshold of the novelty score. For each dataset, we report the mean AUROC (mAUROC) across its 10 classes.\n\n\\noindent \\textbf{Attack setting.}\nWe test adversarial robustness against five white-box attacks, inclduing FGSM \\cite{goodfellow2015explaining}, PGD \\cite{madry2018towards}, MI-FGSM \\cite{dong2018boosting}, MultAdv \\cite{lo2020multav} and AF \\cite{zajac2019adversarial}, where PGD is the default attack if not otherwise specified. A black-box attack and two adaptive attacks \\cite{papernot2017practical,tramer2020adaptive} are also considered. All the attacks are implemented based on the formulation in Sec.~\\ref{sec3}.\n\nFor FGSM, PGD and MI-FGSM, we set $\\epsilon$ to $25\/255$ for MNIST, $16\/255$ for F-MNIST, and $8\/255$ for CIFAR-10. For MultAdv, we set $\\epsilon_m$ to $1.25$ for MNIST, $1.16$ for F-MNIST, and $1.08$ for CIFAR-10. For AF, we set $\\epsilon$ to $160\/255$, $120\/255$ and $80\/255$ for MNIST, F-MNIST and CIFAR-10, respectively. The framing width $w_{AF}$ is set to $1$. The number of attack iterations $t_{max}$ is set to $1$ for FGSM and $5$ for the other attacks.\n\n\\noindent \\textbf{Baseline defenses.}\nTo the best of our knowledge, APAE \\cite{goodge2020robustness} might be the only present defense designed for anomaly detection. In addition to APAE, we implement five commonly-used defenses, which are originally designed for classification tasks, in the context of novelty detection. They are PGD-AT \\cite{madry2018towards}, FD \\cite{Xie_2019_CVPR}, SAT \\cite{xie2020smooth}, RotNet-AT \\cite{hendrycks2019selfsupervised} and SOAP \\cite{shi2021online}, where FD, SAT and RotNet-AT incorporate PGD-AT. We use Gaussian non-local means \\cite{buades2005non} for FD, Swish \\cite{hendrycks2016gaussian} for SAT, and RotNet \\cite{gidaris2018unsupervised} for SOAP. These are their well-performing versions.\n\n\\noindent \\textbf{Benchmark novelty detectors.}\nWe apply PLS to six novelty detection methods, including a vanilla AE, VAE \\cite{kingma2013auto}, AAE \\cite{makhzani2015adversarial}, ALOCC \\cite{sabokrou2018adversarially}, GPND \\cite{pidhorskyi2018generative} and ARAE \\cite{salehi2020arae}, where the vanilla AE is the default novelty detector if not otherwise specified. PLS is added after the last layer of the novelty detection models' encoder.\n\nIn order to evenly evaluate the adversarial robustness of these approaches, we unify their AE backbones into the following archirecture. The encoder consists of four 3 $\\times$ 3 convolutional layers, where each of the first three layers are followed by a 2 $\\times$ 2 max-pooling with stride 2. We use a base channel size of 64, and increase the number of channels by a factor of 2. The decoder mirrors the encoder but replaces every max-pooling by a bilinear interpolation with a factor of 2. All the convolutional layers are followed by a batch normalization layer \\cite{ioffe2015batch} and ReLU. \n\n\n\\noindent \\textbf{Implementation details.}\nAll the models are trained by Adam optimizer \\cite{kingma2014adam} with initial learning rate $5e^{-5}$ and weight decay $1e^{-4}$, where the learning rate is decreased by a factor of 10 at the 20th and 40th epochs. The batch size is 128. For PLS, we set $k_V$ to $1$, $k_S$ to $8$, initial $\\eta_V$ to $0.1$ and initial $\\eta_S$ to $0.001$, where $\\eta_V$ and $\\eta_S$ are also decreased by a factor of 10 at the 20th and 40th epochs.\n\n\\setlength{\\tabcolsep}{5.5pt}\n\\begin{table*}[htp!]\n\t\\begin{center}\n\t\t\\caption{The mAUROC of models under PGD attack. Various novelty detectors are used.}\n\t\t\\label{table:main_results_2}\n\t\t\\begin{tabular}{r | r | c | cccccc}\n\t\t\t\\hline \\noalign{\\smallskip} \\noalign{\\smallskip}\n\t\t\tDataset & Defense & Test type & AE & VAE \\cite{kingma2013auto} & AAE \\cite{makhzani2015adversarial} & ALOCC \\cite{sabokrou2018adversarially} & GPND \\cite{pidhorskyi2018generative} & ARAE \\cite{salehi2020arae} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\t\t\t& No Defense & Clean & 0.964 & 0.979 & 0.973 & 0.961 & 0.946 & 0.965 \\\\\n\t\t\t& No Defense & PGD & 0.051 & 0.087 & 0.056 & 0.141 & 0.128 & 0.133 \\\\\n\t\t\t\\noalign{\\smallskip} \\cline{2-9} \\noalign{\\smallskip}\n\t\t\t& PGD-AT \\cite{madry2018towards} & & 0.357 & 0.521 & 0.427 & 0.312 & 0.582 & 0.341 \\\\\n\t\t\tMNIST & FD \\cite{Xie_2019_CVPR} & & 0.366 & 0.525 & 0.419 & 0.319 & 0.551 & 0.350 \\\\\n\t\t\t\\cite{lecun2010mnist} & SAT \\cite{xie2020smooth} & & 0.295 & 0.485 & 0.470 & 0.330 & 0.527 & 0.254 \\\\\n\t\t\t& RotNet-AT \\cite{hendrycks2019selfsupervised} & PGD & 0.333 & 0.501 & 0.507 & 0.361 & 0.551 & 0.314 \\\\\n\t\t\t& SOAP \\cite{shi2021online} & & 0.504 & 0.608 & 0.398 & 0.606 & 0.425 & 0.522 \\\\\n\t\t\t& APAE \\cite{goodge2020robustness} & & 0.104 & 0.155 & 0.240 & 0.202 & 0.229 & 0.191 \\\\\n\t\t\t& PLS (ours) & & \\textbf{0.678} & \\textbf{0.739} & \\textbf{0.608} & \\textbf{0.693} & \\textbf{0.741} & \\textbf{0.695} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\t\t\t& No Defense & Clean & 0.892 & 0.914 & 0.912 & 0.901 & 0.915 & 0.901 \\\\\n\t\t\t& No Defense & PGD & 0.088 & 0.223 & 0.152 & 0.177 & 0.177 & 0.262 \\\\\n\t\t\t\\noalign{\\smallskip} \\cline{2-9} \\noalign{\\smallskip}\n\t\t\t& PGD-AT \\cite{madry2018towards} & & 0.368 & 0.538 & 0.512 & 0.367 & 0.539 & 0.420 \\\\\n\t\t\tF-MNIST & FD \\cite{Xie_2019_CVPR} & & 0.379 & 0.533 & 0.513 & 0.370 & 0.542 & 0.428 \\\\\n\t\t\t\\cite{xiao2017fashion} & SAT \\cite{xie2020smooth} & & 0.306 & 0.504 & 0.499 & 0.332 & 0.530 & 0.351 \\\\\n\t\t\t& RotNet-AT \\cite{hendrycks2019selfsupervised} & PGD & 0.375 & 0.542 & 0.509 & 0.365 & 0.524 & 0.396 \\\\\n\t\t\t& SOAP \\cite{shi2021online} & & 0.475 & 0.509 & 0.313 & 0.477 & 0.386 & 0.548 \\\\\n\t\t\t& APAE \\cite{goodge2020robustness} & & 0.174 & 0.366 & 0.300 & 0.246 & 0.398 & 0.310 \\\\\n\t\t\t& PLS (ours) & & \\textbf{0.600} & \\textbf{0.604} & \\textbf{0.599} & \\textbf{0.612} & \\textbf{0.626} & \\textbf{0.599} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\t\t\t& No Defense & Clean & 0.550 & 0.552 & 0.555 & 0.551 & 0.559 & 0.578 \\\\\n\t\t\t& No Defense & PGD & 0.034 & 0.073 & 0.051 & 0.037 & 0.027 & 0.087 \\\\\n\t\t\t\\noalign{\\smallskip} \\cline{2-9} \\noalign{\\smallskip}\n\t\t\t& PGD-AT \\cite{madry2018towards} & & 0.145 & 0.177 & 0.195 & 0.146 & 0.182 & 0.157 \\\\\n\t\t\tCIFAR-10 & FD \\cite{Xie_2019_CVPR} & & 0.147 & 0.180 & 0.206 & 0.150 & 0.187 & 0.152 \\\\\n\t\t\t\\cite{krizhevsky2009learning} & SAT \\cite{xie2020smooth} & & 0.141 & 0.170 & 0.186 & 0.141 & 0.181 & 0.107 \\\\\n\t\t\t& RotNet-AT \\cite{hendrycks2019selfsupervised} & PGD & 0.139 & 0.163 & 0.161 & 0.105 & 0.147 & 0.101 \\\\\n\t\t\t& SOAP \\cite{shi2021online} & & 0.131 & 0.094 & 0.043 & 0.172 & 0.075 & 0.117 \\\\\n\t\t\t& APAE \\cite{goodge2020robustness} & & 0.097 & 0.179 & 0.171 & 0.095 & 0.062 & 0.154 \\\\\n\t\t\t& PLS (ours) & & \\textbf{0.245} & \\textbf{0.247} & \\textbf{0.252} & \\textbf{0.244} & \\textbf{0.242} & \\textbf{0.245} \\\\\t\n\t\t\t\\noalign{\\smallskip} \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table*}\n\n\n\\subsection{Robustness} \\label{robustness}\n\n\\subsubsection{White-box attacks}\nThe robustness of one-class novelty detection against various white-box attacks is reported in Table~\\ref{table:main_results_1}, where the vanilla AE is used. Without a defense, mAUROC scores drop significantly under all the white-box attacks, which shows the vulnerability of novelty detectors to the adversarial examples. PGD-AT improves adversarial robustness to a great extent. FD makes a slight improvement upon PGD-AT in most cases. SAT and Rot-AT seem not effective upon PGD-AT in the context of novelty detection. SOAP performs well in some cases but not uniformly. Compared to other methods, APAE generally shows less robustness. The proposed method, PLS, significantly increases mAUROC with PGD-AT, leading the other defenses by a decent margin. Moreover, PLS is consistently better across all the five white-box attacks on three datasets.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.46\\textwidth]{adaptive.pdf}\n\t\\caption{The mAUROC of PLS under PLS-knowledgeable attacks with varied trade-off parameters. (a) Knowledgeable A. (b) Knowledgeable B.}\n\t\\label{fig:adaptive}\n\\end{figure}\n\n\\noindent \\textbf{PLS-knowledgeable attacks.}\nAs discussed above, in a white-box attack, attackers are aware of the presence of the defense mechanism, i.e., PLS (it is differentiable at inference time, see Sec.~\\ref{method}). However, they count on only the novelty detection objective (i.e., MSE loss, see Eq.~\\eqref{mse_loss}) to generate adversarial examples. In this subsection, we follow the practice of the most recent adversarial defense studies such as \\cite{shi2021online}, to thoroughly evaluate the proposed defense mechanism. More precisely, we try to find an adaptive attack \\cite{papernot2017practical,tramer2020adaptive} by giving the full knowledge of the PLS defense mechanism to the attacker. We refer to this type of attack as \\textit{PLS-knowledgeable attack} in the paper.\n\nWe construct two PLS-knowledgeable attacks, Knowledgeable A and Knowledgeable B. They jointly optimize Eq.~\\eqref{mse_loss} and an auxiliary loss developed with the knowledge of PLS. Knowledgeable A attempts to minimize the $L_2$-norm between the latent space before and after the PLS transformation. The intuition is to void PLS such that the input and the output latent space of PLS become closer. In other words, Knowledgeable A replaces Eq.~\\eqref{mse_loss} with the following equation:\n\\begin{equation}\n\\label{adaptive_1}\n\\mathcal{L} = y \\parallel \\hat{\\mathbf{X}}^t - \\mathbf{X}^t \\parallel_2 - \\lambda_A \\parallel \\mathbf{Z}^t_{PLS} - \\mathbf{Z}_{adv}^t \\parallel_2, \n\\end{equation}\nwhere, $\\lambda_A$ is a trade-off parameter. Knowledgeable B attempts to maximize the $L_2$-norm between the latent space of the current adversarial example $\\mathbf{X}^t$ and its clean counterpart $\\mathbf{X}^0$ after the PLS transformation. The intuition is to keep the adversarial latent space away from the clean one. In other words, Knowledgeable B replaces Eq.~\\eqref{mse_loss} with the following equation:\n\\begin{equation}\n\\label{adaptive_2}\n\\mathcal{L} = y \\parallel \\hat{\\mathbf{X}}^t - \\mathbf{X}^t \\parallel_2 + \\lambda_B \\parallel \\mathbf{Z}^t_{PLS} - \\mathbf{Z}^0_{PLS} \\parallel_2, \n\\end{equation}\nwhere, $\\lambda_B$ is a trade-off parameter. When $\\lambda_A=0$ or $\\lambda_B=0$, the PLS-knowledgeable attacks reduce to the conventional white-box attacks.\n\nIn Fig.~\\ref{fig:adaptive}, we can observe that mAUROC monotonously increases as $|\\lambda_A|$ or $|\\lambda_B|$ increases. That is, these PLS-knowledgeable attacks cannot further reduce PLS's mAUROC, and the additional auxiliary loss terms would attenuate the MSE loss gradients. This indicates that attackers cannot straightforwardly benefit from the knowledge of PLS. Hence, the conventional white-box attack still has the greatest attacking strength. This result shows that it is not easy to find a stronger attack to break PLS, even with the full knowledge of the PLS mechanism.\n\n\\subsubsection{Black-box attacks}\nThe robustness against black-box attacks \\cite{papernot2017practical} is shown in the last column of Table~\\ref{table:main_results_1}. Here, we consider a naturally trained (i.e., train with only clean data) GPND as a substitute model and apply MI-FGSM, which has better transferability, to generate black-box adversarial examples for target models. As we can see, the defenses with PGD-AT degrade black-box robustness, which is identical to the observation in classification tasks \\cite{tramer2018ensemble}. SOAP, which is without using AT, shows better black-box robustness. PLS greatly improves the black-box robustness even with PGD-AT, and it is consistently better across all datasets. Naturally trained PLS achieves 0.907, 0.742 and 0.332 mAUROC on MNIST, F-MNIST and CIFAR-10, respectively, under the black-box attack.\n\n\\subsubsection{Generalizability}\nTable~\\ref{table:main_results_2} shows the adversarial robustness of various state-of-the-art novelty detection models. All of them are susceptible to adversarial attacks. We attach the PLS module to these models to protect them. We can see that PLS uniformly robustifies all of these novelty detectors and significantly outperforms the other defense approaches. This confirms that PLS is applicable to a wide variety of the present novelty detection methods, demonstrating its excellent generalizability.\n \n\n\\setlength{\\tabcolsep}{7pt}\n\\begin{table}\n\t\\begin{center}\n\t\t\\caption{The mAUROC of models under clean data.}\n\t\t\\label{table:clean_performance}\n\t\t\\begin{tabular}{r | ccc}\n\t\t\t\\hline \\noalign{\\smallskip} \\noalign{\\smallskip}\n\t\t\tDefense & MNIST & F-MNIST & CIFAR-10 \\\\\n\t\t\t\\hline \\noalign{\\smallskip} \\noalign{\\smallskip}\n\t\t\tNo Defense & 0.964 & 0.892 & 0.550 \\\\\n\t\t\tFD \\cite{Xie_2019_CVPR} & 0.965 & 0.892 & 0.551 \\\\\n\t\t\tSAT \\cite{xie2020smooth} & 0.949 & 0.883 & 0.543 \\\\\n\t\t\tRotNet-AT \\cite{hendrycks2019selfsupervised} & 0.963 & 0.897 & 0.554 \\\\\n\t\t\tSOAP \\cite{shi2021online} & 0.940 & 0.876 & 0.546 \\\\\n\t\t\tAPAE \\cite{goodge2020robustness} & 0.925 & 0.861 & 0.552 \\\\\n\t\t\tPLS (ours) & \\textbf{0.973} & \\textbf{0.922} & \\textbf{0.578} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\n\\subsection{Performance on clean data}\nWe also evaluate the performance of PLS on clean data. In this experiment, all the models are naturally trained. As shown in Table~\\ref{table:clean_performance}, PLS improves the performance upon the original network architecture (No Defense), while, the other defenses do not make obvious improvements. This shows that PLS generalizes better for both clean data and adversarial examples. PLS enjoys this benefit because the principal latent components are learned from only the latent space of the known class. Due to this, when transforming the latent space of any novel class image, PLS projects it into the known class space defined by the principal latent component. This brings the transformed latent space closer to the latent space of the known class, resulting in the decoder trying to reconstruct it into a known class image. Subsequently, this produces high reconstruction error for the novel class images while barely affecting the reconstruction of the known class images.\n\n\n\n\\subsection{Ablation study} \\label{sec_ablation}\n\n\\noindent \\textbf{PLS components.}\nTable~\\ref{table:ablation} reports the results of different PLS variants. First, Vector-PCA alone significantly improves the robustness upon PGD-AT. This shows that the mechanism of replacing perturbed latent vectors by the incrementally-trained principal latent vector is effective. As discussed earlier, in PLS the adversaries can stay only on the scaling factors of the principal latent vector. Next, we further remove the adversaries with the help of denoising operation on the spatial dimension. We try to deploy a feature denoising block \\cite{Xie_2019_CVPR} after the forward Vector-PCA. This baseline is denoted as Vector-PCA+FD. This makes a slight improvement over Vector-PCA baseline. Finally, the complete PLS uses Spatial-PCA for this purpose instead, achieveing great mAUROC increase. This shows Spatial-PCA's advantage over FD in our case.\n\n\\setlength{\\tabcolsep}{7pt}\n\\begin{table}\n\t\\begin{center}\n\t\t\\caption{The mAUROC of PLS variants under PGD attack.}\n\t\t\\label{table:ablation}\n\t\t\\begin{tabular}{r | ccc}\n\t\t\t\\hline \\noalign{\\smallskip} \\noalign{\\smallskip}\n\t\t\tDefense & MNIST & F-MNIST & CIFAR-10 \\\\\n\t\t\t\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\t\t\tPGD-AT \\cite{madry2018towards} & 0.357 & 0.368 & 0.145 \\\\\n\t\t\tVector-PCA & 0.566 & 0.499 & 0.215 \\\\\n\t\t\tVector-PCA+FD & 0.582 & 0.505 & 0.215 \\\\\n\t\t\tPLS (ours) & \\textbf{0.678} & \\textbf{0.600} & \\textbf{0.245} \\\\\n\t\t\t\\noalign{\\smallskip} \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.46\\textwidth]{feature_diff.pdf}\n\t\\caption{Mean $L_2$-norm between the latent space of PGD adversarial examples and that of their clean counterpart on different defenses. The values are the mean over an entire dataset.}\n\t\\label{fig:feature_diff}\n\\end{figure}\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.46\\textwidth]{histogram.pdf}\n\t\\caption{Histograms of reconstruction errors. (a) No Defense under clean data. (b) No Defense under PGD attack. (c) PGD-AT under PGD attack. (d) PLS under PGD attack. Digit 0 of MNIST is set to normal data, and the other digits are anomalous.}\n\t\\label{fig:histogram}\n\\end{figure}\n\n\\noindent \\textbf{Stability of latent space.}\nWe compute the mean $L_2$-norm between the latent space of adversarial examples and that of their clean counterpart: $\\parallel \\mathbf{Z}_{adv} - \\mathbf{Z} \\parallel_2$. As can be seen in Fig.~\\ref{fig:feature_diff}, PLS's mean $L_2$-norm is three orders of magnitude smaller than the other defenses. This indicates that PLS's latent space are barely affected by adversaries, showing PLS's effectiveness in adversary removal.\n\n\\noindent \\textbf{Reconstruction errors.} For an AE-style novelty detection model, normal data and anomalous data are expected to get low and high reconstruction errors, respectively. The model follows this behavior given clean data, as shown in Fig.~\\ref{fig:histogram}(a). When an attacker attempts to maximize the reconstruction errors of normal data and minimize that of anomalous data, the model would make wrong predictions, shown in Fig.~\\ref{fig:histogram}(b). Fig.~\\ref{fig:histogram}(c) shows that PGD-AT pulls back the enlarged reconstruction errors of normal data, but they still overlap for the anomalous data. In Fig.~\\ref{fig:histogram}, it can be observed that PLS pushes the reconstruction errors of anomalous data with better margin. Although the reconstruction errors of normal data also increases, the gap between normal and anomalous data is retained resulting in PLS performing better under attacks.\n\n\\noindent \\textbf{Reconstructed images.} Fig.~\\ref{fig:visual} compares the reconstructed images of No Defense model and PLS under PGD attack. Digit 2 of MNIST is used as the known class. We can see that No Defense model captures the shape of the adversarial anomalous data and thus produces fair reconstructions. In other words, the reconstruction error gap between normal data and anomalous data is insufficiently large. Such observation is consistent with the quantitative results that it is not adversarially robust. In contrast, PLS reconstructs every data into the known class of digit 2. Hence, even under attacks, PLS can obtain very high reconstruction errors from anomalous data and low errors from normal data.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.46\\textwidth]{visual.pdf}\n\t\\caption{Reconstructions under PGD attack with $\\epsilon = 0.3$. Digit 2 is set to normal data, and the other digits are anomalous.}\n\t\\label{fig:visual}\n\\end{figure}\n\n\n\\section{Conclusion}\nIn this paper, we study the adversarial robustness in the context of one-class novelty detection problem. We show that existing novelty detection models are vulnerable to adversarial perturbations and then propose a defense method referred to as Principal Latent Space (PLS). Specifically, PLS purifies the latent space by the incrementally-trained cascade PCA process. Moreover, we construct a generic evaluation framework to fully test the effectiveness of the proposed PLS. We perform extensive experiments on multiple datasets with multiple existing novelty detection models and consider various attacks to show that PLS improves the robustness consistently across different attacks and datasets.\n\n\n\n\n\\ifCLASSOPTIONcompsoc\n \n \\section*{Acknowledgments}\n\\else\n \n \\section*{Acknowledgment}\n\\fi\n\nThis work was supported by the DARPA GARD Program HR001119S0026-GARD-FP-052.\n\n\n\\ifCLASSOPTIONcaptionsoff\n \\newpage\n\\fi\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nNetwork slicing is an important technique in 5G to enable flexibility and customization. Based on software defined networks and network function virtualization techniques, network slicing can define various virtual network slices over a single physical network infrastructure \\cite{b1}. To realize the network slicing, resource allocation is an important part to guarantee service level agreements of slices. The network operator needs to allocate limited resources between slices. For example, enhanced Mobile Broad Band (eMBB) slice usually requires a high throughput, while Ultra Reliable Low Latency Communications (URLLC) slice needs a low latency and a high reliability. Compared with core network slicing, the RAN slicing is more complicated due to limited resources and dynamic channel states \\cite{b2}. In \\cite{b3}, a risk sensitive model for the resource allocation of URLLC slice is presented. A two-layer method is introduced in \\cite{b4} to realize an efficient and low complexity RAN slicing. \\cite{b5} shows a RAN slicing scheme by considering both rate and latency demands of various traffic types, and the scheme is tested in a industry 4.0 case. \n\nThe aforementioned works mainly focus on the radio resource allocation. However, the evolving network architecture requires a joint resource allocation scheme, such as radio and computation resources. Indeed, to support the emerging computation intensive applications, deploying mobile edge computing (MEC) servers becomes an ideal solution\\cite{b5-1}. The computation tasks can be processed in the MEC server instead of the central cloud, thus a lower delay is achieved. The work of \\cite{b6} shows that joint resources allocation slicing has a better performance than slicing one single resource, and \\cite{b7} introduced a mathematical model to jointly slice mobile network and edge computation resource. \n\nAlthough incorporating computation capability into RAN will bring significant benefits, it also leads to a higher network management complexity, especially when multiple slices are involved. To this end, machine learning methods provide a good opportunity for network management \\cite{b8}. For example, in reinforcement learning, the agent interacts with environment to maximize the long term reward based on Markov decision process (MDP), and the complexity of defining a dedicated optimization model is avoided. However, the reinforcement learning algorithms that are applied in most existing works, such as Q-learning and deep Q-learning, generally require a huge number of samples to train the algorithm, which consequently lead to a long convergence time. In addition, the low training efficiency will unavoidably affect the system performance, especially for tasks with tight delay budgets. \nTo this end, we propose a knowledge transfer based resource allocation (KTRA) method in this paper. Different with existing algorithms such as reinforcement learning or deep reinforcement learning, the proposed KTRA method has a knowledge transfer capability, and the agent can leverage the knowledge of other expert agents to improve its own performance on the target task \\cite{b9}. With the prior knowledge of experts, it requires less samples when exploring the target task, which means a higher exploration efficiency. The proposed KTRA is compared with Q-learning based resource allocation (QLRA), and the simulation shows that KTRA achieves a 18.4\\% lower delay for URLLC slice and a 30.1\\% higher throughput for eMBB slice as well as a faster convergence.\n\nThe rest of this work is organized as follows. Section \\ref{s2} presents the related work. Section \\ref{s3} introduces the system model and problem formulation, and Section \\ref{s4} defines the KTRA scheme and baseline algorithm. We show simulation results in Section \\ref{s5}, and conclude this work in Section \\ref{s6}. \n\n\n\\section{Related Work}\n\\label{s2}\nRecently machine learning techniques have been extensively studied for wireless network applications, and various techniques are proposed for resource allocation of 5G networks. \\cite{b10} proposed a reinforcement learning based method for joint power and radio resource allocation for URLLC and eMBB users. A Q-learning based solution is presented in \\cite{b11} to maximize the network utility by satisfying network slicing requests under network resources constraints. \\cite{b12} introduced a RAN slicing method to dynamically allocate radio and computation resources, and a constrained learning scheme is defined. A decentralized deep reinforcement learning method is presented in \\cite{b13} for network slicing, which ensures service level agreements under networking and computation resources constraints. The joint RAN slicing and computation offloading problem is investigated in \\cite{b14}, and multi-agent deep Q-learning is applied to maximize the communication and computation resources utilization. Furthermore, \\cite{b14-1} proposed a actor-critic network based deep reinforcement learning method for joint radio and computation resource allocation of virtualized RAN. \n\nAlthough various machine learning methods have been proposed for resource allocation of 5G, including reinforcement learning\\cite{b10,b11,b12}, deep reinforcement learning\\cite{b13,b14-1}, and multi-agent deep reinforcement learning\\cite{b14}, these methods usually require a huge amount of samples for training, which means a long exploration phase. Deep reinforcement learning is considered as a breakthrough, but the time-consuming network training is a well known issue \\cite{b15}. We propose a correlated Q-learning based method for radio resource allocation of network slicing in \\cite{b16}, but the knowledge transfer capability is still not considered. To this end, we propose a KTRA scheme for the joint radio and computation resources allocation of 5G networks in this work. Based on the knowledge transfer strategy, our proposed method can utilize the prior knowledge of expert agent to achieve a faster convergence speed or a higher average reward. \n\n\n\\section{System Model and Problem Formulation}\n\\label{s3}\n\\subsection{Network Architecture}\n\nThe proposed system model is shown as Fig.\\ref{fig1}. We assume the base station (BS) is equipped with a MEC server to process computation tasks of eMBB and URLLC slices. We apply a two-step resource allocation scheme. In the inter-slice phase, the BS intelligently allocates radio and computation resources between different slices. Then these resources are utilized within each slice in the intra-slice phase. For example, the URLLC slice operator will distribute radio resource between attached UEs, and use available MEC server capacity to process the computation tasks of its UEs. In this work, we mainly focus on the inter-slice resource allocation. Given this architecture, the delay experienced by a computation task is:\n\\begin{equation} \\label{eq1}\nd=d^{tx}+d^{rtx}+d^{que}+\\alpha d^{edge}+(1-\\alpha)d^{cloud},\n\\end{equation}\nwhere $d^{tx}$ and $d^{rtx}$ are the task transmission and retransmission delays, respectively. $d^{que}$ is the queuing delay in BS, which is considered as the scheduling delay. $d^{edge}$ is the processing delay in MEC server, and $d^{cloud}$ is the processing delay in central cloud. $\\alpha$ is a binary variable. $\\alpha=1$ if the task is computed in the BS, and $\\alpha=0$ means the task is forwarded to the central cloud. We presume the intelligent BS will decide whether to process the task in MEC server or offload it to the cloud. Eq. (\\ref{eq1}) shows that the delay is affected by both radio and computation resources. The radio resource allocation will affect the transmission delay $d^{tx}$, and computation resource allocation can change the processing delay $d^{edge}$. Meanwhile, the scheduling efficiency will affect the queuing delay $d^{que}$. Following we will explain the communication and computation model. \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=9cm,height=7cm]{f1.jpg}\n\\caption{Proposed system architecture.}\n\\label{fig1}\n\\vspace{-10pt}\n\\end{figure}\n\n\\subsection{Communication Model}\nWe consider resource blocks (RBs) as the smallest time-frequency resource that is distributed to users. The transmission delay $d^{tx}$ between BS and user equipment (UE) is:\n\\begin{equation} \\label{eq2}\nd^{tx}=\\frac{L_{u}}{E_{j,u}},\n\\end{equation}\nwhere $L_{u}$ is the transmitted packet size of UE $u$, $E_{j,u}$ is the link capacity between the BS $j$ and the UE $u$. The link capacity depends on the number of RBs that are allocated to this transmission:\n\\begin{equation}\n\\resizebox{0.9\\hsize}{!}{$\\begin{split}\n \\label{eq3}\nE_{j,u}=&\\sum _{r\\in{\\mathcal{N}_{u}}}b_{RB}log(1+\\\\ &\\frac{p_{j,r}x_{j,u,r}g_{j,u,r}}{b^{RB}N_{0}+\\sum\\limits_{j'\\in \\mathcal{J}_{-j}}\\sum\\limits_{u'\\in \\mathcal{U}_{j'}}\\sum\\limits_{r'\\in \\mathcal{N}_{j'}}{p_{j',r'}x_{j',u',r'}g_{j',u',r'}}}),\n\\end{split}$}\n\\end{equation}\nwhere $\\mathcal{N}_{u}$ denotes the set of RBs that is allocated to UE $u$, $b_{RB}$ denotes the bandwidth of one RB, $N_{0}$ denotes the noise power density, $p_{j,r}$ denotes the transmission power of RB $r$ in BS $j$. $x_{j,u,r}$ is a binary variable. $x_{j,u,r}=1$ means the RB $r$ is allocated to UE $u$; otherwise $x_{j,u,r}=0$. $g_{j,u,r}$ denotes the channel gain between BS $j$ and UE $u$. $\\mathcal{J}_{-j}$ denotes the set of BSs except $j^{th}$ BS, $\\mathcal{U}_{j'}$ denotes the set of UEs in BS $j'$, and $\\mathcal{N}_{j'}$ denotes the set of total RBs in BS $j'$. \n\n\\subsection{Computation Model}\nFor a computation task from UEs, we assume it requires certain computation resources to complete the task (denoted by number of CPU cycles). Then the processing delay in MEC server $d^{edge}$ is:\n\\begin{equation} \\label{eq4}\nd^{edge}=\\frac{c_{u,q}}{\\beta C_{j}},\n\\end{equation}\nwhere $c_{u,q}$ denotes required computation resources of task $q$ from UE $u$, $\\beta$ denotes the proportion of computation resources allocated to this task ($0\\leq \\beta \\leq 1$), and $C_{j}$ denotes the total computation capacity of MEC server in BS $j$. \n\nOn the other hand, BS may decide to offload the task to central cloud computation servers. The $d^{cloud}$ in Eq. (\\ref{eq1}) is described as:\n\\begin{equation} \\label{eq4-1}\nd^{cloud}=d^{up}+d^{down}+d^{c,que}+d^{c,computation}, \n\\end{equation}\nwhere $d^{up}$ and $d^{down}$ are upload and download transmission delay of the computation task, respectively. $d^{c,que}$ is the cloud queuing delay, and $d^{c,computation}$ is cloud computation delay. The $d^{down}$ and $d^{c,computation}$ can be omitted because: i) the downloaded packet size after computation is usually much smaller than input packet; ii) the central cloud usually has a very high computation capacity \\cite{b16-2}. Then \nEq. (\\ref{eq4-1}) can be rewritten as:\n\\begin{equation} \\label{eq5}\nd^{cloud}=\\frac{1}{\\frac{B}{L_{s}}-\\lambda}+d^{c, que},\n\\end{equation}\nwhere $B$ is the backhaul capacity, $B\/L_{u}$ denote the service rate, and $\\lambda$ is the packet arrival rate. We apply the M\/M\/1 queue model to describe the upload delay $d^{up}=\\frac{1}{\\frac{B}{L_{s}}-\\lambda}$. Meanwhile, this work mainly focuses the RAN resource allocation, and it is reasonable to assume a fixed cloud queuing delay $d^{c, que}$. Finally, we assume: i) the task is preferred to be processed in the MEC server of the BS due to the potential benefit of MEC; ii) BS will offloaded the task to central cloud if the queuing time expires the preset target delay \\cite{b5-1}. \n\n\\subsection{Problem Formulation}\n\nHere we consider two typical slices: eMBB and URLLC slices. The eMBB slice intends to maximize the throughput, while the URLLC slice requires a lower latency. The intelligent BS needs to balance the requirements of both slices, then we define the problem formulation as following: \n\\begin{subequations}\\label{e2:main}\n\\begin{align}\n\\text{max} \\qquad & w^{embb}b^{embb,avg}_{j}+w^{urllc}(d^{tar}-d^{urllc,avg}_{j})& \\tag{\\ref{e2:main}} \\\\\n \\text{s.t.} \\qquad & b^{embb,avg}_{j}=\\frac{\\sum\\limits_{u\\in \\mathcal{M}^{embb}_{j}}b^{embb}_{j,u}}{|\\mathcal{M}^{embb}_{j}|} & \\label{e2:a} \\\\\n& b^{urllc,avg}_{j}=\\frac{\\sum\\limits_{v\\in\\mathcal{M}^{urllc}_{j}}d^{urllc}_{j,v}}{|\\mathcal{M}^{urllc}_{j}|} & \\label{e2:b} \\\\\n & (\\ref{eq1})\\, (\\ref{eq2})\\, (\\ref{eq3}) \\, (\\ref{eq4})\\, (\\ref{eq5}) & \\label{e2:c} \\\\\n&\\sum\\limits_{u\\in \\mathcal{M}^{embb}_{j}}{x_{j,u,r'}}+\\sum\\limits_{v\\in \\mathcal{M}^{urllc}_{j}}{x_{j,v,r'}}=1 & \\label{e2:d}\\\\\n \\sum\\limits_{r'\\in \\mathcal{N}_{j'}} & (\\sum\\limits_{u\\in \\mathcal{M}^{embb}_{j}}{x_{j,u,r'}}+\\sum\\limits_{v\\in \\mathcal{M}^{urllc}_{j}}{x_{j,v,r'}})\\leq |\\mathcal{N}_{j}|& \\label{e2:e}\\\\\n& C^{embb}_{j}+C^{urllc}_{j}\\leq C_{j} & \\label{e2:f}\n\\end{align}\n\\end{subequations}\nwhere $b^{embb,avg}_{j}$ and $d^{urllc,avg}_{j}$ denote the average throughput and latency of eMBB and URLLC slices, respectively, which are calculated by eq. (\\ref{e2:a}) and (\\ref{e2:b}). $w^{embb}$ and $w^{urllc}$ are weighting factors to balance two metrics and form a overall objective. $d^{tar}$ is the preset target delay of URLLC slice, which will bring a positive reward if $d^{tar}>d^{urllc,avg}_{j}$.\n$b^{embb}_{j,u}$ is the throughput of UE $u$ in the eMBB slice of BS $j$, and $d^{urllc}_{j,v}$ is the latency of UE $v$ in the URLLC slice. $\\mathcal{M}^{embb}_{j}$ and $\\mathcal{M}^{urllc}_{j}$ denote the UE set of eMBB and URLLC slices in BS $j$, respectively. In eq. (\\ref{e2:main}), a higher eMBB throughput and a lower URLLC latency are expected to maximize the objective. The eq. (\\ref{e2:d}) guarantees one RB can only be allocated to one UE. Eq. (\\ref{e2:e}) and (\\ref{e2:f}) mean that total allocated RBs and computation capacity should not exceed the available resources in the BS $j$. \n\n\\section{Knowledge Transfer based Radio and Computation Resource Allocation }\n\\label{s4}\nIn this section, we first introduce the proposed KTRA method, including the knowledge transfer strategy, MDP definition, and the knowledge transfer reinforcement learning. Then we introduce the Q-learning based resource allocation scheme as a baseline algorithm. \n\n\\subsection{Knowledge Transfer Strategy and MDP definition.}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=9cm,height=5cm]{f2-1.jpg}\n\\caption{Learning strategy comparison .}\n\\label{fig2}\n\\vspace{-10pt}\n\\end{figure}\n\nFirst, we compare the knowledge transfer reinforcement learning with reinforcement learning to better explain the knowledge transfer strategy. The interaction between agent and task can be described by MDP $$, where $S$, $A$, $T$, $R$ are the set of states, set of actions, transition probability, and reward function, respectively. As shown in Fig.\\ref{fig2}, in reinforcement learning, the agent selects an action, receives reward and arrives new state. The agent starts from scratch to explore the task, since no prior knowledge is available in the learning phase.\n\nBy contrast, in knowledge transfer reinforcement learning, two agents are involved, namely learner and expert agents. Compared with single learning phase in RL, knowledge transfer reinforcement learning has two phases: knowledge transfer phase and learning phase. In knowledge transfer phase, considering different MDP definitions of two agents, a map function is needed to transform the expert agent's knowledge to the leaner agent. In the learning phase, learner agent utilizes the knowledge of expert agent to improve its own performance on target task. Learner agent is expected to achieve a higher exploration efficiency, since it already has some prior knowledge at the beginning of exploration. \n\nIn this work, we define a learner agent for joint radio and computation resources allocation, and an expert agent for radio resource allocation. The expert agent has no knowledge for computation resource allocation, but it's knowledge of radio resource allocation can be used by learner agent. Following we will define the state, action and reward of two agents. \n\n\\begin{itemize}\n \\item \\textbf{State}: The states of both expert and learner agents are $(q^{embb},q^{urllc})$. $q^{embb}$ denotes the number of tasks in the queue of eMBB slice, and $q^{urllc}$ is defined similarly. $(q^{embb},q^{urllc})$ represents the demands of two slices, and agent can select actions accordingly. \n \\item \\textbf{Action}: For the expert agent, it only implements the radio resource allocation, and then the action is defined as $(r^{embb},r^{urllc})$, which represents the number of radio resources allocated to eMBB and URLLC slices. For the learner agent, it considers both radio and computation resource allocation, and then the action is defined as $(r^{embb},r^{urllc},c^{embb},c^{urllc})$, where $c^{embb}$ and $c^{urllc}$ denote the computation capacity allocated to eMBB and URLLC slices, respectively. \n \\item \\textbf{Reward}: The reward is defined by the objective of slices, which is denoted by (\\ref{e2:main}). We also apply a penalty to guarantee the packet drop rate. \n\\end{itemize}\n\n\\subsection{Knowledge Transfer Reinforcement Learning}\nHere we will introduce the knowledge transfer reinforcement learning and define the map function for knowledge transfer. In reinforcement learning, to maximize the expected reward, the agent needs to improve its policy $\\pi(s_{t})$ to select the best action at state $s_{t}$, and arrives a new state $s_{t+1}$. Then the state value is defined to represent the potential reward of arriving a new state:\n\\begin{equation} \\label{eu_eqn}\nV_{\\pi}(s_{t+1}) =\\mathbb{E}_{\\pi}(\\sum_{n=0}^{\\infty}\\gamma^{n} r_{t+1}|s=s_{t+1}),\n\\end{equation}\nwhere $V_{\\pi}(s_{t+1})$ is the state value to describe the expected reward if the agent arrives $s_{t+1}$, $r_{t+1}$ is the reward at time $t+1$, and $\\gamma$ is the discount factor $(0<\\gamma<1)$. Furthermore, we need to define the state-action value to describe the expected reward of taking action $a_{t}$ under state $s_{t}$. In Q-learning, the Q-values are updated by:\n\\begin{equation} \\label{eq7}\n\\begin{aligned}\nQ^{new}(s_{t},a_{t}) &= Q^{old}(s_{t},a_{t})+\\\\\n&\\alpha(r+\\gamma \\max\\limits_{a} Q(s_{t+1},a)-Q^{old}(s_{t},a_{t})),\n\\end{aligned}\n\\end{equation}\nwhere $Q^{old}$ and $Q^{old}$ denote old and new Q-values, respectively, $a_{t}$ is the action at time $t$, $r$ is the reward, and $\\alpha$ is the learning rate ($0< \\alpha <1$). \n\n\\begin{algorithm}[!t]\n\t\\caption{KTRA algorithm}\n\t\\begin{algorithmic}[1]\n\t\t\\STATE \\textbf{Initialize:} Wireless network parameters and Q-table of the expert.\n\t\t\\FOR{$TTI=1$ to $T$}\n\t\t\\FOR{Every BS}\n\t\t\\STATE With probability $\\epsilon$ choose actions randomly, otherwise select $a_{l,t}$ by greedy policy.\n\t\t\\STATE BS allocates radio and computation resource between slices. Slices process computation tasks by allocated MEC server capacity, and allocate RBs by proportional fairness algorithm. \n\t\t\\STATE Calculating reward based on received metrics. \n\t\t\\STATE Updating state $s_{l,t}$, find $Q^{T}(\\mathcal{F}(s_{l,t}),\\mathcal{F'}(a_{l,t}))$.\n\t\t\\STATE Updating Q-values by eq.(\\ref{eq8}).\n\t\t\\ENDFOR\n\t\t\\ENDFOR\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}[!t]\n\t\\caption{QLRA algorithm}\n\t\\begin{algorithmic}[1]\n\t\t\\STATE \\textbf{Initialize:} Wireless network parameters. \n\t\t\\FOR{$TTI=1$ to $T$}\n\t\t\\FOR{Every BS}\n\t\t\\STATE With probability $\\epsilon$ choose actions randomly, otherwise select $a^{l,t}$ by greedy policy.\n\t \\STATE BS allocates radio and computation resource between slices. Slices process computation tasks by allocated MEC server capacity, and allocate RBs by proportional fairness algorithm.\n\t\t\\STATE Calculating reward based on received metrics. \n\t\t\\STATE Updating state $s_{t}$.\n\t\t\\STATE Updating Q-values by eq.(\\ref{eq7}).\n\t\t\\ENDFOR\n\t\t\\ENDFOR\n\t\\end{algorithmic}\n\\end{algorithm}\n\nIn Q-learning, the agent needs to explore the state-action space to find the optimal action sequence. This exploration usually takes a large number of iterations, because the agent needs to try huge amounts of action combinations without any prior knowledge. On the contrary, in knowledge transfer reinforcement learning, we use the prior knowledge of experts to improve the exploration efficiency\\cite{b17}. The Q-values are updated by:\n\\begin{equation} \\label{eq8}\n\\resizebox{0.89\\hsize}{!}{$\\begin{aligned}\nQ^{new}(s_{l,t},a_{l,t})= &Q^{T}(\\mathcal{F}(s_{l,t}),\\mathcal{F'}(a_{l,t}))+Q^{old}(s_{l,t},a_{l,t})+\\\\\n&\\alpha(r+\\gamma \\max\\limits_{a} Q(s_{l,t+1},a)-Q^{old}(s_{l,t},a_{l,t})),\n\\end{aligned}$}\n\\end{equation}\nwhere $s_{l,t}$ and $a_{l,t}$ are learner's state and action at time $t$, respectively. $Q^{T}(\\mathcal{F}(s_{l,t}),\\mathcal{F'}(a_{l,t}))$ is the mapped Q-values as an extra reward of selecting $a_{l,t}$ under state $s_{l,t}$, which aims to guide the exploration of learner agent. $\\mathcal{F}$ and $\\mathcal{F'}$ are state and action map functions, respectively. $Q^{T}(\\mathcal{F}(s_{l,t}),\\mathcal{F'}(a_{l,t}))$ can be generated by:\n\\begin{equation} \\label{eq9}\n\\begin{aligned}\nQ^{T}(\\mathcal{F}(s_{l,t}),\\mathcal{F'}(a_{l,t}))= Q^{e}(s_{e},a_{e}),\n\\end{aligned}\n\\end{equation}\nwhere $Q^{e}$ is the Q-values of expert agent, $s_{e}$ and $a_{e}$ are state and action of expert agent. The goal of eq. (\\ref{eq9}) is to find a specific Q-value of expert agent to represent the potential reward of taking $a_{l,t}$ under $s_{l,t}$ in learner. Thus we need to find similar states and actions in expert agent's Q-table. Note that the expert and learner agents have the same state definition, then we can always find a $s_{e}$ that satisfy $s_{l,t}=s_{e}$, then $\\mathcal{F}$ can be easily defined. Meanwhile, based on the definition of actions, for any $a_{l}=(r^{eM},r^{UR},c^{eM},c^{UR})$, we consider $a_{e}=(r^{eM},r^{UR})$ as a similar action for mapping, and $\\mathcal{F'}$ can be defined accordingly.\n\nThe KTRA algorithm is summarized as Algorithm 1. Noting that we apply a classic proportional fairness algorithm for intra-slice radio resource allocation, because this work mainly focus on inter-slice level scheduling \\cite{b16}, and there is no intra-slice computation resource allocation. \n\n\n\\subsection{Baseline Algorithms: QLRA}\nWe apply QLRA as a baseline algorithm. Q-learning is the most generally applied reinforcement learning algorithm. The MDP definition of QLRA is the same with KTRA, but there is no prior knowledge. QLRA is given in Algorithm 2. \n\n\\section{Performance Evaluation}\n\\label{s5}\n\\subsection{Parameter Settings}\nWe consider three different cases, including:\n\n\\begin{itemize}\n \\item \\textbf{Case I}: Q-learning based radio resource allocation. It works as an expert for the Case II. \n \\item \\textbf{Case II}: KTRA based joint radio and computation resource allocation. As a learner, BSs in Case II can utilize the Q-tables of expert agent as prior knowledge. \n \\item \\textbf{Case III}: QLRA based joint radio and computation resource allocation. It is considered as a baseline algorithm without any prior knowledge for the task. \n\\end{itemize}\n\nEach case contains 5 BSs with 500 m inter-site distance, and each BS is considered as an independent agent to implement the proposed strategy. For example, all BSs in Case II will implement the KTRA independently to achieve its own goal. We assume each BS has one eMBB slice with 5 UEs and one URLLC slice with 10UEs. The avaliable bandwidth of one BS is 20 MHz, which contains 100 RBs. We assume there are 13 resource block groups (RBGs) to reduce the allocation complexity. The first 12 RBGs contains 8 RBs each, while the last RBG has 4 RBs. 200 CPU cycles are required to process 1 bit data \\cite{b12}. Other parameters are shown as Table \\ref{tab2}.\n\n\\begin{table}[!t]\n\\vspace{-5pt}\n\\caption{Parameters Settings}\n\\centering\n\\renewcommand\\arraystretch{1.4}\n\\begin{tabular}{|p{4cm}<{\\centering}||p{3.8cm}<{\\centering}|}\n\\hline\n \\textbf{5G Networking} & \\textbf{Computation Settings}\\\\\n\\hline\n3GPP Urban Macro network & Computation capacity: 3 GHz\\\\ \n 2 OFDM symbols for each TTI & CPU cycles required per bit: 200\\\\\n\\cline{2-2}\n Tx\/Rx antenna gain: 15 dB. & \\textbf{Traffic Model} \\\\\n\\cline{2-2}\n \\quad \\, Number of subcarriers \\quad \\quad \\, in each RB: 12 & URLLC$\\backslash$eMBB traffic: Poisson distribution\\\\\n Subcarrier bandwidth: 15kHz & URLLC packet size: 50 Bytes \\\\ \n Transmission power: 40 dBm & eMBB packet size: 100 Bytes \\\\ \n \\cline{2-2}\n Backhaul capacity: 10 Mpbs& \\textbf{Problem Formulation}\\\\\n\\cline{1-2}\n \\textbf{Propagation Model} & eMBB$\\backslash$URLLC weight factor: 1 \\\\\n \\cline{1-1}\n 128.1+37.6log(distance(km)) & URLLC target delay: 2 ms \\\\\nLog-Normal shadowing: 8 dB.& Fixed cloud queuing delay: 1 ms \\\\\n \\cline{1-2}\n \\textbf{Retransmission Settings} & \\textbf{Learning Settings}\\\\\n\\cline{1-2}\nMax number of retransmissions: 1 & Learning rate: 0.9 \\\\\nRound trip delay: 4 TTIs & Discount factor: 0.5 \\\\\nProtocol: asynchronous HARQ. & Epsilon value: 0.05 \\\\\n\\hline\n\\end{tabular}\n\\label{tab2}\n\\vspace{-10pt}\n\\end{table}\n\n\\begin{figure*}[t!]\n\\centering\n\\label{f3}\n\\vspace{0pt}\n\\subfigure[ URLLC latency distribution \\text{[ms]} under 2 Mbps eMBB and URLLC traffic, and 3 GHz MEC server per cell.]{\n\\includegraphics[width=7.4cm,height=5.8cm]{fig4.jpg}\n}\n\\quad\n\\subfigure[ Average URLLC latency \\text{[ms]} under various URLLC traffic, 2 Mbps eMBB traffic and 3 GHz MEC server per cell.]{\n\\includegraphics[width=7.4cm,height=5.8cm]{fig5.jpg}\n}\n\\quad\n\\subfigure[eMBB throughput \\text{[Mbps]} per cell under various URLLC traffic, 2 Mbps eMBB traffic and 3 GHz MEC server.]{\n\\includegraphics[width=7.4cm,height=5.8cm]{fig6.jpg}\n}\n\\quad\n\\subfigure[Average URLLC latency \\text{[ms]} under various MEC server capacities, 2 Mbps eMBB and URLLC traffic per cell.]{\n\\includegraphics[width=7.4cm,height=5.8cm]{fig7.jpg}\n}\n\\quad\n\\subfigure[eMBB throughput \\text{[Mbps]} per cell under various MEC server capacities, 2 Mbps eMBB and URLLC traffic.]{\n\\includegraphics[width=7.4cm,height=5.8cm]{fig8.jpg}\n}\n\\quad\n\\subfigure[Convergence performance of KTRA and QLRA under 2 Mbps eMBB and URLLC traffic, and 2 GHz computation capacity per cell.]{\n\\includegraphics[width=7.4cm,height=5.8cm]{fig9.jpg}\n}\n\\caption{Simulation results comparison.}\n\\label{f3}\n\\end{figure*}\n\n\n\\subsection{Simulation Results}\n\nFirstly, Fig.\\ref{f3} (a) shows the empirical cumulative distribution function (ECDF) of URLLC latency of three cases under 2 Mbps eMBB and URLLC traffic, and 3 GHz MEC server per cell. The result shows that expert case has the highest delay distribution, and the reason is that we assume MEC servers are not deployed in expert. Thus all computation tasks need to be processed in the central cloud, which leads to a higher delay. Meanwhile, the proposed KTRA method outperforms QLRA by a better delay distribution, as indicated by a lower ECDF curve in Fig.\\ref{f3} (a), and it can be explained by the knowledge transfer capability of KTRA. \n\nFig.\\ref{f3} (b) presents the average delay experienced by URLLC UEs, and Fig.\\ref{f3} (c) shows the average eMBB throughput per cell. Here the eMBB traffic load is fixed to 2 Mbps per cell, and URLLC traffic varies from 1 Mbps to 4 Mbps. Without MEC servers, expert case still has the highest delay and the lowest throughput. Meanwhile, KTRA method achieves a lower URLLC delay and higher eMBB throughput than QLRA. KTRA has a 18.4\\% lower URLLC delay and 30.1\\% higher eMBB throughput under 2 Mbps URLLC traffic. \n\nShown by Fig.\\ref{f3} (d) and (e), we investigate the network performance under various MEC server capacities, which is indicated by CPU cycles per second. Considering expert case has no MEC capability, we focus on the performance of KTRA and QLRA to further present the advantage of proposed method. As expected, both algorithms have a lower URLLC delay and a higher eMBB throughput by increasing the MEC server capacity, because higher computation capacity means lower task processing delay. KTRA still outperforms QLRA by a 15.1\\% lower URLLC delay and a 33.8\\% higher eMBB throughput under 3 GHz MEC server capacity. \n\nFurthermore, we compare the convergence performance in Fig.\\ref{f3} (f). Based on prior knowledge of expert, KTRA has a significantly higher exploration efficiency, which is indicated by a shorter exploration period and a higher average reward. On the contrary, QLRA suffers a longer exploration phase and a lower average reward, because it needs to explore the task from scratch. To summarize, KTRA achieves a better performance in both network metrics (higher URLLC delay and lower eMBB throughput) and machine learning metrics (better convergence speed and higher average reward). \n\nFinally, the packet drop rate of KTRA and QLRA are 0.042\\% and 0.048\\%, respectively, under 2 Mpbs eMBB traffic and 4 Mbps URLLC traffic. The satisfying packet drop rate is because we apply a penalty in both algorithms to prevent dropping packet. \n\n\\section{Conclusion}\n\\label{s6}\nThe evolving network architecture requires more efficient solutions for network resource allocation. In this work, we propose a KTRA method for joint radio and computation resource allocation. Compared with existing works, the main difference is that the proposed method has a knowledge transfer capability. The proposed KTRA method is compared with Q-learning based resource allocation, and KTRA presents a 18.4\\% lower URLLC delay and 30.1\\% higher eMBB throughput as well as a faster convergence. In the future, we will consider the knowledge transfer of tasks with different state definition.\n\n\\section*{Acknowledgment}\nThis work is supported by Natural Sciences and Engineering Research Council of Canada (NSERC), Collaborative Research and Training Experience Program (CREATE) under Grant 497981 and Canada Research Chairs Program.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe ongoing technological progress in the fabrication and\ncontrol of nanoscale electronic circuits, such as quantum dots,\nhas stimulated detailed studies of various quantum-impurity\nmodels, where a few local degrees of freedom are coupled to a\ncontinuum. Of particular interest are models with\nexperimentally verifiable universal properties. One of the best\nstudied examples is the Anderson single impurity\nmodel,~\\cite{Anderson61} which describes successfully\nelectronic correlations in small quantum\ndots~\\cite{NgLee88,GlazmanRaikh88}. The experimental control of\nmost of the parameters of this model, e.g., the impurity energy\nlevel position or the level broadening due to hybridization\nwith the continuum, allows for detailed\ninvestigations~\\cite{DGG98,vanderWiel00} of the universal\nlow-temperature behavior of the Anderson model.\n\n\\begin{figure*}\n\\includegraphics[width=14cm]{fig1.eps}\n\\caption{A schematic representation of the double-dot system,\n along with its reduction in the local-moment regime\n to an effective Kondo model with a tilted magnetic\n field.\n (a) The model system: two localized levels coupled\n by tunnelling matrix elements to one another and\n to two separate leads. A constant magnetic flux\n induces phase factors on those elements. Spinless\n electrons residing on the two levels experience a\n repulsive interaction.\n (b) The mapping onto a spinful generalized Anderson\n model, with a tilted magnetic field and different\n tunnelling elements for spin-up and spin-down\n electrons.\n (c) The low-energy behavior of the generalized\n Anderson model is mapped onto an anisotropic Kondo\n model with a tilted magnetic field,\n $\\vec{h}_{\\text{tot}}$.\n} \\label{fig:models}\n\\end{figure*}\n\nIn this paper we study the low-energy behavior of a generic\nmodel, depicted in Fig.~\\ref{fig:models}a, which pertains\neither to a single two-level quantum dot or to a double quantum\ndot where each dot harbors only a single level. The spin\ndegeneracy of the electrons is assumed to be lifted by an\nexternal magnetic field. Several variants of this model have\nbeen studied intensely in recent years, in conjunction with a\nplethora of phenomena, such as many-body resonances in the\nspectral density,~\\cite{Boese01} phase lapses in the\ntransmission phase,~\\cite{Silva02,Golosov06} charge\noscillations,~\\cite{Gefen04,Sindel05} and correlation-induced\nresonances in the conductance~\\cite{Meden06PRL,Karrasch06}.\nAlbeit being described by the same model, no clear linkage has\nbeen established between these seemingly different effects. The\nreason is in part due to the large number of model parameters\ninvolved, which so far obscured a clear physical picture. While\nsome exact statements can be made, these are restricted to\ncertain solvable limits,~\\cite{Boese01} and are apparently\nnongeneric~\\cite{Meden06PRL}. Here we construct a framework\nwhich encompasses all parameter regimes of the model, and\nenables a unified description of the various phenomena alluded\nto above, exposing their common physical origin. For the most\ninteresting regime of strong fluctuations between the two\nlevels, we are able to give: (i) explicit analytical\nconditions for the\n occurrence of transmission phase lapses;\n(ii) an explanation of the population inversion and the\n charge oscillations~\\cite{Gefen04,Sindel05,Silvestrov00}\n (including a Kondo enhancement of the latter);\n(iii) a complete account of the correlation-induced\n resonances~\\cite{Meden06PRL} as a disguised Kondo phenomenon.\n\nAfter introducing the details of the double-dot Hamiltonian in\nSec.~\\ref{sec:Model}, we begin our analysis by constructing a\nlinear transformation of the dot operators, \\emph{and} a\nsimultaneous (generally different) linear transformation of the\nlead operators, such that the 2$\\times$2 tunnelling matrix\nbetween the two levels on the dot and the leads becomes\ndiagonal (with generally different eigenvalues). As a result,\nthe electrons acquire a pseudo-spin degree of freedom which is\nconserved upon tunnelling between the dot and the continuum, as\nshown schematically in Fig.~\\ref{fig:models}b. Concomitantly,\nthe transformation generates a local Zeeman magnetic field. In\nthis way the original double-dot model system is transformed\ninto a generalized Anderson impurity model in the presence of a\n(generally tilted) external magnetic field. This first stage is\ndetailed in Sec.~\\ref{sec:ModelAnderson} and\nAppendix~\\ref{App:SVDdetails}.\n\nWe next analyze in Sec.~\\ref{sec:LocalMoment} the low-energy\nproperties of our generalized Anderson model. We confine\nourselves to the local moment regime, in which there is a\nsingle electron on the impurity. The fluctuations of the\npseudo-spin degree of freedom (which translate into charge\nfluctuations between the two localized levels in the original\nmodel) are determined by two competing effects: the polarizing\neffect of the local magnetic field, and the Kondo screening by\nthe itinerant electrons. In order to quantitatively analyze\nthis competition, we derive an effective low-energy Kondo\nHamiltonian, using Haldane's scaling\nprocedure,~\\cite{HaldanePRL78} together with the\nSchrieffer-Wolff~\\cite{Wolff66} transformation and Anderson's\npoor man's scaling~\\cite{Anderson70}. This portion of the\nderivation resembles recent studies of the Kondo effect in the\npresence of ferromagnetic leads,~\\cite{Martinek03PRL} although\nthe physical context and implications are quite different.\n\nAs is mentioned above, the tunnelling between the impurity and\nthe continuum in the generalized Anderson model is (pseudo)\nspin dependent. This asymmetry results in two important\neffects:\n(a) different renormalizations of the two local levels,\n which in turn generates an additional local magnetic\n field~\\cite{Martinek03PRL}. This field is not necessarily\n aligned with the original Zeeman field that is present\n in the generalized Anderson model.\n(b) An anisotropy of the exchange coupling between the\n conduction electrons and the local moment in the Kondo\n Hamiltonian.\nHowever, since the scaling equations for the anisotropic Kondo\nmodel~\\cite{Anderson70,AndersonYuvalHamann70} imply a flow\ntowards the \\emph{isotropic} strong coupling fixed point, the\nlow-energy behavior of the generalized Anderson model can be\nstill described in terms of two competing energy scales, the\nKondo temperature, $T^{}_{K}$, and the renormalized magnetic\nfield, $h_{\\text{tot}}$. Our two-stage mapping, double-dot\n$\\Rightarrow$ generalized Anderson model $\\Rightarrow$\nanisotropic Kondo model (see Fig.~\\ref{fig:models}), allows us\nto obtain analytic expressions for the original model\nproperties in terms of those of the Kondo model. We derive in\nSec.~\\ref{sec:observables} the occupation numbers on the two\nlocalized levels by employing the Bethe \\emph{ansatz} solution\nof the magnetization of a Kondo spin in a finite magnetic\nfield~\\cite{AndreiRMP83,WiegmannA83}. This solution also\nresults in a highly accurate expression for the conductance\nbased upon the Friedel-Langreth sum rule~\\cite{Langreth66}.\nPerhaps most importantly, it provides a single coherent picture\nfor the host of phenomena to which our model has been applied.\n\nExamples of explicit results stemming from our general analysis\nare presented in Sec.~\\ref{sec:results}. First, we consider the\ncase in which the tunnelling is isotropic, being the same for\nspin-up and spin-down electrons. Then the model is exactly\nsolvable by direct application of the Bethe \\emph{ansatz} to\nthe Anderson Hamiltonian~\\cite{WiegmannC83,WiegmannA83}. We\nsolve the resulting equations~\\cite{Okiji82,WiegmannC83}\nnumerically and obtain the occupation numbers for arbitrary\nparameter values of the model, and in particular, for arbitrary\nvalues of the local Zeeman field. By comparing with the\noccupation numbers obtained in Sec.~\\ref{sec:observables} from\nthe Kondo version of the model, we are able to test the\naccuracy of the Schrieffer-Wolff mapping onto the Kondo\nHamiltonian. We find that this mapping yields extremely precise\nresults over the entire local-moment regime. This exactly\nsolvable example has another virtue. It clearly demonstrates\nthe competition between the Kondo screening of the local spin,\nwhich is governed by $T_K$, and the polarizing effect of the\nlocal field $h_{\\text{tot}}$. This competition is reflected\nin the charging process of the quantum dot described by the\noriginal Hamiltonian. We next proceed to apply our general\nmethod to the features for which the anisotropy in the\ntunnelling is relevant, notably the transmission phase lapses\nand the correlation-induced resonances~\\cite{Meden06PRL}. In\nparticular, we derive analytical expressions for the occupation\nnumbers and the conductance employing the mapping onto the\nKondo Hamiltonian. These analytical expressions give results\nwhich are in a very good agreement with the data presented by\nMeden and Marquardt,~\\cite{Meden06PRL} which was obtained by\nthe functional and numerical renormalization-group methods\napplied to the original model.\n\nAs our treatment makes extensive usage of the exact Bethe\n\\emph{ansatz} solutions for the impurity magnetization in the\nisotropic Kondo and Anderson models with a finite magnetic\nfield, all relevant details of the solutions are concisely\ngathered for convenience in Appendix~\\ref{app:Bethe}.\n\n\\section{The double-dot system as a generalized Anderson model}\n\\label{sec:secII}\n\n\\subsection{The model\\label{sec:Model}}\n\nWe consider spinless electrons in a system of two distinct\nenergy levels (a `quantum dot'), labelled $i = 1, 2$, which are\nconnected by tunnelling to two leads, labelled $\\alpha = L, R$.\nThis quantum dot is penetrated by a (constant) magnetic flux.\nThe total Hamiltonian of the system reads\n\\begin{eqnarray}\n\\mathcal{H} = \\mathcal{H}_l + \\mathcal{H}_d + \\mathcal{H}_{ld} \\, ,\n\\label{IHAM}\n\\end{eqnarray}\nin which $\\mathcal{H}_{l}$ is the Hamiltonian of the leads,\n$\\mathcal{H}_{d}$ is the Hamiltonian of the isolated dot, and\n$\\mathcal{H}_{ld}$ describes the coupling between the dot and\nthe leads. The system is portrayed schematically in\nFig.~\\ref{fig:models}a.\n\nEach of the leads is modelled by a continuum of noninteracting\nenergy levels lying within a band of width $2D$, with a\nconstant density of states $\\rho$~\\cite{Comm-on-equal-rho}. The\ncorresponding Hamiltonian is given by\n\\begin{eqnarray}\n\\mathcal{H}_l = \\sum_{ k\\alpha} \\varepsilon^{}_{k}\n c_{k\\alpha}^{\\dagger} c^{}_{k\\alpha} \\, ,\n\\end{eqnarray}\nwhere $c^{\\dagger}_{ k\\alpha}$ ($c^{}_{ k\\alpha}$) creates\n(annihilates) an electron of wave vector $k$ on lead $\\alpha$.\nThe two leads are connected to two external reservoirs, held at\nthe same temperature $T$ and having different chemical\npotentials, $\\mu_L$ and $\\mu_R$, respectively. We take the\nlimit $\\mu_L \\!\\! \\to \\!\\! \\mu_R = 0$ in considering\nequilibrium properties and the linear conductance.\n\n\nThe isolated dot is described by the Hamiltonian\n\\begin{equation}\n\\mathcal{H}_d = \\left [\n \\begin{array}{cc}\n d^{\\dagger}_{1} & d^{\\dagger}_{2}\n \\end{array}\n \\right ] \\cdot \\hat{\\mathcal{E}}_d \\cdot\n \\left [\n \\begin{array}{c}\n d_{1} \\\\ d_{2}\n \\end{array}\n \\right ] + U \\, n_{1} n_{2} \\, ,\n\\label{HDOT}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\hat{\\mathcal{E}}_{d} = \\frac{1}{2}\n \\left [\n \\begin{array}{cc}\n 2 \\, \\epsilon_0 + \\Delta &\n b \\, e^{i(\\varphi_{L}-\\varphi_{R})\/2} \\\\\n b \\, e^{-i(\\varphi_{L}-\\varphi_{R})\/2} &\n 2 \\, \\epsilon_0 -\\Delta\n \\end{array}\n \\right ] \\, .\n\\label{HPS}\n\\end{eqnarray}\nHere, $d^{\\dagger}_{i}$ ($d^{}_{i}$) creates (annihilates) an\nelectron on the $i$th level, $n_i \\equiv d^{\\dagger}_i d^{}_i$\nare the occupation-number operators (representing the local\ncharge), $U>0$ denotes the Coulomb repulsion between electrons\nthat occupy the two levels, $\\epsilon_{0} \\pm \\Delta \/2$ are\nthe (single-particle) energies on the levels, and $b\/2$ is the\namplitude for tunnelling between them. The phases $\\varphi_{L}$\nand $\\varphi_{R}$, respectively, represent the Aharonov-Bohm\nfluxes (measured in units of the flux quantum $2 \\pi \\hbar c\/e$)\nin the left and in the right hopping loops, such that the total\nflux in the two loops is $\\varphi \\equiv \\varphi_{L} +\n\\varphi_{R}$ [see Fig.~\\ref{fig:models}a].\n\nGauge invariance grants us the freedom to distribute the\nAharonov-Bohm phases among the inter-dot coupling $b$ and the\ncouplings between the dot levels and the leads. With the\nconvention of Eq.~(\\ref{HPS}), the coupling between the quantum\ndot and the leads is described by the Hamiltonian\n\\begin{eqnarray}\n\\mathcal{H}_{ld} = \\sum_{k}\n \\left [\n \\begin{array}{cc}\n c^{\\dagger}_{kL} & c^{\\dagger}_{kR}\n \\end{array}\n \\right ] \\cdot \\hat{A} \\cdot\n \\left [\n \\begin{array}{c}\n d_1 \\\\ d_2\n \\end{array}\n \\right ] + \\text{H.c.} \\, ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\hat{A} =\n \\left [\n \\begin{array}{cc}\n a_{L1}e^{i\\varphi \/2} & a_{L2}\\\\\n a_{R1} & a_{R2}e^{i\\varphi \/2}\n \\end{array}\n \\right ]\\, , \\quad\n \\varphi = \\varphi_{L} + \\varphi_{R} \\, .\n\\label{AA}\n\\end{eqnarray}\nHere the real (possibly negative) coefficients $a_{\\alpha i}$\nare the tunnelling amplitudes for transferring an electron from\nthe level $i$ to lead $\\alpha$. Note that the Hamiltonian\ndepends solely on the total Aharonov-Bohm flux $\\varphi$ when\nthe interdot coupling $b$ vanishes. Also, the tunnelling matrix\n$\\hat{A}$ is assumed to be independent of the wave vector $k$.\nThis assumption considerably simplifies the analysis while\nkeeping the main physical picture intact.\n\n\n\\subsection{Mapping onto a generalized Anderson model}\n\\label{sec:ModelAnderson}\n\nThe analysis of the model defined in Sec.~\\ref{sec:Model}\nemploys an {\\it exact} mapping of the Hamiltonian of\nEq.~(\\ref{IHAM}) onto a generalized Anderson Hamiltonian, which\npertains to a single-level quantum dot, coupled to a\nspin-degenerate band of conduction electrons. We show in\nAppendix~\\ref{App:SVDdetails} that the model depicted in\nFig.~\\ref{fig:models}a is fully described by the Hamiltonian\n\\begin{widetext}\n\\begin{equation}\n\\mathcal{H} = \\sum_{k, \\sigma}\n \\varepsilon_k \\, c_{k\\sigma}^{\\dagger} c^{}_{k\\sigma}\n + \\sum_{\\sigma}\n \\Bigl (\n \\epsilon_0 - \\sigma \\frac{h}{2} \\cos \\theta\n \\Bigr )\n n_{\\sigma}\n - \\bigl ( d_{\\uparrow}^{\\dagger} d^{}_{\\downarrow} +\n d_{\\downarrow}^{\\dagger} d^{}_{\\uparrow} \\bigr ) \\,\n \\frac{h}{2} \\, \\sin \\theta\n + U n_{\\uparrow} n_{\\downarrow}\n + \\sum_{k, \\sigma} V^{}_{\\sigma}\n \\Bigl (\n c_{k\\sigma}^{\\dagger} d^{}_{\\sigma} + \\text{H.c.}\n \\Bigr ) \\, ,\n\\label{eq:Hand}\n\\end{equation}\n\\end{widetext}\nschematically sketched Fig.~\\ref{fig:models}b, which\ngeneralizes the original Anderson model~\\cite{Anderson61} in\ntwo aspects. Firstly, it allows for spin-dependent coupling\nbetween the dot and the conduction band. A similar variant of\nthe Anderson model has recently attracted much theoretical and\nexperimental attention in connection with the Kondo effect for\nferromagnetic\nleads~\\cite{Martinek03PRL,MartinekNRGferro,Pasupathy04,MartinekPRB05,\nComment-on-FM}. Secondly, it allows for a Zeeman field whose\ndirection is inclined with respect to the ``anisotropy'' axis\n$z$. For spin-independent tunnelling, one can easily realign\nthe field along the $z$ axis by a simple rotation of the\ndifferent operators about the $y$ axis. This is no longer the\ncase once $V_{\\uparrow} \\neq V_{\\downarrow}$, which precludes the\nuse of some of the exact results available for the Anderson\nmodel. As we show below, the main effect of spin-dependent\ntunnelling is to modify the effective field seen by electrons\non the dot, by renormalizing its $z$-component.\n\nThe derivation of Eq.~(\\ref{eq:Hand}) is accomplished by a\ntransformation known as the singular-value\ndecomposition,~\\cite{Golub96} which allows one to express the\ntunnelling matrix $\\hat{A}$ in the form\n\\begin{equation}\n\\hat{A} = R_l^{\\dagger} \\cdot\n \\left [\n \\begin{array}{cc}\n V_{\\uparrow} & 0 \\\\\n 0 & V_{\\downarrow}\n \\end{array}\n \\right ] \\cdot R^{}_d \\, .\n\\end{equation}\nHere $R_{l}$ and $R_{d}$ are unitary 2$\\times$2 matrices, which\nare used to independently rotate the lead and the dot operators\naccording to\n\\begin{eqnarray}\n \\left [\n \\begin{array}{c}\n d_{\\uparrow} \\\\ d_{\\downarrow}\n \\end{array}\n \\right ] \\equiv R_{d} \\cdot\n \\left [\n \\begin{array}{c}\n d_{1} \\\\ d_{2}\n \\end{array}\n \\right ]\\, , \\quad\n \\left [\n \\begin{array}{c}\n c_{k\\uparrow} \\\\ c_{k\\downarrow}\n \\end{array}\n \\right ] \\equiv R_{l} \\cdot\n \\left [\n \\begin{array}{c}\n c_{kL} \\\\ c_{kR}\n \\end{array}\n \\right ]\\, .\n\\label{eq:SVDdef}\n\\end{eqnarray}\nTo make contact with the conventional Anderson impurity model,\nwe have labelled the linear combinations of the original\noperators [defined through Eqs.~(\\ref{eq:SVDdef})] by the\n``spin'' index $\\sigma = \\uparrow$ ($+1$) and $\\sigma =\n\\downarrow$ ($-1$).\n\nThe transformation (\\ref{eq:SVDdef}) generalizes the one in\nwhich the \\emph{same} rotation $R$ is applied to both the dot\nand the lead operators. It is needed in the present, more\ngeneral, case since the matrix $\\hat{A}$ generically lacks an\northogonal basis of eigenvectors. The matrices $R_d$ and $R_l$\ncan always be chosen uniquely (up to a common overall phase)\nsuch that~\\cite{Comment-on-uniqueness} (a) the tunnelling\nbetween the dot and the continuum is\n diagonal in the spin basis (so that the tunnelling\n conserves the spin);\n(b) the amplitudes $V_{\\uparrow} \\ge V_{\\downarrow} \\ge 0$\n are real; and\n(c) the part of the Hamiltonian of Eq.~(\\ref{eq:Hand})\n pertaining to the dot has only real matrix elements\n with $h \\sin \\theta \\geq 0$.\nThe explicit expressions for the rotation matrices $R_d$\nand $R_l$ as well as for the model parameters appearing\nin Eq.~(\\ref{eq:Hand}) in terms of those of the original\nHamiltonian are given in Appendix~\\ref{App:SVDdetails}.\n\nIt should be emphasized that partial transformations involving\nonly one rotation matrix, either $R_d$ or $R_l$, have\npreviously been applied in this context (see, e.g.,\nRefs.~~\\onlinecite{Boese01} and~~\\onlinecite{Glazman01}).\nHowever, excluding special limits, both $R_d$ and $R_l$ are\nrequired to expose the formal connection to the Anderson model.\nA first step in this direction was recently taken by Golosov\nand Gefen~\\cite{Golosov06}, yet only on a restricted\nmanifold for the tunnelling amplitudes $a_{\\alpha i}$. In the\nfollowing section we discuss in detail the low-energy physics\nof the Hamiltonian of Eq.~(\\ref{eq:Hand}), focusing on the\nlocal-moment regime. Explicit results for the conductance and\nthe occupations of the levels are then presented in\nSecs.~\\ref{sec:observables} and \\ref{sec:results}.\n\n\n\\section{The local-moment regime\\label{sec:LocalMoment}}\n\nThere are two limits where the model of Eq.~(\\ref{IHAM}) has an\nexact solution:~\\cite{Boese01} (i) when the spin-down state is\ndecoupled in Eq.~(\\ref{eq:Hand}), i.e., when $V_{\\downarrow} =\nh\\sin \\theta = 0$; (ii) when the coupling is isotropic, i.e.,\n$V_{\\uparrow} = V_{\\downarrow}$. In the former case,\n$n_{\\downarrow}$ is conserved. The Hilbert space separates then\ninto two disconnected sectors with $n_{\\downarrow} = 0$ and\n$n_{\\downarrow} = 1$. Within each sector, the Hamiltonian can be\ndiagonalized independently as a single-particle problem. In the\nlatter case, one can always align the magnetic field $h$ along\nthe $z$ axis by a simple rotation of the different operators\nabout the $y$ axis. The model of Eq.~(\\ref{eq:Hand}) reduces\nthen to a conventional Anderson model in a magnetic field, for\nwhich an exact Bethe {\\em ansatz} solution is\navailable~\\cite{WiegmannA83}. (This special case will be\nanalyzed in great detail in Sec.~\\ref{sec:ResultsIsotropic}.)\n\n\nIn terms of the model parameters appearing in the original\nHamiltonian, the condition $V_{\\downarrow} = 0$ corresponds to\n\\begin{equation}\n |a_{L1} a_{R2}| = |a_{R1} a_{L2}|, \\ \\ \\text{and}\n \\ \\ \\varphi = \\beta \\!\\!\\!\\! \\mod 2\\pi ,\n\\label{exact-a}\n\\end{equation}\nwhereas $V_{\\uparrow} = V_{\\downarrow} = V$ corresponds to\n\\begin{equation}\n |a_{L1}| = |a_{R2}| , \\\n |a_{L2}| = |a_{R1}| , \\\n \\text{and} \\\n \\varphi = (\\pi + \\beta) \\!\\!\\!\\! \\mod 2\\pi \\, .\n\\label{exact-b}\n\\end{equation}\nHere\n\\begin{equation}\n\\beta = \\left \\{\n \\begin{array}{cc}\n 0 & \\text{if} \\ \\ a_{L1} a_{L2} a_{R1} a_{R2} > 0 \\\\ \\\\\n \\pi & {\\rm if}\\ \\ a_{L1} a_{L2} a_{R1} a_{R2} < 0\n \\end{array}\n \\right .\n\\end{equation}\nrecords the combined signs of the four coefficients\n$a_{\\alpha i}$~\\cite{Comment-on-phi}.\n\nExcluding the two cases mentioned above, no exact solutions to\nthe Hamiltonian of Eq.~(\\ref{IHAM}) are known. Nevertheless,\nwe shall argue below that the model displays generic low-energy\nphysics in the ``local-moment'' regime, corresponding to the\nKondo effect in a finite magnetic field. To this end we focus\nhereafter on $\\Gamma_{\\uparrow}, \\Gamma_{\\downarrow}, h \\ll\n-\\epsilon_0, U + \\epsilon_0$, and derive an effective\nlow-energy Hamiltonian for general couplings. Here\n$\\Gamma_{\\sigma} = \\pi \\rho V_{\\sigma}^2$ is half the\ntunnelling rate between the spin state $\\sigma$ and the leads.\n\n\n\\subsection{Effective low-energy Hamiltonian}\n\nAs is mentioned above, when $V_{\\uparrow} = V_{\\downarrow}$ one\nis left with a conventional Kondo effect in the presence of a\nfinite magnetic field. Asymmetry in the couplings,\n$V_{\\uparrow} \\neq V_{\\downarrow}$, changes this situation in\nthree aspects. Firstly, the effective magnetic field seen by\nelectrons on the dot is modified, acquiring a renormalized\n$z$-component. Secondly, the elimination of the charge\nfluctuations by means of a Schrieffer-Wolff\ntransformation,~\\cite{Wolff66} results in an anisotropic\nspin-exchange interaction. Thirdly, a new interaction term is\nproduced, coupling the spin and the charge. Similar aspects\nhave been previously discussed in the context of the Kondo\neffect in the presence of ferromagnetic\nleads,~\\cite{Martinek03PRL} where the source of the asymmetry\nis the inequivalent density of states for conduction electrons\nwith opposite spin~\\cite{Comment-on-FM}. Below we elaborate on\nthe emergence of these features in the present case.\n\nBefore turning to a detailed derivation of the effective\nlow-energy Hamiltonian, we briefly comment on the physical\norigin of the modified magnetic field. As is well known, the\ncoupling to the continuum renormalizes the bare energy levels\nof the dot. For $\\Gamma_{\\uparrow}, \\Gamma_{\\downarrow}, h \\ll\n-\\epsilon_0, U + \\epsilon_0$, these renormalizations can be\naccurately estimated using second-order perturbation theory in\n$V_{\\sigma}$. For $V_{\\uparrow} \\neq V_{\\downarrow}$, each of\nthe bare levels $\\epsilon_{\\sigma} = \\epsilon_0 - \\frac{1}{2}\n\\sigma h \\cos \\theta$ is shifted by a different amount, which\nacts in effect as an excess magnetic field. Explicitly, for $T\n= 0$ and $D \\gg |\\epsilon_0|, U$ one\nobtains~\\cite{Martinek03PRL,Silvestrov00}\n\\begin{equation}\n\\Delta h_z =\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}{\\pi}\n \\ln \\frac{\\epsilon_0 + U}{|\\epsilon_0|} \\, .\n\\label{Delta-h_z}\n\\end{equation}\nAs $\\epsilon_0$ is swept across $-U\/2$, $\\Delta h_z \\propto\n\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}$ changes sign. Had\n$|\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}|$ exceeded $h$ this\nwould have dictated a sign-reversal of the $z$-component of the\ncombined field as $\\epsilon_0$ is tuned across the\nCoulomb-blockade valley. As originally noted by Silvestrov and\nImry,~\\cite{Silvestrov00} this simple but insightful\nobservation underlies the population inversion discussed in\nRefs.~\\onlinecite{Gefen04,Sindel05}\nand~\\onlinecite{Silvestrov00} for a singly occupied dot. We\nshall return to this important point in greater detail later\non.\n\n\nA systematic derivation of the effective low-energy Hamiltonian\nfor $\\Gamma_{\\uparrow}, \\Gamma_{\\downarrow}, h \\ll -\\epsilon_0, U\n+ \\epsilon_0$ involves the combination of Anderson's poor-man's\nscaling~\\cite{Anderson70} and the Schrieffer-Wolff\ntransformation~\\cite{Wolff66}. For $|\\epsilon_0| \\sim U +\n\\epsilon_0$, the elimination of high-energy excitations\nproceeds in three steps. First Haldane's perturbative scaling\napproach~\\cite{HaldanePRL78} is applied to progressively reduce\nthe bandwidth from its bare value $D$ down to $D_{\\rm SW} \\sim\n|\\epsilon_0| \\sim U + \\epsilon_0$. Next a Schrieffer-Wolff\ntransformation is carried out to eliminate charge fluctuations\non the dot. At the conclusion of this second step one is left\nwith a generalized Kondo Hamiltonian [Eq.~(\\ref{H-Kondo})\nbelow], featuring an anisotropic spin-exchange interaction and\nan additional interaction term that couples spin and charge.\nThe Kondo Hamiltonian also includes a finite magnetic field\nwhose direction is inclined with respect to the anisotropy axis\n$z$. In the third and final stage, the Kondo Hamiltonian is\ntreated using Anderson's poor-man's scaling~\\cite{Anderson70}\nto expose its low-energy physics.\n\nThe above procedure is further complicated in the case where\n$|\\epsilon_0|$ and $U + \\epsilon_0$ are well separated in\nenergy. This situation requires two distinct Schrieffer-Wolff\ntransformations: one at $D_{\\rm SW}^{\\rm up} \\sim \\max \\{\n|\\epsilon_0|, U + \\epsilon_0\\}$ and the other at $D_{\\rm\nSW}^{\\rm down} \\sim \\min \\{ |\\epsilon_0|, U + \\epsilon_0\\}$.\nReduction of the bandwidth from $D_{\\rm SW}^{\\rm up}$ to\n$D_{\\rm SW}^{\\rm down}$ is accomplished using yet another\n(third) segment of the perturbative scaling. It turns out that\nall possible orderings of $|\\epsilon_0|$ and $U + \\epsilon_0$\nproduce the same Kondo Hamiltonian, provided that\n$\\Gamma_{\\uparrow}$, $\\Gamma_{\\downarrow}$ and $h$ are\nsufficiently small. To keep the discussion as concise as\npossible, we therefore restrict the presentation to the case\n$|\\epsilon_0| \\sim U + \\epsilon_0$.\n\nConsider first the energy window between $D$ and $D_{\\rm SW}$,\nwhich is treated using Haldane's perturbative\nscaling~\\cite{HaldanePRL78}. Suppose that the bandwidth has\nalready been lowered from its initial value $D$ to some value\n$D' = D e^{-l}$ with $0 < l < \\ln (D\/D_{SW})$. Further reducing\nthe bandwidth to $D'(1 - \\delta l)$ produces a renormalization\nof each of the energies $\\epsilon_{\\uparrow}$,\n$\\epsilon_{\\downarrow}$, and $U$. Specifically, the\n$z$-component of the magnetic field, $h_z \\equiv\n\\epsilon_{\\downarrow} - \\epsilon_{\\uparrow}$, is found to obey\nthe scaling equation\n\\begin{equation}\n\\frac{d h_z}{d l} =\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}{\\pi}\n \\left [\n \\frac{1}{1 - e^{l} \\epsilon_0\/D} -\n \\frac{1}{1 + e^{l} (U + \\epsilon_0)\/D}\n \\right ] .\n\\label{dh_z-dl}\n\\end{equation}\nHere we have retained $\\epsilon_0$ and $U + \\epsilon_0$ in the\ndenominators, omitting corrections which are higher-order in\n$\\Gamma_{\\uparrow}$, $\\Gamma_{\\downarrow}$, and $h$ (these\ninclude also the small renormalizations of $\\epsilon_{\\sigma}$\nand $U$ that are accumulated in the course of the scaling). The\n$x$-component of the field, $h_x = h \\sin \\theta$, remains\nunchanged throughout the procedure. Upon reaching $D' = D_{\\rm\nSW}$, the renormalized field $h_z$ becomes\n\\begin{equation}\nh_z^{\\ast} = h \\cos \\theta +\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}{\\pi}\n \\ln \\frac{D_{\\rm SW} + U + \\epsilon_0}\n {D_{\\rm SW} - \\epsilon_0} \\, ,\n\\end{equation}\nwhere we have assumed $D \\gg |\\epsilon_0|, U$.\n\nOnce the scale $D_{\\rm SW}$ is reached, charge fluctuations on\nthe dot are eliminated via a Schrieffer-Wolff\ntransformation,~\\cite{Wolff66} which generates among other\nterms also further renormalizations of $\\epsilon_{\\sigma}$.\nNeglecting $h$ in the course of the transformation, one arrives\nat the following Kondo-type Hamiltonian,\n\\begin{eqnarray}\n\\mathcal{H}_K &=& \\sum_{k, \\sigma}\n \\varepsilon^{}_k c_{k\\sigma}^{\\dagger} c^{}_{k\\sigma}\n + J_{\\perp} \\left ( S_x s_x + S_y s_y \\right )\n + J_z S_z s_z\n\\nonumber\\\\\n &+& v_{\\rm sc} S^z \\sum_{k, k', \\sigma}\\!\\!\n :\\! c^{\\dagger}_{k \\sigma} c^{}_{k' \\sigma}\\!:\n + \\sum_{k, k', \\sigma}\\! (v_+ + \\sigma v_-)\\!\n :\\! c^{\\dagger}_{k \\sigma} c^{}_{k' \\sigma}\\!:\n\\nonumber\\\\\n &-& \\tilde{h}_z S_z - \\tilde{h}_x S_x .\n\\label{H-Kondo}\n\\end{eqnarray}\nHere we have represented the local moment on the dot by\nthe spin-$\\frac{1}{2}$ operator\n\\begin{equation}\n\\vec{S} = \\frac{1}{2} \\sum_{\\sigma, \\sigma'}\n \\vec{\\tau}^{}_{\\sigma \\sigma'}\n d^{\\dagger}_{\\sigma} d^{}_{\\sigma'}\n\\end{equation}\n($\\vec{\\tau}$ being the Pauli matrices), while\n\\begin{equation}\n\\vec{s} = \\frac{1}{2} \\sum_{k, k'} \\sum_{\\sigma, \\sigma'}\n \\vec{\\tau}^{}_{\\sigma \\sigma'}\n c^{\\dagger}_{k \\sigma} c^{}_{k' \\sigma'}\n\\end{equation}\nare the local conduction-electron spin densities. The symbol\n$:\\!c^{\\dagger}_{k \\sigma} c^{}_{k' \\sigma}\\!\\!:\\ = c^{\\dagger}_{k\n\\sigma} c^{}_{k' \\sigma} - \\delta_{k, k'} \\theta(-\\epsilon_k)$\nstands for normal ordering with respect to the filled Fermi sea.\nThe various couplings that appear in Eq.~(\\ref{H-Kondo}) are given\nby the explicit expressions\n\\begin{equation}\n\\rho J_{\\perp} =\n \\frac{2 \\sqrt{\\Gamma_{\\uparrow} \\Gamma_{\\downarrow}}}\n {\\pi}\n \\left (\n \\frac{1}{|\\epsilon_0|}\n + \\frac{1}{U + \\epsilon_0}\n \\right ) ,\n\\label{J-perp}\n\\end{equation}\n\\begin{equation}\n\\rho J_{z} =\n \\frac{\\Gamma_{\\uparrow} + \\Gamma_{\\downarrow}}\n {\\pi}\n \\left (\n \\frac{1}{|\\epsilon_0|}\n + \\frac{1}{U + \\epsilon_0}\n \\right ) ,\n\\label{J-z}\n\\end{equation}\n\\begin{equation}\n\\rho v_{\\rm sc} =\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}\n {4 \\pi}\n \\left (\n \\frac{1}{|\\epsilon_0|}\n + \\frac{1}{U + \\epsilon_0}\n \\right ) ,\n\\end{equation}\n\\begin{equation}\n\\rho v_{\\pm} =\n \\frac{\\Gamma_{\\uparrow} \\pm \\Gamma_{\\downarrow}}\n {4 \\pi}\n \\left (\n \\frac{1}{|\\epsilon_0|}\n - \\frac{1}{U + \\epsilon_0}\n \\right ) ,\n\\label{v-pm}\n\\end{equation}\n\\begin{equation}\n\\tilde{h}_z = h \\cos \\theta +\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}\n {\\pi}\n \\ln \\frac{U + \\epsilon_0}{|\\epsilon_0|} \\, ,\n\\label{h_z-tilde}\n\\end{equation}\nand\n\\begin{equation}\n\\tilde{h}_x = h \\sin \\theta .\n\\label{h_x-tilde}\n\\end{equation}\n\nEquations~(\\ref{J-perp})--(\\ref{h_x-tilde}) are correct to\nleading order in $\\Gamma_{\\uparrow}$, $\\Gamma_{\\downarrow}$,\nand $h$, in accordance with the inequality $\\Gamma_{\\uparrow},\n\\Gamma_{\\downarrow}, h \\ll |\\epsilon_0|, U + \\epsilon_0$. In\nfact, additional terms are generated in Eq.~(\\ref{H-Kondo})\nwhen $h$ is kept in the course of the Schrieffer-Wolff\ntransformation. However, the neglected terms are smaller than\nthe ones retained by a factor of $h\/\\min \\{|\\epsilon_0|, U +\n\\epsilon_0 \\} \\ll 1$, and are not expected to alter the\nlow-energy physics in any significant way. We also note that\n$\\tilde{h}_z$ accurately reproduces the second-order correction\nto $h_z$ detailed in Eq.~(\\ref{Delta-h_z}). As emphasized\nabove, the same effective Hamiltonian is obtained when\n$|\\epsilon_0|$ and $U + \\epsilon_0$ are well separated in\nenergy, although the derivation is notably more cumbersome. In\nunifying the different possible orderings of $|\\epsilon_0|$ and\n$U + \\epsilon_0$, the effective bandwidth in\nEq.~(\\ref{H-Kondo}) must be taken to be $D_0 \\sim \\min\n\\{|\\epsilon_0|, U + \\epsilon_0\\}$.\n\n\n\\subsection{Reduction to the Kondo effect in a finite\n magnetic field}\n\nIn addition to spin-exchange anisotropy and a tilted magnetic\nfield, the Hamiltonian of Eq.~(\\ref{H-Kondo}) contains a new\ninteraction term, $v_{\\rm sc}$, which couples spin and charge.\nIt also includes spin-dependent potential scattering,\nrepresented by the term $v_{-}$ above. As is well known,\nspin-exchange anisotropy is irrelevant for the conventional\nspin-$\\frac{1}{2}$ single-channel Kondo problem. As long as one\nlies within the confines of the antiferromagnetic domain, the\nsystem flows to the same strong-coupling fixed point no matter\nhow large the exchange anisotropy is. SU(2) spin symmetry is\nthus restored at low energies. A finite magnetic field $h$ cuts\noff the flow to isotropic couplings, as does the temperature\n$T$. However, the residual anisotropy is negligibly small if\n$h$, $T$ and the bare couplings are small. That is,\nlow-temperature thermodynamic and dynamic quantities follow a\nsingle generic dependence on $T\/T_K$ and $h\/T_k$, where $T_K$\nis the Kondo temperature. All relevant information on the bare\nspin-exchange anisotropy is contained for weak couplings in the\nmicroscopic form of $T_K$.\n\nThe above picture is insensitive to the presence of weak\npotential scattering, which only slightly modifies the\nconduction-electron phase shift at the Fermi energy. As we show\nbelow, neither is it sensitive to the presence of the weak\ncouplings $v_{\\rm sc}$ and $v_{-}$ in Eq.~(\\ref{H-Kondo}). This\nobservation is central to our discussion, as it enables a very\naccurate and complete description of the low-energy physics of\n$\\mathcal{H}_K$ in terms of the conventional Kondo model in a\nfinite magnetic field. Given the Kondo temperature $T_K$ and\nthe direction and magnitude of the renormalized field\npertaining to Eq.~(\\ref{H-Kondo}), physical observables can be\nextracted from the exact Bethe {\\em ansatz} solution of the\nconventional Kondo model. In this manner, one can accurately\ncompute the conductance and the occupation of the levels, as\ndemonstrated in Secs.~\\ref{sec:observables} and\n\\ref{sec:results}.\n\nTo establish this important point, we apply poor-man's\nscaling~\\cite{Anderson70} to the Hamiltonian of\nEq.~(\\ref{H-Kondo}). Of the different couplings that appear in\n$\\mathcal{H}_K$, only $J_z$, $J_{\\perp}$, and $\\tilde{h}_z$ are\nrenormalized at second order. Converting to the dimensionless\nexchange couplings $\\tilde{J}_z = \\rho J_z$ and\n$\\tilde{J}_{\\perp} = \\rho J_{\\perp}$, these are found to obey\nthe standard scaling\nequations~\\cite{AndersonYuvalHamann70,Anderson70}\n\\begin{eqnarray}\n\\frac{ d\\tilde{J}_z }{dl} &=& \\tilde{J}_{\\perp}^2 \\, ,\n\\label{scaling-J_z} \\\\\n\\frac{ d\\tilde{J}_{\\perp} }{dl} &=&\n \\tilde{J}_z \\tilde{J}_{\\perp} \\, ,\n\\label{scaling-J_perp}\n\\end{eqnarray}\nindependent of $v_{\\rm sc}$ and $v_{\\pm}$. Indeed, the\ncouplings $v_{\\rm sc}$ and $v_{\\pm}$ do not affect the scaling\ntrajectories in any way, other than through a small\nrenormalization to $\\tilde{h}_z$:\n\\begin{equation}\n\\frac{d \\tilde{h}_z}{d l} =\n D_0 \\, e^{-l}\n \\left (\n \\tilde{J}_z \\tilde{v}_{-} +\n 2 \\tilde{v}_{\\rm sc} \\tilde{v}_{+}\n \\right ) 8 \\ln 2 .\n\\label{scaling-h_z}\n\\end{equation}\nHere $\\tilde{v}_{\\mu}$ are the dimensionless couplings $\\rho\nv_{\\mu}$ ($\\mu = {\\rm sc}, \\pm$), and $l$ equals $\\ln (D_0\n\/D')$ with $D'$ the running bandwidth.\n\nAs stated above, the scaling equations\n(\\ref{scaling-J_z})--(\\ref{scaling-J_perp}) are identical to\nthose obtained for the conventional anisotropic Kondo model.\nHence, the Kondo couplings flow toward strong coupling along\nthe same scaling trajectories and with the same Kondo\ntemperature as in the absence of $v_{\\rm sc}$ and $v_{\\pm}$.\nStraightforward integration of\nEqs.~(\\ref{scaling-J_z})--(\\ref{scaling-J_perp}) yields\n\\begin{equation}\nT_K = D_0 \\exp\n \\left (\n -\\frac{1}{\\rho \\, \\xi} \\tanh^{-1}\\!\n \\frac{\\xi}{J_{z}}\n \\right )\n\\label{scaling-T_K-1}\n\\end{equation}\nwith $\\xi = \\sqrt{J_z^2 - J_{\\perp}^2}$. Here we have exploited\nthe hierarchy $J_z \\geq J_{\\perp} > 0$ in deriving\nEq.~(\\ref{scaling-T_K-1}). In terms of the original model\nparameters appearing in Eq.~(\\ref{eq:Hand}),\nEq.~(\\ref{scaling-T_K-1}) takes the form\n\\begin{equation}\nT_K = D_0 \\exp\n \\left [\n \\frac{\\pi \\epsilon_0 (U + \\epsilon_0)}\n {2U(\\Gamma_{\\uparrow}-\\Gamma_{\\downarrow})}\n \\ln\\!\n \\frac{\\Gamma_{\\uparrow}}{\\Gamma_{\\downarrow}}\n \\right ] \\, .\n\\label{scaling-T_K-2}\n\\end{equation}\nEquation~(\\ref{scaling-T_K-2}) was obtained within second-order\nscaling, which is known to overestimate the pre-exponential\nfactor that enters $T_K$. We shall not seek an improved\nexpression for $T_K$ encompassing all parameter regimes of\nEq.~(\\ref{eq:Hand}). More accurate expressions will be given\nfor the particular cases of interest, see\nSec.~\\ref{sec:results} below. Much of our discussion will not\ndepend, though, on the precise form of $T_K$. We shall only\nassume it to be sufficiently small such that the renormalized\nexchange couplings can be regarded isotropic starting at\nenergies well above $T_K$.\n\nThe other competing scale which enters the low-energy physics\nis the fully renormalized magnetic field: $\\vec{h}_{\\text{tot}}\n= h^x_{\\text{tot}}\\, \\hat{x} + h^z_{\\text{tot}}\\, \\hat{z}$.\nWhile the transverse field $h^x_{\\text{tot}}$ remains given by\n$h \\sin \\theta$, the longitudinal field $h^z_{\\text{tot}}$ is\nobtained by integration of Eq.~(\\ref{scaling-h_z}), subject to\nthe initial condition of Eq.~(\\ref{h_z-tilde}). Since the\nrunning coupling $\\tilde{J}_z$ is a slowly varying function of\n$l$ in the range where\nEqs.~(\\ref{scaling-J_z})--(\\ref{scaling-h_z}) apply, it can be\nreplaced for all practical purposes by its bare value in\nEq.~(\\ref{scaling-h_z}). Straightforward integration of\nEq.~(\\ref{scaling-h_z}) then yields\n\\begin{eqnarray}\nh_{\\text{tot}}^{z} &=&\n h \\cos \\theta +\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}\n {\\pi}\n \\ln \\frac{U + \\epsilon_0}{|\\epsilon_0|}\n\\nonumber\\\\\n &+& 3 \\ln(2) \\, D_0\n \\frac{\\Gamma^2_{\\uparrow}-\\Gamma^2_{\\downarrow}}\n {\\pi^2} \\times\n \\frac{U (U + 2 \\epsilon_0)}\n {(U + \\epsilon_0)^2 \\epsilon_0^2} \\, ,\n\\label{h-total}\n\\end{eqnarray}\nwhere we have used Eqs.~(\\ref{J-z})--(\\ref{v-pm}) for $J_z$,\n$v_{\\rm sc}$, and $v_{\\pm}$. Note that the third term on the\nright-hand side of Eq.~(\\ref{h-total}) is generally much\nsmaller than the first two terms, and can typically be\nneglected.\n\nTo conclude this section, we have shown that the Hamiltonian of\nEq.~(\\ref{eq:Hand}), and thus that of Eq.~(\\ref{IHAM}), is\nequivalent at sufficiently low temperature and fields to the\nordinary \\emph{isotropic} Kondo model with a tilted magnetic\nfield, provided that $\\Gamma_{\\uparrow}, \\Gamma_{\\downarrow}\n\\ll |\\epsilon_0|, U + \\epsilon_0$. The relevant Kondo\ntemperature is approximately given by\nEq.~(\\ref{scaling-T_K-2}), while the components of\n$\\vec{h}_{\\text{tot}} = h^x_{\\text{tot}}\\, \\hat{x} +\nh^z_{\\text{tot}}\\, \\hat{z}$ are given by $h^x_{\\text{tot}} = h\n\\sin \\theta$ and Eq.~(\\ref{h-total}).\n\n\n\\section{Physical observables\n\\label{sec:observables}}\n\nHaving established the intimate connection between the\ngeneralized Anderson Hamiltonian, Eq.~(\\ref{eq:Hand}), and the\nstandard Kondo model with a tilted magnetic field, we now\nemploy well-known results of the latter model in order to\nobtain a unified picture for the conductance and the occupation\nof the levels of our original model, Eq.~(\\ref{IHAM}). The\nanalysis extends over a rather broad range of parameters. For\nexample, when $U + 2\\epsilon_0 = 0$, then the sole requirement\nfor the applicability of our results is for $\\sqrt{\\Delta^2 +\nb^2}$ to be small. The tunnelling matrix $\\hat{A}$ can be\npractically arbitrary as long as the system lies deep in the\nlocal-moment regime. The further one departs from the middle of\nthe Coulomb-blockade valley the more restrictive the condition\non $\\hat{A}$ becomes in order for $\\vec{h}_{\\rm tot}$ to stay\nsmall. Still, our approach is applicable over a surprisingly\nbroad range of parameters, as demonstrated below. Unless stated\notherwise, our discussion is restricted to zero temperature.\n\n\n\\subsection{Conductance}\n\nAt zero temperature, a local Fermi liquid is formed in the\nKondo model. Only elastic scattering takes place at the Fermi\nenergy, characterized by the scattering phase shifts for the\ntwo appropriate conduction-electron modes. For a finite\nmagnetic field $h$ in the $z$-direction, single-particle\nscattering is diagonal in the spin index. The corresponding\nphase shifts, $\\delta_{\\uparrow}(h)$ and\n$\\delta_{\\downarrow}(h)$, are given by the Friedel-Langreth\nsum rule,~\\cite{Langreth66,Comment-on-Langreth}\n$\\delta_{\\sigma}(h) = \\pi \\qav{n_{\\sigma}}$, which when applied\nto the local-moment regime takes the form\n\\begin{equation}\n\\delta_{\\sigma}(h) = \\frac{\\pi}{2} + \\sigma \\pi M(h) \\, .\n\\label{phase-shift-1}\n\\end{equation}\nHere $M(h)$ is the spin magnetization, which\nreduces~\\cite{Comment-on-M_K} in the scaling\nregime to a universal function of $h\/T_K$,\n\\begin{equation}\n\\label{eq:MhUniversal}\nM(h) = M_K(h\/T_K) \\, .\n\\end{equation}\nThus, Eq.~(\\ref{phase-shift-1}) becomes\n$\\delta_{\\sigma}(h) = \\pi\/2 + \\sigma \\pi M_K(h\/T_K)$,\nwhere $M_K(h\/T_K)$ is given by Eq.~(\\ref{eq:MKfullWiegmann})\n\nTo apply these results to the problem at hand, one first needs\nto realign the tilted field along the $z$ axis. This is\nachieved by a simple rotation of the different operators about\nthe $y$ axis. Writing the field $\\vec{h}_{\\text{tot}}$ in the\npolar form\n\\begin{align}\n\\label{eq:htotExplicit}\n\\vec{h}_{\\text{tot}} & \\equiv h_{\\text{tot}}\n \\left (\n \\sin \\theta_h \\hat{x} +\n \\cos \\theta_h \\hat{z}\n \\right ) \\nonumber\\\\\n & \\approx h \\sin \\theta \\, \\hat{x} +\n \\left ( h \\cos \\theta +\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}\n {\\pi}\n \\ln \\frac{U + \\epsilon_0}{|\\epsilon_0|}\n \\right ) \\hat{z} \\, ,\n\\end{align}\nthe lead and the dot operators are rotated according to\n\\begin{equation}\n \\left [\n \\begin{array}{c}\n \\tilde{c}_{k\\uparrow} \\\\\n \\tilde{c}_{k\\downarrow}\n \\end{array}\n \\right ] = R_{h} \\cdot\n \\left [\n \\begin{array}{c}\n c_{k\\uparrow} \\\\ c_{k\\downarrow}\n \\end{array}\n \\right ] = R_{h} R_{l} \\cdot\n \\left [\n \\begin{array}{c}\n c_{kL} \\\\ c_{kR}\n \\end{array}\n \\right ]\n\\end{equation}\nand\n\\begin{equation}\n \\left [\n \\begin{array}{c}\n \\tilde{d}_{\\uparrow} \\\\\n \\tilde{d}_{\\downarrow}\n \\end{array}\n \\right ] = R_{h} \\cdot\n \\left [\n \\begin{array}{c}\n d_{\\uparrow} \\\\ d_{\\downarrow}\n \\end{array}\n \\right ] = R_h R_{d} \\cdot\n \\left [\n \\begin{array}{c}\n d_{1} \\\\ d_{2}\n \\end{array}\n \\right ]\\ , \\label{eq:RhRddef}\n\\end{equation}\nwith\n\\begin{equation}\nR_{h} = e^{i (\\theta_h\/2) \\tau_y} =\n \\left [\n \\begin{array}{cc}\n \\ \\ \\cos (\\theta_h\/2) &\\\n \\sin (\\theta_h\/2) \\\\\n - \\sin (\\theta_h\/2) &\\\n \\cos (\\theta_h\/2)\n \\end{array}\n \\right ] \\, . \\label{eq:Rhdef}\n\\end{equation}\nHere $R_{l}$ and $R_{d}$ are the unitary matrices used in\nEq.~(\\ref{eq:SVDdef}) to independently rotate the lead and the\ndot operators. Note that since $\\sin \\theta \\ge 0 $, the range\nof $\\theta_h$ is $\\theta_h \\in [0; \\pi]$.\n\nThe new dot and lead degrees of freedom have their spins\naligned either parallel ($\\tilde{d}_{\\uparrow}$ and\n$\\tilde{c}_{k \\uparrow}$) or antiparallel\n($\\tilde{d}_{\\downarrow}$ and $\\tilde{c}_{k \\downarrow}$) to\nthe field $\\vec{h}_{\\text{tot}}$. In this basis the\nsingle-particle scattering matrix is diagonal,\n\\begin{equation}\n\\tilde{S} =\n - \\left [\n \\begin{array}{cc}\n e^{i 2 \\pi M_K(h_{\\text{tot}}\/T_K)} &\n 0 \\\\\n 0 &\n e^{-i 2 \\pi M_K(h_{\\text{tot}}\/T_K)}\n \\end{array}\n \\right ] \\, .\n\\label{Scatt-mat-hat}\n\\end{equation}\nThe conversion back to the original basis set of left- and\nright-lead electrons is straightforward,\n\\begin{equation}\nS = R^{\\dagger}_{l} R^{\\dagger}_{h} \\tilde{S}\n R_{h}^{} R_{l}^{} \\equiv\n \\left [\n \\begin{array}{cc}\n r & t' \\\\\n t & r'\n \\end{array}\n \\right ] \\, ,\n\\label{Scatt-mat}\n\\end{equation}\nproviding us with the zero-temperature conductance\n$G = (e^2\/2 \\pi \\hbar) |t|^2$.\n\nEquations~(\\ref{Scatt-mat-hat}) and (\\ref{Scatt-mat})\nwere derived employing the mapping of Eq.~(\\ref{IHAM})\nonto an effective isotropic Kondo model with a tilted\nmagnetic field, in the $v_{\\rm sc}, v_{\\pm} \\to 0$\nlimit. Within this framework, Eqs.~(\\ref{Scatt-mat-hat})\nand (\\ref{Scatt-mat}) are exact in the scaling regime,\n$T_K\/D_0 \\ll 1$. The extent to which these equations\nare indeed valid can be appreciated by considering the special case\n$h \\sin \\theta = 0$, for which there exists an exact (and\nindependent) solution for the scattering matrix $S$ in terms of the\ndot ``magnetization'' $M = \\langle n_{\\uparrow} - n_{\\downarrow}\n\\rangle\/2$ [see Eq.~(\\ref{Scatt-mat-Langreth}) below]. That\nsolution, which is based on the Friedel-Langreth sum\nrule~\\cite{Langreth66} applied directly to a spin-conserving\nAnderson model, reproduces Eqs.~(\\ref{Scatt-mat-hat}) and\n(\\ref{Scatt-mat}) in the Kondo regime.\n\n\n\\subsubsection{Zero Aharonov-Bohm fluxes}\n\nOf particular interest is the case where no Aharonov-Bohm\nfluxes are present, where further analytic progress can\nbe made. For $\\varphi_L = \\varphi_R = 0$, the parameters\nthat appear in the Hamiltonian of Eq.~(\\ref{IHAM}) are\nall real. Consequently, the rotation matrices $R_{d}$\nand $R_{l}$ acquire the simplified forms given by\nEqs.~\\eqref{R_d-no-AB} and \\eqref{R_l-no-AB}\n(see Appendix~\\ref{App:SVDdetails} for details). Under\nthese circumstances, the matrix product $R_{h} R_{l}$\nbecomes $\\pm e^{i \\tau_y (\\theta_h + s_R \\, \\theta_l)\/2}\ne^{i \\pi \\tau_z (1 - s_R)\/4}$, and the elements of the\nscattering matrix [see Eq.~(\\ref{Scatt-mat})] are\n\\begin{align}\nt = t' =& - i \\sin[2\\pi M_K(h_{\\text{tot}}\/T_K)]\n \\sin (\\theta_l + s_R \\theta_h) \\ ,\n\\nonumber\\\\\nr = (r')^{\\ast} =&\n - \\cos[2\\pi M_K(h_{\\text{tot}}\/T_K)]\n\\nonumber\\\\\n &- i \\sin[2\\pi M_K(h_{\\text{tot}}\/T_K)]\n \\cos (\\theta_l + s_R \\theta_h) \\ .\n\\end{align}\nHence, the conductance is\n\\begin{align}\nG = \\frac{e^2}{2 \\pi \\hbar}\n \\sin^2[2\\pi M_K(h_{\\text{tot}}\/T_K)]\n \\sin^2(\\theta_l + s_R \\theta_h) \\, ,\n\\label{G-no-flux}\n\\end{align}\nwhere the sign $s_R$ and angle $\\theta_l$ are given by\nEqs.~\\eqref{eq:sR} and \\eqref{eq:theta-no-AB}, respectively.\nAll dependencies of the conductance on the original model\nparameters that enter Eq.~(\\ref{IHAM}) are combined in\nEq.~(\\ref{G-no-flux}) into two variables alone, $\\theta_l + s_R\n\\theta_h$ and the reduced field $h_{\\text{tot}}\/T_K$. In\nparticular, $\\theta_l$ is determined exclusively by the\ntunnelling matrix $\\hat{A}$, while $s_R$ depends additionally\non the two dot parameters $\\Delta$ and $b$.\n\nThe conditions for a phase lapse to occur are particularly\ntransparent from Eq.~(\\ref{G-no-flux}). These lapses correspond\nto zeroes of $t$, and, in turn, of the conductance. There are\ntwo possibilities for $G$ to vanish: either $h_{\\text{tot}}$ is\nzero, or $\\theta_l + s_R \\theta_h$ equals an integer multiple\nof $\\pi$. For example, when the Hamiltonian of\nEq.~(\\ref{eq:Hand}) is invariant under the particle-hole\ntransformation $d_{\\sigma} \\to d_{\\sigma}^{\\dagger}$ and $c_{k\n\\sigma} \\to -c_{k \\sigma}^{\\dagger}$ (which happens to be the\ncase whenever $\\sqrt{ \\Delta^2 + b^2} = 0$ and $U + 2\\epsilon_0\n= 0$), then $h_{\\text{tot}}$ vanishes, and consequently the\nconductance vanishes as well. A detailed discussion of the\nramifications of Eq.~(\\ref{G-no-flux}) is held in\nSec.~\\ref{sec:ResultsAnisotropic} below.\n\n\n\\subsubsection{Parallel-field configuration}\n\nFor $h \\sin \\theta = 0$, spin is conserved by the\nHamiltonian of Eq.~(\\ref{eq:Hand}). We refer to this\ncase as the ``parallel-field'' configuration, since the\nmagnetic field is aligned with the anisotropy axis $z$.\nFor a parallel field, one can easily generalize the\nFriedel-Langreth sum rule~\\cite{Langreth66} to the\nHamiltonian of Eq.~(\\ref{eq:Hand}).~\\cite{MartinekNRGferro}\nApart from the need to consider\neach spin orientation separately, details of the\nderivation are identical to those for the ordinary\nAnderson model,~\\cite{Langreth66} and so is the\nformal result for the $T = 0$ scattering phase\nshift: $\\delta_{\\sigma} = \\pi \\Delta N_{\\sigma}$,\nwhere $\\Delta N_{\\sigma}$ is the number of\ndisplaced electrons in the spin channel $\\sigma$.\nIn the wide-band limit, adopted throughout our\ndiscussion, $\\Delta N_{\\sigma}$ reduces to the\noccupancy of the corresponding dot level,\n$\\langle n_{\\sigma} \\rangle$. The exact\nsingle-particle scattering matrix then becomes\n\\begin{equation}\nS = e^{i\\pi \\langle n_{\\uparrow} + n_{\\downarrow} \\rangle}\n R^{\\dagger}_{l} \\cdot\n \\left [\n \\begin{array}{cc}\n e^{i 2 \\pi M} & 0 \\\\\n 0 & e^{-i 2 \\pi M}\n \\end{array}\n \\right ] \\cdot R_{l} \\, ,\n\\label{Scatt-mat-Langreth}\n\\end{equation}\nwhere $M = \\langle n_{\\uparrow}-n_{\\downarrow} \\rangle\/2$\nis the dot ``magnetization.''\n\nEquation~(\\ref{Scatt-mat-Langreth}) is quite general. It covers\nall physical regimes of the dot, whether empty, singly occupied\nor doubly occupied, and extends to arbitrary fluxes $\\varphi_L$\nand $\\varphi_R$. Although formally exact, it does not specify\nhow the dot ``magnetization'' $M$ and the total dot occupancy\n$\\langle n_{\\uparrow} + n_{\\downarrow} \\rangle$ relate to the\nmicroscopic model parameters that appear in\nEq.~(\\ref{eq:Hand}). Such information requires an explicit\nsolution for these quantities. In the Kondo regime considered\nabove, $\\langle n_{\\uparrow} + n_{\\downarrow} \\rangle$ is\nreduced to one and $M$ is replaced by $\\pm\nM_K(h_{\\text{tot}}\/T_K)$. Here the sign depends on whether the\nfield $\\vec{h}_{\\text{tot}}$ is parallel or antiparallel to the\n$z$ axis (recall that $h_{\\text{tot}} \\ge 0$ by definition). As\na result, Eq.~(\\ref{Scatt-mat-Langreth}) reproduces\nEqs.~(\\ref{Scatt-mat-hat})--(\\ref{Scatt-mat}).\n\nTo carry out the rotation in Eq.~(\\ref{Scatt-mat-Langreth}), we\nrewrite it in the form\n\\begin{equation}\nS = e^{i\\pi \\langle n_{\\uparrow} + n_{\\downarrow} \\rangle}\n R^{\\dagger}_{l}\n \\left [\n \\cos (2 \\pi M) + i\\sin (2 \\pi M) \\tau_z\n \\right ]\n R_{l} \\, .\n\\end{equation}\nUsing the general form of Eq.~(\\ref{eq:phasel}) for the\nrotation matrix $R_{l}$, the single-particle scattering matrix\nis written as $S = e^{i\\pi \\langle n_{\\uparrow} +\nn_{\\downarrow} \\rangle} \\bar{S}$, where\n\\begin{eqnarray}\n\\bar{S} &=& \\cos (2 \\pi M)\n + i \\sin (2 \\pi M) \\cos \\theta_l\\, \\tau_z\n\\nonumber \\\\\n &+&\n i \\sin (2 \\pi M) \\sin \\theta_l\n \\left [\n \\cos \\phi_l\\, \\tau_x + \\sin \\phi_l\\, \\tau_y\n \\right ] \\, .\n\\end{eqnarray}\nThe zero-temperature conductance,\n$G = (e^2\/2 \\pi \\hbar)|t|^2$, takes then the exact form\n\\begin{equation}\nG = \\frac{e^2}{2 \\pi \\hbar}\n \\sin^2(2\\pi M) \\sin^2 \\theta_l \\, .\n\\label{G-parallel-field}\n\\end{equation}\n\nTwo distinct properties of the conductance are apparent form\nEq.~(\\ref{G-parallel-field}). Firstly, $G$ is bounded by\n$\\sin^2 \\theta_l $ times the conductance quantum unit $e^2\/2\n\\pi \\hbar$. Unless $\\theta_{l}$ happens to equal $\\pm \\pi\/2$,\nthe maximal conductance is smaller than $e^2\/2 \\pi \\hbar$.\nSecondly, $G$ vanishes for $M = 0$ and is maximal for $M = \\pm\n1\/4$. Consequently, when $M$ is tuned from $M \\approx -1\/2$ to\n$M \\approx 1\/2$ by varying an appropriate control parameter\n(for example, $\\epsilon_0$ when $\\Gamma_{\\uparrow} \\gg\n\\Gamma_{\\downarrow}$), then $G$ is peaked at the points where\n$M = \\pm 1\/4$. In the Kondo regime, when $M \\to \\pm\nM_K(h_{\\text{tot}}\/T_K)$, this condition is satisfied for\n$h_{\\text{tot}} \\approx 2.4 T_K$. As we show in\nSec.~\\ref{sec:ResultsAnisotropic}, this is the physical origin\nof the correlation-induced peaks reported by Meden and\nMarquardt.~\\cite{Meden06PRL} Note that for a given fixed\ntunnelling matrix $\\hat{A}$ in the parallel-field\nconfiguration, the condition for a phase lapse to occur is\nsimply for $M$ to vanish.\n\n\n\\subsection{Occupation of the dot levels}\n\nSimilar to the zero-temperature conductance, one can exploit\nexact results of the standard Kondo model to obtain the\noccupation of the levels at low temperatures and fields.\nDefining the two reduced density matrices\n\\begin{equation}\nO_d =\n \\begin{bmatrix}\n \\langle d^{\\dagger}_1 d^{}_1 \\rangle &\n \\langle d^{\\dagger}_2 d^{}_1 \\rangle \\\\\n \\langle d^{\\dagger}_1 d^{}_2 \\rangle &\n \\langle d^{\\dagger}_2 d^{}_2 \\rangle\n \\end{bmatrix}\n\\end{equation}\nand\n\\begin{equation}\n\\tilde{O}_d =\n \\begin{bmatrix}\n \\langle\n \\tilde{d}^{\\dagger}_{\\uparrow}\n \\tilde{d}^{}_{\\uparrow}\n \\rangle &\n \\langle\n \\tilde{d}^{\\dagger}_{\\downarrow}\n \\tilde{d}^{}_{\\uparrow}\n \\rangle \\\\\n \\langle\n \\tilde{d}^{\\dagger}_{\\uparrow}\n \\tilde{d}^{}_{\\downarrow}\n \\rangle &\n \\langle\n \\tilde{d}^{\\dagger}_{\\downarrow}\n \\tilde{d}^{}_{\\downarrow}\n \\rangle\n \\end{bmatrix}\n\\, ,\n\\end{equation}\nthese are related through\n\\begin{equation}\nO_d^{} = R^{\\dagger}_{d} R^{\\dagger}_{h} \\tilde{O}_d^{}\n R^{}_{h} R^{}_{d} \\, .\n\\label{O_d-via-t-O_d}\n\\end{equation}\nHere $R_{h} R_{d}$ is the overall rotation matrix pertaining to\nthe dot degrees of freedom, see Eq.~\\eqref{eq:RhRddef}.\n\nAt low temperatures, the mapping onto an isotropic Kondo model\nimplies\n\\begin{equation}\n\\tilde{O}_d = \\begin{bmatrix}\n \\qav{\\tilde{n}_{\\uparrow}} & 0 \\\\\n 0 & \\qav{\\tilde{n}_{\\downarrow}} \\\\\n \\end{bmatrix} \\, ,\n\\label{t-O_d}\n\\end{equation}\nwhere\n\\begin{equation}\n\\qav{\\tilde{n}_{\\sigma}} = n_{\\text{tot}}\/2 +\n \\sigma \\tilde{M} \\, .\n\\label{eq:ntildeseparated}\n\\end{equation}\nHere we have formally separated the occupancies\n$\\qav{\\tilde{n}_{\\sigma}}$ into the sum of a spin component and\na charge component. The spin component involves the\nmagnetization $\\tilde{M}$ along the direction of the total\neffective field $\\vec{h}_{\\text{tot}}$. The latter is well\ndescribed by the universal magnetization curve\n$M_K(h_{\\text{tot}}\/T_K)$ of the Kondo model [see\nEq.~\\eqref{eq:MKfullWiegmann}]. As for the total dot occupancy\n$n_{\\text{tot}}$, deep in the local-moment regime charge\nfluctuations are mostly quenched at low temperatures, resulting\nin the near integer valance $n_{\\text{tot}} \\approx 1$. One can\nslightly improve on this estimate of $n_{\\text{tot}}$ by\nresorting to first-order perturbation theory in\n$\\Gamma_{\\sigma}$ (and zeroth order in $h$):\n\\begin{align}\nn_{\\text{tot}} & \\approx\n 1 + \\frac{\\Gamma_{\\uparrow} +\\Gamma_{\\downarrow}}{2 \\pi}\n \\left (\n \\frac{1}{\\epsilon_0}+\\frac{1}{U+\\epsilon_0}\n \\right )\n = 1 -2 \\rho v_{+} \\, .\n\\label{eq:n0PT}\n\\end{align}\nThis low-order process does not enter the Kondo\neffect, and is not contained in\n$M_K(h_{\\text{tot}}\/T_K)$.~\\cite{Comment-on-charge-fluc}\nWith the above approximations, the combination of\nEqs.~(\\ref{O_d-via-t-O_d}) and (\\ref{t-O_d}) yields\na general formula for the reduced density matrix\n\\begin{equation}\nO_d = n_{\\text{tot}}\/2 + M_K(h_{\\rm tot}\/T_K)\n R^{\\dagger}_{d} R^{\\dagger}_{h}\n \\tau_z R^{}_{h} R^{}_{d} \\, .\n\\label{O_d-general}\n\\end{equation}\n\n\\subsubsection{Zero Aharonov-Bohm fluxes}\n\nAs in the case of the conductance, Eq.~(\\ref{O_d-general})\nconsiderably simplifies in the absence of Aharonov-Bohm\nfluxes, when the combined rotation $R_{h} R_{d}$ equals\n$(s_R s_{\\theta})^{1\/2} e^{i \\tau_y (\\theta_h +\ns_{\\theta} \\theta_d)\/2} e^{i \\pi \\tau_z (1 - s_{\\theta})\/4}$\n[see Eqs.~\\eqref{eq:Rhdef} and \\eqref{R_d-no-AB}].\nExplicitly, Eq.~(\\ref{O_d-general}) becomes\n\\begin{eqnarray}\nO_d = n_{\\text{tot}}\/2 &+&\n M_K(h_{\\rm tot}\/T_K)\n \\cos (\\theta_d + s_{\\theta} \\theta_h) \\tau_z\n\\nonumber \\\\\n &+& M_K(h_{\\rm tot}\/T_K)\n \\sin (\\theta_d + s_{\\theta} \\theta_h) \\tau_x \\, ,\n\\label{O_d-no-flux}\n\\end{eqnarray}\nwhere the sign $s_{\\theta}$ and angle $\\theta_d$ are given\nby Eqs.~\\eqref{eq:stheta} and \\eqref{eq:theta-no-AB},\nrespectively.\n\nSeveral observations are apparent from Eq.~(\\ref{O_d-no-flux}).\nFirstly, when written in the original ``spin'' basis\n$d^{\\dagger}_1$ and $d^{\\dagger}_2$, the reduced density matrix\n$O_d$ contains the off-diagonal matrix element $M_K(h_{\\rm\ntot}\/T_K) \\sin (\\theta_d + s_{\\theta} \\theta_h)$. The latter\nreflects the fact that the original ``spin'' states are\ninclined with respect to the anisotropy axis dynamically\nselected by the system. Secondly, similar to the conductance of\nEq.~(\\ref{G-no-flux}), $O_{d}$ depends on two variables alone:\n$\\theta_d + s_{\\theta} \\theta_h$ and the reduced field\n$h_{\\text{tot}}\/T_K$. Here, again, the angle $\\theta_d$ depends\nsolely on the tunnelling matrix $\\hat{A}$, while the sign\n$s_{\\theta}$ depends additionally on $\\Delta$ and $b$. Thirdly,\nthe original levels $d^{\\dagger}_1$ and $d^{\\dagger}_2$ have\nthe occupation numbers\n\\begin{subequations}\n\\label{eq:Actualn1n2}\n\\begin{align}\n\\qav{n_1} &=\n n_{\\text{tot}}\/2 + M_K(h_{\\text{tot}}\/T_K)\n \\cos (\\theta_d + s_{\\theta} \\theta_h) \\, , \\\\\n\\qav{n_2} &=\n n_{\\text{tot}}\/2 - M_K(h_{\\text{tot}}\/T_K)\n \\cos (\\theta_d + s_{\\theta} \\theta_h) \\, .\n\\end{align}\n\\end{subequations}\nIn particular, equal populations $\\langle n_1 \\rangle = \\langle\nn_2 \\rangle$ are found if either $h_{\\text{tot}}$ is zero or if\n$\\theta_d +s_{\\theta} \\theta_d$ equals $\\pi\/2$ up to an integer\nmultiple of $\\pi$. This provides one with a clear criterion for\nthe occurrence of population\ninversion,~\\cite{Gefen04,Sindel05,Silvestrov00} i.e., the\ncrossover from $\\qav{n_1} > \\qav{ n_2}$ to\n$\\qav{n_2} > \\qav{n_1}$ or vice versa.\n\n\\subsubsection{Parallel-field configuration}\n\\label{sec:occupany-PF}\n\nIn the parallel-field configuration, the angle $\\theta_h$ is\neither zero or $\\pi$, depending on whether the magnetic field\n$\\vec{h}_{\\text{tot}}$ is parallel or antiparallel to the $z$\naxis (recall that $h \\sin \\theta = h_{\\text{tot}} \\sin\n\\theta_h=0$ in this case). The occupancies $\\langle n_1\n\\rangle$ and $\\langle n_2 \\rangle$ acquire the exact\nrepresentation\n\\begin{subequations}\n\\label{eq:Actualn1n2-PF}\n\\begin{align}\n\\langle n_1 \\rangle &=\n n_{\\text{tot}}\/2 + M \\cos \\theta_d \\, , \\\\\n\\langle n_2 \\rangle &=\n n_{\\text{tot}}\/2 - M \\cos \\theta_d \\, ,\n\\end{align}\n\\end{subequations}\nwhere $n_{\\text{tot}}$ is the exact total occupancy of the dot\nand $M = \\langle n_{\\uparrow} - n_{\\downarrow} \\rangle\/2$ is\nthe dot ``magnetization,'' defined and used previously (not to\nbe confused with $\\tilde{M} = \\pm M$). As with the conductance,\nEqs.~(\\ref{eq:Actualn1n2-PF}) encompass all regimes of the dot,\nand extend to arbitrary Aharonov-Bohm fluxes. They properly\nreduce to Eqs.~(\\ref{eq:Actualn1n2}) in the Kondo regime, when\n$n_{\\text{tot}} \\approx 1$ [see Eq.~\\eqref{eq:n0PT}] and $M \\to\n\\pm M_K(h_{\\text{tot}}\/T_K)$. [Note that Eqs.~(\\ref{eq:Actualn1n2})\nhave been derived for zero Aharonov-Bohm fluxes.]\n\nOne particularly revealing observation that follows from\nEqs.~(\\ref{eq:Actualn1n2-PF}) concerns the connection between\nthe phenomena of population inversion and phase lapses in the\nparallel-field configuration. For a given fixed tunnelling\nmatrix $\\hat{A}$ in the parallel-field configuration, the\ncondition for a population inversion to occur is identical to\nthe condition for a phase lapse to occur. Both require that $M\n= 0$. Thus, these seemingly unrelated phenomena are synonymous\nin the parallel-field configuration. This is not generically\nthe case when $h_{\\text{tot}}^x\\neq 0$, as can be seen, for\nexample, from Eqs.~(\\ref{G-no-flux}) and (\\ref{eq:Actualn1n2}).\nIn the absence of Aharonov-Bohm fluxes, the conductance is\nproportional to $\\sin^2(\\theta_l + s_R \\theta_h)$. It therefore\nvanishes for $h_{\\text{tot}}^x\\neq 0$ only if $\\theta_l + s_R\n\\theta_h = 0\\!\\!\\mod\\!\\pi$. By contrast, the difference in\npopulations $\\langle n_1 - n_2 \\rangle$ involves the unrelated\nfactor $\\cos (\\theta_d + s_{\\theta}\\theta_h)$, which generally\ndoes not vanish together with $\\sin(\\theta_l + s_R \\theta_h)$.\n\nAnother useful result which applies to the parallel-field\nconfiguration is an exact expression for the $T = 0$\nconductance in terms of the population difference $\\langle n_1\n- n_2 \\rangle$. It follows from Eqs.~(\\ref{eq:Actualn1n2-PF})\nthat $M = \\qav{n_1 - n_2}\/( 2 \\cos \\theta_d)$. Inserting this\nrelation into Eq.~(\\ref{G-parallel-field}) yields\n\\begin{align}\nG = \\frac{e^2}{2 \\pi \\hbar}\n \\sin^2 \\left (\n \\frac{\\pi \\langle n_1-n_2 \\rangle}\n {\\cos \\theta_d}\n \\right )\n \\sin^2 \\theta_l \\, .\n\\label{G-parallel}\n\\end{align}\nThis expression will be used in Sec.~\\ref{sec:results} for\nanalyzing the conductance in the presence of isotropic\ncouplings, and for the cases considered by Meden and\nMarquardt~\\cite{Meden06PRL}.\n\n\n\\section{Results}\n\\label{sec:results}\n\nUp until this point we have developed a general framework for\ndescribing the local-moment regime in terms of two competing\nenergy scales, the Kondo temperature $T_K$ and the renormalized\nmagnetic field $h_{\\text{tot}}$. We now turn to explicit\ncalculations that exemplify these ideas. To this end, we begin\nin Sec.~\\ref{sec:ResultsIsotropic} with the exactly solvable\ncase $V_{\\uparrow} = V_{\\downarrow}$, which corresponds to the\nconventional Anderson model in a finite magnetic\nfield~\\cite{Boese01}. Using the exact Bethe \\emph{ansatz}\nsolution of the Anderson model,~\\cite{WiegmannA83} we present a\ndetailed analysis of this special case with three objectives in\nmind: (i) to benchmark our general treatment against\n rigorous results;\n(ii) to follow in great detail the delicate interplay\n between the two competing energy scales that\n govern the low-energy physics;\n(iii) to set the stage for the complete explanation of the\n charge oscillations~\\cite{Gefen04,Sindel05,Silvestrov00}\n and the correlation-induced resonances in the\n conductance of this device~\\cite{Meden06PRL,Karrasch06}.\n\nWe then proceed in Sec.~\\ref{sec:ResultsAnisotropic} to the\ngeneric anisotropic case $V_{\\uparrow} \\neq V_{\\downarrow}$. Here\na coherent explanation is provided for the ubiquitous phase\nlapses,~\\cite{Golosov06} population\ninversion,~\\cite{Gefen04,Sindel05} and correlation-induced\nresonances~\\cite{Meden06PRL,Karrasch06} that were reported\nrecently in various studies of two-level quantum dots. In\nparticular, we expose the latter resonances as a disguised\nKondo phenomenon. The general formulae of\nSec.~\\ref{sec:observables} are quantitatively compared to the\nnumerical results of Ref.~\\onlinecite{Meden06PRL}. The detailed\nagreement that is obtained nicely illustrates the power of the\nanalytical approach put forward in this paper.\n\n\n\\subsection{Exact treatment of $V_{\\uparrow}=V_{\\downarrow}$}\n\\label{sec:ResultsIsotropic}\n\nAs emphasized in Sec.~\\ref{sec:LocalMoment}, all tunnelling\nmatrices $\\hat{A}$ which satisfy Eq.~\\eqref{exact-b} give rise\nto equal amplitudes $V_{\\uparrow} = V_{\\downarrow} = V$ within\nthe Anderson Hamiltonian description of Eq.~\\eqref{eq:Hand}.\nGiven this extra symmetry, one can always choose the unitary\nmatrices $R_{l}$ and $R_{d}$ in such a way that the magnetic\nfield $h$ points along the $z$ direction [namely, $\\cos \\theta\n= 1$ in Eq.~\\eqref{eq:Hand}]. Perhaps the simplest member in\nthis class of tunnelling matrices is the case where $a_{L1} =\n-a_{L2} = a_{R1} = a_{R2} = V\/\\sqrt{2}$, $\\varphi_L = \\varphi_R\n= 0$ and $b=0$. One can simply convert the conduction-electron\noperators to even and odd combinations of the two leads,\ncorresponding to choosing $\\theta_l = \\pi\/2 + \\theta_d$.\nDepending on the sign of $\\Delta$, the angle $\\theta_d$ is\neither zero (for $\\Delta < 0$) or $\\pi$ (for $\\Delta > 0$),\nwhich leaves us with a conventional Anderson impurity in the\npresence of the magnetic field $\\vec{h} = |\\Delta| \\, \\hat{z}$.\nAll other rotation angle that appear in Eqs.~\\eqref{eq:phased}\nand \\eqref{eq:phased} (i.e., $\\chi$'s and $\\phi$'s) are equal\nto zero. For concreteness we shall focus hereafter on this\nparticular case, which represents, up to a simple rotation of\nthe $d^{\\dagger}_{\\sigma}$ and $c^{\\dagger}_{k \\sigma}$\noperators, all tunnelling matrices $\\hat{A}$ in this category\nof interest. Our discussion is restricted to zero temperature.\n\n\\subsubsection{Impurity magnetization}\n\\label{sec:ResTests}\n\nWe have solved the exact Bethe \\emph{anstaz} equations\nnumerically using the procedure outlined in\nAppendix~\\ref{app:Bethe}. Our results for the occupation\nnumbers $\\qav{n_{\\sigma}}$ and the magnetization $M =\n\\qav{n_{\\uparrow} - n_{\\downarrow}}\/2$ are summarized in\nFigs.~\\ref{fig:MethodCompare} and \\ref{fig:LargeFeature}.\nFigure~\\ref{fig:MethodCompare} shows the magnetization of the\nAnderson impurity as a function of the (average) level position\n$\\epsilon_0$ in a constant magnetic field, $h = \\Delta =\n10^{-3} U$. The complementary regime $\\epsilon_0 < -U\/2$ is\nobtained by a simple reflection about $\\epsilon_0 = -U\/2$, as\nfollows from the particle-hole transformation $d_{\\sigma} \\to\nd_{-\\sigma}^{\\dagger}$ and $c_{k \\sigma} \\to -c_{k\n-\\sigma}^{\\dagger}$. The Bethe \\emph{ansatz} curve accurately\ncrosses over from the perturbative domain at large $\\epsilon_0\n\\gg \\Gamma$ (when the dot is almost empty) to the local-moment\nregime with a fully pronounced Kondo effect (when the dot is\nsingly occupied). In the latter regime, we find excellent\nagreement with the analytical magnetization curve of the Kondo\nmodel, Eq.~\\eqref{eq:MKfullWiegmann}, both as a function of\n$\\epsilon_0$ and as a function of the magnetic field $\\Delta$\n(lower left inset to Fig.~\\ref{fig:MethodCompare}). The\nagreement with the universal Kondo curve is in fact quite\nsurprising in that it extends nearly into the mixed-valent\nregime. As a function of field, the Kondo curve of\nEq.~(\\ref{eq:MKfullWiegmann}) applies up to fields of the order\nof $h \\sim \\sqrt{\\Gamma U} \\gg T_K$.\n\n\\begin{figure}[t]\n\\includegraphics[width=7cm]{fig2.eps}\n\\caption{(Color online) Magnetization of the isotropic\n case as a function of $\\epsilon_0$: exact Bethe\n \\emph{ansatz} curve and comparison with different\n approximation schemes.\n Black symbols show the magnetization $M$\n derived from the exact Bethe \\emph{ansatz}\n equations; the dashed (red) line marks the result\n of first-order perturbation theory in $\\Gamma$\n (Ref.~\\onlinecite{Gefen04}, divergent at\n $\\epsilon_0 = 0$); the thick (blue) line is the\n analytical formula for the magnetization in the\n Kondo limit, Eq.~\\eqref{eq:MKfullWiegmann}, with\n $T_K$ given by Eq.~\\eqref{eq:TKAndersonAccurate}.\n The model parameters are $\\Gamma\/U = 0.05$,\n $\\Delta\/U = 10^{-3}$ and $T = 0$.\n The upper right inset shows the same data but\n on a linear scale.\n The lower left inset shows the magnetization $M$\n as a function of the magnetic field $h = \\Delta$\n at fixed $\\epsilon_0\/U = -0.2$. The universal\n magnetization curve of the Kondo model well\n describes the exact magnetization up to\n $M \\approx 0.42$ (lower fields not shown),\n while first-order perturbation theory in\n $\\Gamma$ fails from $M \\approx 0.46$ downwards.}\n\\label{fig:MethodCompare}\n\\end{figure}\n\n\n\\subsubsection{Occupation numbers and charge oscillations}\n\\label{sec:ResCharging}\n\n\\begin{figure}[t]\n\\includegraphics[width=7.5cm]{fig3.eps}\n\\caption{(Color online) The occupation numbers $\\qav{n_1}$\n [solid (blue) lines] and $\\qav{n_2}$ [dotted (red)\n lines] versus $\\epsilon_0$, as obtained from\n the solution of the exact Bethe \\emph{ansatz}\n equations. In going from the inner-most to the\n outer-most pairs of curves, the magnetic field\n $h = \\Delta$ increases by a factor of $10$\n between each successive pair of curves, with\n the inner-most (outer-most) curves corresponding\n to $\\Delta\/U = 10^{-5}$ ($\\Delta\/U = 0.1$).\n The remaining model parameters are\n $\\Gamma\/U = 0.05$ and $T=0$.\n Nonmonotonicities are seen in the process\n of charging. These are most pronounced for\n intermediate values of the field. The evolution\n of the nonmonotonicities with increasing field\n is tracked by arrows. The dashed black lines\n show the approximate values calculated from\n Eqs.~\\eqref{eq:Actualn1n2} and (\\ref{eq:n0PT})\n based on the mapping onto the Kondo Hamiltonian\n (here $\\theta_h = 0$ and $\\theta_d = \\pi$).}\n\\label{fig:LargeFeature}\n\\end{figure}\n\nFigure~\\ref{fig:LargeFeature} displays the individual\noccupation numbers $\\qav{n_1}$ and $\\qav{n_2}$ as a function of\n$\\epsilon_0$, for a series of constant fields $h = \\Delta$. In\ngoing from large $\\epsilon_0 \\gg \\Gamma$ to large $-(\\epsilon_0\n+ U) \\gg \\Gamma$, the total charge of the quantum dot increases\nmonotonically from nearly zero to nearly two. However, the\npartial occupancies $\\qav{n_1}$ and $\\qav{n_2}$ display\nnonmonotonicities, which have drawn considerable theoretical\nattention lately~\\cite{Silvestrov00,Gefen04,Sindel05}. As seen\nin Fig.~\\ref{fig:LargeFeature}, the nonmonotonicities can be\nquite large, although no population inversion occurs for\n$\\Gamma_{\\uparrow} = \\Gamma_{\\downarrow}$.\n\nOur general discussion in Sec.~\\ref{sec:LocalMoment} makes it\nis easy to interpret these features of the partial occupancies\n$\\qav{n_{i}}$. Indeed, as illustrated in\nFig.~\\ref{fig:LargeFeature}, there is excellent agreement in\nthe local-moment regime between the exact Bethe \\emph{ansatz}\nresults and the curves obtained from Eqs.~\\eqref{eq:Actualn1n2}\nand (\\ref{eq:n0PT}) based on the mapping onto the Kondo\nHamiltonian. We therefore utilize Eqs.~\\eqref{eq:Actualn1n2}\nfor analyzing the data. To begin with we note that, for\n$\\Gamma_{\\uparrow} = \\Gamma_{\\downarrow}$, there is no\nrenormalization of the effective magnetic field. The latter\nremains constant and equal to $h = \\Delta$ independent of\n$\\epsilon_0$. Combined with the fact that $\\cos(\\theta_d +\ns_\\theta \\theta_h) \\equiv -1$ in Eqs.~\\eqref{eq:Actualn1n2},\nthe magnetization $M = \\qav{n_{\\uparrow} - n_{\\downarrow}}\/2 =\n\\qav{n_{2} - n_{1}}\/2$ depends exclusively on the ratio\n$\\Delta\/T_K$. The sole dependence on $\\epsilon_0$ enters\nthrough $T_K$, which varies according to\nEq.~(\\ref{eq:TKAndersonAccurate}). Thus, $M$ is positive for\nall gate voltages $\\epsilon_0$, excluding the possibility of a\npopulation inversion.\n\nThe nonmonotonicities in the individual occupancies stem from\nthe explicit dependence of $T_K$ on the gate voltage\n$\\epsilon_0$. According to Eq.~(\\ref{eq:TKAndersonAccurate}),\n$T_K$ is minimal in the middle of the Coulomb-blockade valley,\nincreasing monotonically as a function of $|\\epsilon_0 + U\/2|$.\nThus, $\\Delta\/T_K$, and consequently $M$, is maximal for\n$\\epsilon_0 = -U\/2$, decreasing monotonically the farther\n$\\epsilon_0$ departs from $-U\/2$. Since $n_{\\text{tot}} \\approx\n1$ is nearly a constant in the local-moment regime, this\nimplies the following evolution of the partial occupancies:\n$\\qav{n_1}$ decreases ($\\qav{n_2}$ increases) as $\\epsilon_0$\nis lowered from roughly zero to $-U\/2$. It then increases\n(decreases) as $\\epsilon_0$ is further lowered toward $-U$.\nCombined with the crossovers to the empty-impurity and doubly\noccupied regimes, this generates a local maximum (minimum) in\n$\\qav{n_1}$ ($\\qav{n_2}$) near $\\epsilon_0 \\sim 0$ ($\\epsilon_0\n\\sim -U$).\n\nNote that the local extremum in $\\qav{n_i}$ is most pronounced\nfor intermediate values of the field $\\Delta$. This can be\nunderstood by examining the two most relevant energy scales in\nthe problem, namely, the minimal Kondo temperature\n$T_{K}^{\\text{min}} = T_{K}^{}|_{\\epsilon_0=-U\/2}$ and the\nhybridization width $\\Gamma$. These two energies govern the\nspin susceptibility of the impurity in the middle of the\nCoulomb-blockade valley (when $\\epsilon_0 = -U\/2$) and in the\nmixed-valent regime (when either $\\epsilon_0 \\approx 0$ or\n$\\epsilon \\approx -U$), respectively. The charging curves of\nFig.~\\ref{fig:LargeFeature} stem from an interplay of the three\nenergy scales $\\Delta$, $T_K^{\\text{min}}$ and $\\Gamma$ as\ndescribed below.\n\n\\begin{figure}\n\\includegraphics[width=7.5cm]{fig4.eps}\n\\caption{(Color online) The exact conductance $G$\n [in units of $e^2\/(2\\pi\\hbar)$] versus\n $\\epsilon_0$, as obtained from the Bethe\n \\emph{ansatz} magnetization $M$ and\n Eq.~(\\ref{G-parallel}) with\n $\\theta_l = 3\\pi\/2$ and $\\theta_d = \\pi$.\n Here $\\Delta\/U$ equals $10^{-5}$ [full\n (black) line], $10^{-4}$ [dotted (red)\n line], $10^{-3}$ [dashed (green) line]\n and $0.1$ [dot-dashed (blue) line]. The\n remaining model parameters are\n $\\Gamma\/U = 0.05$ and $T = 0$. Once $\\Delta$\n exceeds the critical field $h_c^{} \\approx\n 2.4 T_K^{\\text{min}}$, the single peak at\n $\\epsilon_0 = -U\/2$ is split into two\n correlation-induced peaks, which cross\n over to Coulomb-blockade peaks at large\n $\\Delta$.}\n\\label{fig:isoCIR-1}\n\\end{figure}\n\nWhen $\\Delta \\ll T_K^{\\text{min}}$, exemplified by\nthe pair of curves corresponding to the smallest field\n$\\Delta = 10^{-5} U \\approx 0.24 T^{\\text{min}}_K$\nin Fig.~\\ref{fig:LargeFeature}, the magnetic field\nremains small throughout the Coulomb-blockade valley\nand no significant magnetization develops. The two\nlevels are roughly equally populated, showing a\nplateaux at $\\qav{n_{1}} \\approx \\qav{n_{2}}\n\\approx 1\/2$ in the regime where the dot is singly\noccupied. As $\\Delta$ grows and approaches\n$T_K^{\\text{min}}$, the field becomes sufficiently strong\nto significantly polarize the impurity in the vicinity\nof $\\epsilon_0 = -U\/2$. A gap then rapidly develops\nbetween $\\qav{n_{1}}$ and $\\qav{n_{2}}$ near\n$\\epsilon_0 = -U\/2$ as $\\Delta$ is increased. Once\n$\\Delta$ reaches the regime $T_K^{\\text{min}} \\ll\n\\Delta \\ll \\Gamma$, a crossover from $h \\gg T_K$\n(fully polarized impurity) to $h \\ll T_K$ (unpolarized\nimpurity) occurs as $\\epsilon_0$ is tuned away\nfrom the middle of the Coulomb-blockade valley. This\nleads to the development of a pronounced maximum\n(minimum) in $\\qav{n_1}$ ($\\qav{n_2}$), as marked by\nthe arrows in Fig.~\\ref{fig:LargeFeature}. Finally, when\n$h \\gtrsim \\Gamma$, the field is sufficiently large\nto keep the dot polarized throughout the local-moment\nregime. The extremum in $\\qav{n_i}$ degenerates into\na small bump in the vicinity of either $\\epsilon_0\n\\approx 0$ or $\\epsilon_0 \\approx -U$, which is\nthe nonmonotonic feature first discussed in\nRef.~\\onlinecite{Gefen04}. This regime is exemplified\nby the pair of curves corresponding to the largest\nfield $\\Delta = 0.1 U = 2\\Gamma$ in\nFig.~\\ref{fig:LargeFeature}, whose parameters match\nthose used in Fig.~2 of Ref.~\\onlinecite{Gefen04}.\nNote, however, that the perturbative calculations of\nRef.~\\onlinecite{Gefen04} will inevitably miss the\nregime $T_K^{\\text{min}} \\ll \\Delta \\ll \\Gamma$ where\nthis feature is large~\\cite{comm-Sindel-feature}.\n\n\n\\subsubsection{Conductance}\n\\label{Sec:isoCond}\n\n\\begin{figure}\n\\includegraphics[width=7cm]{fig5.eps}\n\\caption{(Color online) The exact occupation numbers\n $\\qav{n_i}$ and conductance $G$ [in units\n of $e^2\/(2 \\pi \\hbar)$] as a function of\n $\\epsilon_2$, for $T = 0$, $\\Gamma\/U = 0.2$\n and fixed $\\epsilon_1\/U = -1\/2$. The\n population inversion at $\\epsilon_2 =\n \\epsilon_1$ leads to a sharp transmission\n zero (phase lapse). Note the general\n resemblance between the functional dependence of\n $G$ on $\\epsilon_2$ and the correlation-induced\n resonances reported by Meden and\n Marquardt~\\cite{Meden06PRL} for\n $\\Gamma_{\\uparrow} \\neq \\Gamma_{\\downarrow}$\n (see Fig.~\\ref{fig:CIR}).}\n\\label{fig:isoCIR-2}\n\\end{figure}\n\nThe data of Fig.~\\ref{fig:LargeFeature} can easily be\nconverted to conductance curves by using the exact\nformula of Eq.~\\eqref{G-parallel} with $\\theta_l = 3\\pi\/2$\nand $\\theta_d = \\pi$.\nThe outcome is presented in Fig.~\\ref{fig:isoCIR-1}.\nThe evolution of $G(\\epsilon_0)$ with increasing\n$\\Delta$ is quite dramatic. When $\\Delta$ is small,\nthe conductance is likewise small with a shallow peak\nat $\\epsilon_0 = -U\/2$. This peak steadily grows with\nincreasing $\\Delta$ until reaching the unitary limit, at\nwhich point it is split in two. Upon further increasing\n$\\Delta$, the two split peaks gradually depart,\napproaching the peak positions $\\epsilon_0 \\approx 0$\nand $\\epsilon_0 \\approx -U$ for large $\\Delta$. The\nconductance at each of the two maxima remains pinned\nat all stages at the unitary limit.\n\nThese features of the conductance can be naturally\nunderstood based on Eqs.~\\eqref{G-parallel} and\n\\eqref{eq:Actualn1n2}. When $\\Delta \\ll T_K^{\\text{min}}$,\nthe magnetization $M \\approx \\Delta\/(2\\pi T_K)$ and the\nconductance $G \\approx (\\Delta\/T_K)^2 e^2\/(2\\pi \\hbar)$\nare uniformly small, with a peak at $\\epsilon_0 = -U\/2$\nwhere $T_K$ is the smallest. The conductance\nmonotonically grows with increasing $\\Delta$ until\nreaching the critical field $\\Delta = h_c^{} \\approx 2.4\nT_K^{\\text{min}}$, where $M|_{\\epsilon_0 = -U\/2} = 1\/4$\nand $G|_{\\epsilon_0 = -U\/2} = e^2\/(2 \\pi \\hbar)$. Upon\nfurther increasing $\\Delta$, the magnetization at\n$\\epsilon_0 = -U\/2$ exceeds $1\/4$, and the associated\nconductance decreases. The unitarity condition\n$M = 1\/4$ is satisfied at two gate voltages\n$\\epsilon^{\\pm}_{\\text{max}}$ symmetric about $-U\/2$,\ndefined by the relation $T_K \\approx \\Delta\/2.4$. From\nEq.~(\\ref{eq:TKAndersonAccurate}) one obtains\n\\begin{equation}\n\\epsilon_{\\pm}^{\\text{max}} = -\\frac{U}{2}\n \\pm \\sqrt{\n \\frac{U^2}{4} - \\Gamma^2 +\n \\frac{2\\Gamma U}{\\pi}\n \\ln \\left (\n \\frac{\\pi \\Delta}\n {2.4 \\sqrt{2 \\Gamma U}}\n \\right )\n } \\, .\n\\end{equation}\nThe width of the two conductance peaks,\n$\\Delta \\epsilon$, can be estimated for\n$T_K^{\\text{min}} \\ll \\Delta \\ll \\Gamma$ from the inverse\nof the derivative $d(\\Delta\/T_K)\/d\\epsilon_0$, evaluated\nat $\\epsilon_0^{} = \\epsilon_{\\text{max}}^{\\pm}$. It\nyields\n\\begin{equation}\n\\Delta \\epsilon \\sim \\frac{\\Gamma U}\n {\\pi | \\epsilon_{\\text{max}}^{\\pm} + U\/2 |} \\, .\n\\end{equation}\nFinally, when $\\Delta > \\Gamma$, the magnetization\nexceeds $1\/4$ throughout the local-moment regime. The\nresonance condition $M = 1\/4$ is met only as charge\nfluctuations become strong, namely, for either\n$\\epsilon_0 \\approx 0$ or $\\epsilon_0 \\approx -U$. The\nresonance width $\\Delta \\epsilon$ evolves continuously in\nthis limit to the standard result for the Coulomb-blockade\nresonances, $\\Delta \\epsilon \\sim \\Gamma$.\n\nUp until now the energy difference $\\Delta$ was kept\nconstant while tuning the average level position\n$\\epsilon_0$. This protocol, which precludes population\ninversion as a function of the control parameter, best\nsuits a single-dot realization of our model, where both\nlevels can be uniformly tuned using a single gate voltage.\nIn the alternative realization of two spatially separated\nquantum dots, each controlled by its own separate gate\nvoltage, one could fix the energy level\n$\\epsilon_1 = \\epsilon_0 + \\Delta\/2$ and sweep the other\nlevel, $\\epsilon_2 = \\epsilon_0 - \\Delta\/2$. This setup\namounts to changing the field $h$ externally, and is\nthus well suited for probing the magnetic response of\nour effective impurity.\n\nAn example for such a protocol is presented in\nFig.~\\ref{fig:isoCIR-2}, where $\\epsilon_1$ is held\nfixed at $\\epsilon_1 = -U\/2$. As $\\epsilon_2$ is\nswept through $\\epsilon_1$, a population inversion\ntakes place, leading to a narrow dip in the conductance.\nThe width of the conductance dip is exponentially\nsmall due to Kondo correlations. Indeed, one can\nestimate the dip width, $\\Delta \\epsilon_{\\text{dip}}$,\nfrom the condition $|\\epsilon_1 - \\epsilon_2| =\nT_K|_{\\epsilon_2 = \\epsilon_1}$, which yields\n\\begin{equation}\n\\Delta \\epsilon_{\\text{dip}} \\sim\n \\sqrt{U \\Gamma} \\exp\n \\left (\n -\\frac{\\pi U}{8 \\Gamma}\n \\right ) \\, .\n\\label{eq:CIRwidthIsotropic}\n\\end{equation}\n\n\\subsection{Anisotropic couplings, $\\Gamma_{\\uparrow}\n \\neq \\Gamma_{\\downarrow}$}\n\\label{sec:ResultsAnisotropic}\n\nAs demonstrated at length in Sec.~\\ref{sec:ResultsIsotropic},\nthe occurrence of population inversion and a transmission\nzero for $\\Gamma_{\\uparrow} = \\Gamma_{\\downarrow}$\nrequires an external modulation of the effective\nmagnetic field. Any practical device will inevitably\ninvolve, though, some tunnelling anisotropy,\n$V_{\\uparrow} \\neq V_{\\downarrow}$. The latter provides\na different route for changing the effective magnetic\nfield, through the anisotropy-induced terms in\nEq.~\\eqref{h-total}. Implementing the same protocol\nas in Sec.~\\ref{sec:ResCharging} (that is, uniformly\nsweeping the average level position $\\epsilon_0$ while\nkeeping the difference $\\Delta$ constant) would now\ngenerically result both in population inversion and a\ntransmission zero due to the rapid change in direction\nof the total field $\\vec{h}_{\\text{tot}}$. As emphasized\nin Sec.~\\ref{sec:occupany-PF}, the two phenomena\nwill generally occur at different gate voltages\nwhen $V_{\\uparrow} \\neq V_{\\downarrow}$.\n\n\\subsubsection{Degenerate levels, $\\Delta = b = 0$}\n\nWe begin our discussion with the case where\n$\\Delta = b = 0$, which was extensively studied in\nRef.~\\onlinecite{Meden06PRL}. It corresponds to a\nparticular limit of the parallel-field configuration where\n$h = 0$. In the parallel-field configuration, the\nconductance $G$ and occupancies $\\qav{n_i}$ take the exact\nforms specified in Eqs.~(\\ref{G-parallel-field}) and\n(\\ref{eq:Actualn1n2-PF}), respectively. These expressions\nreduce in the Kondo regime to Eqs.~(\\ref{G-no-flux}) and\n(\\ref{eq:Actualn1n2}), with $\\theta_h$ either equal to\nzero or $\\pi$, depending on the sign of $h_{\\text{tot}}^z$.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{fig6.eps}\n\\caption{The occupation numbers $\\qav{n_i}$ and conductance\n $G$ [in units of $e^2\/(2 \\pi \\hbar)$] as a\n function of $\\epsilon_0 + U\/2$ [in units of\n $\\Gamma_{\\text{tot}} = (\\Gamma_{\\uparrow} +\n \\Gamma_{\\downarrow})$],\n calculated from Eqs.~(\\ref{G-no-flux}) and\n (\\ref{eq:Actualn1n2}) based on the mapping\n onto the Kondo model. The model parameters\n are identical to those used in Fig.~2 of\n Ref.~\\onlinecite{Meden06PRL}, lower left panel:\n $h = \\varphi = 0$, $U\/\\Gamma_{\\text{tot}} = 6$,\n $\\Gamma_{\\uparrow}\/\\Gamma_{\\text{tot}} = 0.62415$\n and $T = 0$. The\n explicit tunnelling matrix elements are detailed\n in Eq.~(\\ref{eq:AMM}), corresponding to the\n rotation angles $\\theta_l = 2.1698$ and\n $\\theta_d = -0.63434$ (measured in radians).\n The angle $\\theta_h$ equals zero. The\n inset shows functional renormalization-group (fRG)\n data as defined in Ref.~\\onlinecite{Meden06PRL},\n corrected for the renormalization of the\n two-particle vertex~\\cite{Karrasch06,MedenThanks}.\n The small symbols in the inset\n show the conductance as calculated\n from the fRG occupation numbers using our\n Eq.~\\eqref{G-parallel}. The horizontal dotted\n lines in each plot mark the maximal conductance\n predicted by Eq.~\\eqref{G-parallel},\n $(e^2\/2 \\pi \\hbar) \\sin^2 \\theta_l$.}\n\\label{fig:CIR}\n\\end{figure}\n\nFigure~\\ref{fig:CIR} shows the occupation numbers and\nthe conductance obtained from Eqs.~(\\ref{G-no-flux})\nand (\\ref{eq:Actualn1n2}), for $\\Delta = b = 0$ and\nthe particular tunnelling matrix used in Fig.~2 of\nRef.~\\onlinecite{Meden06PRL}:\n\\begin{align}\n\\hat{A} = A_0\n \\begin{bmatrix}\n \\sqrt{0.27} & \\sqrt{0.16} \\\\\n \\sqrt{0.33} & -\\sqrt{0.24}\\\\\n\\end{bmatrix} \\, .\n\\label{eq:AMM}\n\\end{align}\nHere $A_0$ equals $\\sqrt{\\Gamma_{\\text{tot}}\/ (\\pi \\rho)}$,\nwith $\\Gamma_{\\text{tot}} =\n\\Gamma_{\\uparrow}+\\Gamma_{\\downarrow}$ being the combined\nhybridization width. The Coulomb repulsion $U$ is set equal to\n$6 \\Gamma_{\\text{tot}}$, matching the value used in the lower\nleft panel of Fig.~2 in Ref.~\\onlinecite{Meden06PRL}. For\ncomparison, the corresponding functional renormalization-group\n(fRG) data of Ref.~\\onlinecite{Meden06PRL} is shown in the\ninset, after correcting for the renormalization of the\ntwo-particle vertex~\\cite{Karrasch06,MedenThanks}. The accuracy\nof the fRG has been established~\\cite{Meden06PRL,Karrasch06} up\nto moderate values of $U\/\\Gamma_{\\text{tot}} \\sim 10$ through\na comparison with Wilson's numerical renormalization-group\nmethod~\\cite{NrgMethods}. Including the renormalization of\nthe two-particle vertex further improves the fRG data as\ncompared to that of Ref.~\\onlinecite{Meden06PRL}, as reflected,\ne.g., in the improved position of the outer pair of conductance\nresonances.\n\nThe agreement between our analytical approach and the fRG is\nevidently very good in the local-moment regime, despite the\nrather moderate value of $U\/\\Gamma_{\\text{tot}}$ used.\nNoticeable deviations develop in $\\qav{n_i}$ only as the\nmixed-valent regime is approached (for $\\epsilon_0 \\agt\n-\\Gamma_{\\text{tot}}$ or $\\epsilon + U \\alt\n\\Gamma_{\\text{tot}}$), where our approximations naturally break\ndown. In particular, our approach accurately describes the\nphase lapse at $\\epsilon_0 = -U\/2$, the inversion of population\nat the same gate voltage, the location and height of the\ncorrelation-induced resonances, and even the location and\nheight of the outer pair of conductance resonances. Most\nimportantly, our approach provides a coherent analytical\npicture for the physics underlying these various features, as\nelaborated below.\n\nBefore proceeding to elucidate the underlying physics, we\nbriefly quote the relevant parameters that appear in the\nconversion to the generalized Anderson model of\nEq.~(\\ref{eq:Hand}). Using the prescriptions detailed in\nAppendix~\\ref{App:SVDdetails}, the hybridization widths\n$\\Gamma^{}_{\\sigma} = \\pi \\rho V_{\\sigma}^2$ come out to be\n\\begin{equation}\n\\Gamma_{\\uparrow}\/\\Gamma_{\\text{tot}} = 0.62415 \\; ,\n\\;\\;\\;\\;\n\\Gamma_{\\downarrow}\/\\Gamma_{\\text{tot}} = 0.36585 \\, ,\n\\end{equation}\nwhile the angles of rotation equal\n\\begin{equation}\n\\theta_l = 2.1698 \\; , \\;\\;\\;\\;\n\\theta_d = -0.63434 \\, .\n\\end{equation}\nHere $\\theta_l$ and $\\theta_d$ are quoted in radians. Using the\nexact conductance formula of Eq.~(\\ref{G-parallel-field}), $G$\nis predicted to be bounded by the maximal conductance\n\\begin{equation}\nG_{\\text{max}} =\n \\frac{e^2}{2 \\pi \\hbar} \\sin^2 \\theta_l\n = 0.68210 \\frac{e^2}{2 \\pi \\hbar} \\, ,\n\\label{G-max}\n\\end{equation}\nobtained whenever the magnetization $M = \\qav{n_{\\uparrow} -\nn_{\\downarrow}}\/2$ is equal to $\\pm 1\/4$. The heights of the\nfRG resonances are in excellent agreement with\nEq.~(\\ref{G-max}). Indeed, as demonstrated in the inset to\nFig.~\\ref{fig:CIR}, the fRG occupancies and conductance comply\nto within extreme precision with the exact relation of\nEq.~(\\ref{G-parallel}). As for the functional form of the Kondo\ntemperature $T_K$, its exponential dependence on $\\epsilon_0$\nis very accurately described by Eq.~(\\ref{scaling-T_K-2}). In\nthe absence of a precise expression for the pre-exponential\nfactor when $\\Gamma_{\\uparrow} \\neq \\Gamma_{\\downarrow}$, we\nemploy the expression\n\\begin{equation}\nT_K = (\\sqrt{U \\Gamma_{\\text{tot}}}\/\\pi) \\exp\n \\left [\n \\frac{\\pi \\epsilon_0 (U + \\epsilon_0)}\n {2U(\\Gamma_{\\uparrow}-\\Gamma_{\\downarrow})}\n \\ln\\!\n \\frac{\\Gamma_{\\uparrow}}{\\Gamma_{\\downarrow}}\n \\right ] \\, ,\n\\label{T_K-anisotropic}\n\\end{equation}\nwhich properly reduces to Eq.~\\eqref{eq:TKAndersonAccurate} (up\nto the small $\\Gamma^2$ correction in the exponent) when\n$\\Gamma_{\\uparrow} = \\Gamma_{\\downarrow} = \\Gamma$.\n\nThe occupancies and conductance of Fig.~\\ref{fig:CIR} can be\nfully understood from our general discussion in\nSec.~\\ref{sec:LocalMoment}. Both quantities follow from the\nmagnetization $M$, which vanishes at $\\epsilon_0 = -U\/2$ due to\nparticle-hole symmetry. As a consequence, the two levels are\nequally populated at $\\epsilon_0 = -U\/2$ and the conductance\nvanishes [see Eqs.~(\\ref{G-parallel-field}) and\n(\\ref{eq:Actualn1n2-PF})]. Thus, there is a simultaneous phase\nlapse and an inversion of population at $\\epsilon_0 = -U\/2$,\nwhich is a feature generic to $\\Delta = b = 0$ and arbitrary\n$\\hat{A}$. As soon as the gate voltage is removed from $-U\/2$,\ni.e., $\\epsilon_0 = -U\/2 + \\delta \\epsilon$ with $\\delta\n\\epsilon \\neq 0$, a finite magnetization develops due to the\nappearance of a finite effective magnetic field\n$\\vec{h}_{\\text{tot}} = h^z_{\\text{tot}} \\hat{z}$ with\n\\begin{equation}\nh^{z}_{\\text{tot}} \\approx\n \\frac{\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}}{\\pi}\n \\ln \\frac{1 + 2\\delta \\epsilon\/U}\n {1 - 2\\delta \\epsilon\/U}\n\\label{h-z-tot}\n\\end{equation}\n[see Eq.~(\\ref{eq:htotExplicit})]. Note that the sign of\n$h^{z}_{\\text{tot}}$ coincides with that of $\\delta \\epsilon$,\nhence $M$ is positive (negative) for $\\epsilon_0 > -U\/2$\n($\\epsilon_0 < -U\/2$). Since $\\cos \\theta_d > 0$ for the model\nparameters used in Fig.~\\ref{fig:CIR}, it follows from\nEq.~(\\ref{eq:Actualn1n2-PF}) that $\\qav{n_1} > \\qav{n_2}$\n($\\qav{n_1} < \\qav{n_2}$) for $\\epsilon_0 > -U\/2$ ($\\epsilon_0\n< -U\/2$), as is indeed found in Fig.~\\ref{fig:CIR}. Once again,\nthis result is generic to $\\Delta = b = 0$, except for the sign\nof $\\cos \\theta_d$ which depends on details of the tunnelling\nmatrix $\\hat{A}$.\n\nIn contrast with the individual occupancies, the conductance $G$\ndepends solely on the magnitude of $M$, and is therefore a\nsymmetric function of $\\delta \\epsilon$. Similar to the rich\nstructure found for $\\Gamma_{\\uparrow} = \\Gamma_{\\downarrow}$\nand $\\Delta > 0$ in Fig.~\\ref{fig:isoCIR-1}, the intricate\nconductance curve in Fig.~\\ref{fig:CIR} is the result of the\ninterplay between $h_{\\text{tot}}^z$ and $T_K$, and the\nnonmonotonic dependence of $G$ on $|M|$. The basic physical\npicture is identical to that in Fig.~\\ref{fig:isoCIR-1}, except\nfor the fact that the effective magnetic field\n$h_{\\text{tot}}^z$ is now itself a function of the gate voltage\n$\\epsilon_0$.\n\n\nAs a rule, the magnetization $|M|$ first increases with\n$|\\delta \\epsilon|$ due to the rapid increase in\n$h_{\\text{tot}}^z$. It reaches its maximal value\n$M_{\\text{max}}$ at some intermediate $|\\delta \\epsilon|$\nbefore decreasing again as $|\\delta \\epsilon|$ is further\nincreased. Inevitably $|M|$ becomes small again once $|\\delta\n\\epsilon|$ exceeds $U\/2$. The shape of the associated\nconductance curve depends crucially on the magnitude of\n$M_{\\text{max}}$, which monotonically increases as a function\nof $U$. When $M_{\\text{max}} < 1\/4$, the conductance features\ntwo symmetric maxima, one on each side of the particle-hole\nsymmetric point. Each of these peaks is analogous to the one\nfound in Fig.~\\ref{fig:isoCIR-1} for $\\Delta < h_c$. Their\nheight steadily grows with increasing $U$ until the unitarity\ncondition $M_{\\text{max}} = 1\/4$ is met. This latter condition\ndefines the critical repulsion $U_c$ found in\nRef.~\\onlinecite{Meden06PRL}. For $U > U_c$, the maximal\nmagnetization $M_{\\text{max}}$ exceeds one quarter. Hence the\nunitarity condition $M = \\pm 1\/4$ is met at two pairs of gate\nvoltages, one pair of gate voltages on either side of the\nparticle-hole symmetric point $\\epsilon_0 = -U\/2$. Each of the\nsingle resonances for $U < U_c$ is therefore split in two, with\nthe inner pair of peaks evolving into the correlation-induced\nresonances of Ref.~\\onlinecite{Meden06PRL}. The point of\nmaximal magnetization now shows up as a local minimum of the\nconductance, similar to the point $\\epsilon_0 = -U\/2$ in\nFig.~\\ref{fig:isoCIR-1} when $\\Delta > h_c$.\n\nFor large $U \\gg \\Gamma_{\\text{tot}}$, the magnetization $|M|$\ngrows rapidly as one departs from $\\epsilon_0 = -U\/2$, due to\nthe exponential smallness of the Kondo temperature\n$T_K|_{\\epsilon_0 = -U\/2}$. The dot remains polarized\nthroughout the local-moment regime, loosing its polarization\nonly as charge fluctuations become strong. In this limit the\ninner pair of resonances lie exponentially close to $\\epsilon_0\n= -U\/2$ (see below), while the outer pair of resonances\napproach $|\\delta \\epsilon| \\approx U\/2$ (the regime of\nthe conventional Coulomb blockade).\n\nThe description of this regime can be made quantitative by\nestimating the position $\\pm \\delta \\epsilon_{\\text{CIR}}$ of\nthe correlation-induced resonances. Since $M \\to\nM_K(h^{z}_{\\text{tot}}\/T^{}_K)$ deep in the local-moment\nregime, and since $\\delta \\epsilon_{\\text{CIR}} \\ll\n\\Gamma_{\\text{tot}}$ for $\\Gamma_{\\text{tot}} \\ll U$, the\ncorrelation-induced resonances are peaked at the two gate\nvoltages where $h^{z}_{\\text{tot}} \\approx \\pm 2.4\nT^{}_K|_{\\epsilon_0 = -U\/2}$. Expanding Eq.~(\\ref{h-z-tot}) to\nlinear order in $\\delta \\epsilon_{\\text{CIR}}\/U \\ll 1$ and\nusing Eq.~(\\ref{T_K-anisotropic}) one finds\n\\begin{eqnarray}\n\\delta \\epsilon_{\\text{CIR}} &\\approx& 0.6\n \\frac{\\pi U}{\\Gamma_{\\uparrow}-\\Gamma_{\\downarrow}}\n T_K|_{\\epsilon_0 = -U\/2}\n\\nonumber \\\\\n&=& 0.6 \\frac{U \\sqrt{U \\Gamma_{\\text{tot}}}}\n {\\Gamma_{\\uparrow}-\\Gamma_{\\downarrow}}\n \\exp\\!\n \\left [\n \\frac{-\\pi U \\ln(\\Gamma_{\\uparrow}\/\n \\Gamma_{\\downarrow})}\n {8(\\Gamma_{\\uparrow}-\\Gamma_{\\downarrow})}\n \\right ] .\n\\label{eq:CIRwidth}\n\\end{eqnarray}\nHere the pre-exponential factor in the final expression for\n$\\delta \\epsilon_{\\text{CIR}}$ is of the same accuracy as that\nin Eq.~(\\ref{T_K-anisotropic}).\n\nWe note in passing that the shape of the correlation-induced\nresonances and the intervening dip can be conveniently\nparameterized in terms of the peak position $\\delta\n\\epsilon_{\\text{CIR}}$ and the peak conductance\n$G_{\\text{max}}$. Expanding Eq.~(\\ref{h-z-tot}) to linear order\nin $\\delta \\epsilon\/U \\ll 1$ and using\nEq.~(\\ref{G-parallel-field}) one obtains\n\\begin{equation}\nG(\\delta \\epsilon) = G_{\\text{max}} \\sin^2\\!\n \\left [\n 2 \\pi M_K\\!\n \\left (\n \\frac{2.4 \\delta \\epsilon}\n {\\delta \\epsilon_{\\text{CIR}}}\n \\right )\n \\right ] \\, ,\n\\end{equation}\nwhere $M_K(h\/T_K)$ is the universal magnetization curve of the\nKondo model [given explicitly by \\eqref{eq:MKfullWiegmann}].\nThis parameterization in terms of two easily extractable\nparameters may prove useful for analyzing future experiments.\n\nIt is instructive to compare Eq.~(\\ref{eq:CIRwidth}) for\n$\\delta \\epsilon_{\\text{CIR}}$ with the fRG results of\nRef.~\\onlinecite{Meden06PRL}, which tend to overestimate\n$\\delta \\epsilon_{\\text{CIR}}$. For the special case where\n$a_{L 1} = a_{R 1}$ and $a_{L 2} = -a_{R 2}$, an analytic\nexpression was derived for $\\delta \\epsilon_{\\text{CIR}}$ based\non the fRG~\\cite{Meden06PRL}. The resulting expression,\ndetailed in Eq.~(4) of Ref.~\\onlinecite{Meden06PRL}, shows an\nexponential dependence nearly identical to that of\nEq.~(\\ref{eq:CIRwidth}), but with an exponent that is smaller\nin magnitude by a factor of $\\pi^2\/8 \\approx\n1.23$~\\cite{Relating-the-Gamma's}. The same numerical factor\nappears to distinguish the fRG and the numerical\nrenormalization-group data depicted in Fig.~3 of\nRef.~\\onlinecite{Meden06PRL}, supporting the accuracy of our\nEq.~(\\ref{eq:CIRwidth}). It should be emphasized, however, that\nFig.~3 of Ref.~\\onlinecite{Meden06PRL} pertains to the\ntunnelling matrix of Eq.~(\\ref{eq:AMM}) rather than the special\ncase referred to above.\n\nWe conclude the discussion of the case where $\\Delta = b = 0$\nwith accurate results on the renormalized dot levels when the\ndot is tuned to the peaks of the correlation-induced\nresonances. The renormalized dot levels,\n$\\tilde{\\epsilon}_{\\uparrow}$ and\n$\\tilde{\\epsilon}_{\\downarrow}$, can be defined through the $T\n= 0$ retarded dot Green functions at the Fermi energy:\n\\begin{equation}\nG_{\\sigma}(\\epsilon = 0) =\n \\frac{1}{-\\tilde{\\epsilon}_{\\sigma}\n + i\\Gamma_{\\sigma}} \\, .\n\\label{renormalized-levels-def}\n\\end{equation}\nHere, in writing the Green functions of\nEq.~(\\ref{renormalized-levels-def}), we have made use of the\nfact that the imaginary parts of the retarded dot\nself-energies, $-\\Gamma_{\\sigma}$, are unaffected by the\nCoulomb repulsion $U$ at zero temperature at the Fermi energy.\nThe energies $\\tilde{\\epsilon}_{\\sigma}$ have the exact\nrepresentation~\\cite{Langreth66} $\\tilde{\\epsilon}_{\\sigma} =\n\\Gamma_{\\sigma} \\cot \\delta_{\\sigma}$ in terms of the\nassociated phase shifts $\\delta_{\\sigma} = \\pi \\qav{n_{\\sigma}}$.\nSince $M = \\pm 1\/4$ at the peaks of the correlation-induced\nresonances, this implies that $\\delta_{\\sigma} = \\pi\/2 \\pm\n\\sigma \\pi\/4$, where we have set $n_{\\text{tot}} =\n1$~\\cite{Comment-on-renormalized-levels}. Thus, the\nrenormalized dot levels take the form\n$\\tilde{\\epsilon}_{\\sigma} = \\mp\\sigma \\Gamma_{\\sigma}$,\nresulting in\n\\begin{equation}\n\\tilde{\\epsilon}_{\\uparrow} \\tilde{\\epsilon}_{\\downarrow}\n = -\\Gamma_{\\uparrow} \\Gamma_{\\downarrow} \\, .\n\\label{renormalized-levels}\n\\end{equation}\n\nThe relation specified in Eq.~(\\ref{renormalized-levels})\nwas found in Ref.~\\onlinecite{Meden06PRL},\nfor the special case where $a_{L 1} = a_{R 1}$ and\n$a_{L 2} = -a_{R 2}$~\\cite{Relating-the-Gamma's}.\nHere it is seen to be a generic feature of the\ncorrelation-induced resonances for $\\Delta = b = 0$\nand arbitrary $\\hat{A}$.\n\n\\subsubsection{Nondegenerate levels:\n arbitrary $\\Delta$ and $b$}\n\nOnce $\\sqrt{\\Delta^2 + b^2} \\neq 0$, the conductance and the\npartial occupancies can have a rather elaborate dependence on\nthe gate voltage $\\epsilon_0$. As implied by the general\ndiscussion in Sec.~\\ref{sec:LocalMoment}, the underlying\nphysics remains driven by the competing effects of the\npolarizing field $h_{\\text{tot}}$ and the Kondo temperature\n$T_K$. However, the detailed dependencies on $\\epsilon_0$ can be\nquite involving and not as revealing. For this reason we shall\nnot seek a complete characterization of the conductance $G$ and\nthe partial occupancies $\\qav{n_i}$ for arbitrary couplings.\nRather, we shall focus on the case where no Aharonov-Bohm\nfluxes are present and ask two basic questions: (i) under what\ncircumstances is the phenomenon of a phase lapse generic? (ii)\nunder what circumstances is a population inversion generic?\n\nWhen $\\varphi_L = \\varphi_R = 0$, the conductance and the\npartial occupancies are given by Eqs.~(\\ref{G-no-flux}) and\n(\\ref{eq:Actualn1n2}), respectively. Focusing on $G$ and on\n$\\qav{n_1 - n_2}$, these quantities share a common form, with\nfactorized contributions of the magnetization $M_K$ and the\nrotation angles. The factors containing\n$M_K(h_{\\text{tot}}\/T_K)$ never vanish when $h \\sin \\theta \\neq\n0$, since $h_{\\text{tot}}$ always remains positive. This\ndistinguishes the generic case from the parallel-field\nconfiguration considered above, where phase lapses and\npopulation inversions are synonymous with $M = 0$. Instead, the\nconditions for phase lapses and population inversions to occur\nbecome distinct once $h \\sin \\theta \\neq 0$, originating from\nthe independent factors where the rotation angles appear. For a\nphase lapse to develop, the combined angle $\\theta_l + s_R\n\\theta_h$ must equal an integer multiple of $\\pi$. By contrast,\nthe inversion of population requires that $\\theta_d +\ns_{\\theta}\\theta_h = \\pi\/2 \\!\\!\\mod\\!\\pi$. Here the dependence\non the gate voltage $\\epsilon_0$ enters solely through the\nangle $\\theta_h$, which specifies the orientation of the\neffective magnetic field $\\vec{h}_{\\text{tot}}$ [see\nEq.~(\\ref{eq:htotExplicit})]. Since the rotation angles\n$\\theta_l$ and $\\theta_d$ are generally unrelated, this implies\nthat the two phenomena will typically occur, if at all, at\ndifferent gate voltages.\n\nFor phase lapses and population inversions to be ubiquitous,\nthe angle $\\theta_h$ must change considerably as $\\epsilon_0$\nis swept across the Coulomb-blockade valley. In other words,\nthe effective magnetic field $\\vec{h}_{\\text{tot}}$ must nearly\nflip its orientation in going from $\\epsilon_0 \\approx 0$ to\n$\\epsilon_0 \\approx -U$. Since the $x$ component of the field\nis held fixed at $h_{\\text{tot}}^{x} = h \\sin \\theta > 0$, this\nmeans that its $z$ component must vary from $h_{\\text{tot}}^{z}\n\\gg h_{\\text{tot}}^{x}$ to $-h_{\\text{tot}}^{z} \\gg\nh_{\\text{tot}}^{x}$ as a function of $\\epsilon_0$. When this\nrequirement is met, then both a phase lapse and an inversion of\npopulation are essentially guaranteed to occur. Since\n$h_{\\text{tot}}^{z}$ crudely changes by\n\\begin{equation}\n\\Delta h_{\\text{tot}}^{z} \\sim\n \\frac{2}{\\pi} (\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow})\n \\ln ( U\/\\Gamma_{\\text{tot}} )\n\\end{equation}\nas $\\epsilon_0$ is swept across the Coulomb-blockade\nvalley, this leaves us with the criterion\n\\begin{equation}\n(\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow})\n \\ln ( U\/\\Gamma_{\\text{tot}} ) \\gg\n \\sqrt{\\Delta^2 + b^2} \\, .\n\\label{condition-for-PL}\n\\end{equation}\nConversely, if $\\sqrt{\\Delta^2 + b^2}\\gg (\\Gamma_{\\uparrow} -\n\\Gamma_{\\downarrow}) \\ln ( U\/\\Gamma_{\\text{tot}} )$, then\nneither a phase lapse nor an inversion of population will occur\nunless parameters are fine tuned. Thus, the larger $U$ is, the\nmore ubiquitous phase lapses\nbecome~\\cite{Golosov06,Meden06PRL}.\n\nAlthough the logarithm $\\ln(U\/\\Gamma_{\\text{tot}})$ can be made\nquite large, in reality we expect it to be a moderate factor of\norder one. Similarly, the difference in widths\n$\\Gamma_{\\uparrow} - \\Gamma_{\\downarrow}$ is generally expected\nto be of comparable magnitude to $\\Gamma_{\\uparrow}$. Under\nthese circumstances, the criterion specified in\nEq.~(\\ref{condition-for-PL}) reduces to $\\Gamma_{\\uparrow} \\gg\n\\sqrt{\\Delta^2 + b^2}$. Namely, phase lapses and population\ninversions are generic as long as the (maximal) tunnelling rate\nexceeds the level spacing. This conclusion is in line with that\nof a recent numerical study of multi-level quantum\ndots~\\cite{Karrasch06num}.\n\nFinally, we address the effect of nonzero $h = \\sqrt{\\Delta^2 +\nb^2}$ on the correlation-induced resonances. When $h \\gg\n\\Gamma_{\\uparrow} \\ln(U\/\\Gamma_{\\text{tot}})$, the effective\nmagnetic field $h_{\\text{tot}} \\approx h$ is large throughout\nthe local-moment regime, always exceeding\n$\\Gamma_{\\uparrow}$ and $\\Gamma_{\\downarrow}$. Consequently,\nthe dot is nearly fully polarized for all $-U < \\epsilon_0 <\n0$, and the correlation-induced resonances are washed out.\nAgain, for practical values of $U\/\\Gamma_{\\text{tot}}$ this\nregime can equally be characterized by $h \\gg\n\\Gamma_{\\uparrow}$~\\cite{Meden06PRL}.\n\n\n\nThe picture for $\\Gamma_{\\uparrow}\\ln(U\/\\Gamma_{\\text{tot}})\n\\gg h$ is far more elaborate. When $T_K|_{\\epsilon_0 = -U\/2}\n\\gg h$, the magnetic field is uniformly small, and no\nsignificant modifications show up as compared with the case\nwhere $h = 0$. This leaves us with the regime $T_K|_{\\epsilon_0\n= -U\/2} \\ll h \\ll \\Gamma_{\\uparrow}$, where various behaviors\ncan occur. Rather than presenting an exhaustive discussion of\nthis limit, we settle with identifying certain generic features\nthat apply when both components $|h \\cos \\theta|$ and $h\n\\sin\\theta$ exceed $T_K|_{\\epsilon_0 = -U\/2}$. To begin with,\nwhatever remnants of the correlation-induced resonances that\nare left, these are shifted away from the middle of the\nCoulomb-blockade valley in the direction where\n$|h_{\\text{tot}}^z|$ acquires its minimal value. Consequently,\n$h_{\\text{tot}}$ and $T_K$ no longer obtain their minimal\nvalues at the same gate voltage $\\epsilon_0$. This has the\neffect of generating highly asymmetric structures in place of\nthe two symmetric resonances that are found for $h = 0$. The\nheights of these features are governed by the ``geometric''\nfactors $\\sin^2 (\\theta_l + s_R \\theta_h)$ at the corresponding\ngate voltages. Their widths are controlled by the underlying\nKondo temperatures, which can differ substantially in\nmagnitude. Since the entire structure is shifted away from the\nmiddle of the Coulomb-blockade valley where $T_K$ is minimal,\nall features are substantially broadened as compared with the\ncorrelation-induced resonances for $h = 0$. Indeed, similar\ntendencies are seen in Fig.~5 of Ref.~\\onlinecite{Meden06PRL},\neven though the model parameters used in this figure lie on the\nborderline between the mixed-valent and the local-moment\nregimes.\n\n\n\\section{Concluding remarks}\n\\label{sec:conclusions}\n\nWe have presented a comprehensive investigation of the general\ntwo-level model for quantum-dot devices. A proper choice of the\nquantum-mechanical representation of the dot and the lead\ndegrees of freedom reveals an exact mapping onto a generalized\nAnderson model. In the local-moment regime, the latter\nHamiltonian is reduced to an anisotropic Kondo model with a\ntilted effective magnetic field. As the anisotropic Kondo model\nflows to the isotropic strong-coupling fixed point, this\nenables a unified description of all coupling regimes of the\noriginal model in terms of the universal magnetization curve of\nthe conventional isotropic Kondo model, for which exact results\nare available. Various phenomena, such as phase lapses in the\ntransmission phase,~\\cite{Silva02,Golosov06} charge\noscillations,~\\cite{Gefen04,Sindel05} and correlation-induced\nresonances~\\cite{Meden06PRL,Karrasch06} in the conductance, can\nthus be accurately and coherently described within a single\nphysical framework.\n\nThe enormous reduction in the number of parameters in\nthe system was made possible by the key observation that\na general, possibly non-Hermitian tunnelling matrix\n$\\hat{A}$ can always be diagonalized with the help of\ntwo simultaneous unitary transformations, one pertaining\nthe dot degrees of freedom, and the other applied to\nthe lead electrons. This transformation, known as the\nsingular-value decomposition, should have\napplications in other physical problems involving\ntunnelling or transfer matrices without any special\nunderlying symmetries.\n\nAs the two-level model for transport is quite general, it can\npotentially be realized in many different ways. As already\nnoted in the main text, the model can be used to describe\neither a single two-level quantum dot or a double quantum dot\nwhere each dot harbors only a single level. Such realizations\nrequire that the spin degeneracy of the electrons will be\nlifted by an external magnetic field. Alternative realizations\nmay directly involve the electron spin. For example, consider a\nsingle spinful level coupled to two ferromagnetic leads with\n\\emph{non-collinear} magnetizations. Written in a basis with a\nparticular \\emph{ad hoc} local spin quantization axis, the\nHamiltonian of such a system takes the general form of\nEq.~\\eqref{IHAM}, after properly combining the electronic\ndegrees of freedom in both leads. As is evident from our\ndiscussion, the local spin will therefore experience an\neffective magnetic field that is not aligned with either of the\ntwo magnetizations of the leads. This should be contrasted with\nthe simpler configurations of parallel and antiparallel\nmagnetizations, as considered, e.g., in\nRefs.~\\onlinecite{Martinek03PRL,MartinekNRGferro}\nand~~\\onlinecite{MartinekPRB05}.\n\nAnother appealing system for the experimental observation of the\nsubtle correlation effects discussed in the present paper is a\ncarbon nanotube-based quantum dot. In such a device both charging\nenergy and single-particle level spacing can be\nsufficiently large~\\cite{Buitelaar02} to provide a set of\nwell-separated discrete electron states. Applying external\nmagnetic field either perpendicular~\\cite{Nygard00} or\/and\nparallel~\\cite{HerreroSU4} to the nanotube gives great\nflexibility in tuning the energy level structure, and thus\nturns the system into a valuable testground for probing\nthe Kondo physics addressed in this study.\n\n\nThroughout this paper we confined ourselves to spinless\nelectrons, assuming that spin degeneracy has been lifted by an\nexternal magnetic field. Our mapping can equally be applied to\nspinful electrons by implementing an identical singular-value\ndecomposition to each of the two spin orientations separately\n(assuming the tunnelling term is diagonal in and independent of\nthe spin orientation). Indeed, there has been considerable\ninterest lately in spinful variants of the Hamiltonian of\nEq.~(\\ref{IHAM}), whether in connection with lateral quantum\ndots,~\\cite{Kondo2stage,HofstetterZarand04} capacitively\ncoupled quantum dots,~\\cite{KondoSU4,LeHuretal05,Galpinetal06}\nor carbon nanotube devices~\\cite{AguadoPRL05}. Among the\nvarious phenomena that have been discussed in these contexts,\nlet us mention SU(4) variants of the Kondo\neffect~\\cite{KondoSU4,LeHuretal05,AguadoPRL05}, and\nsinglet-triplet transitions with two-stage screening on the\ntriplet side~\\cite{Kondo2stage,HofstetterZarand04}.\n\nSome of the effects that have been predicted for the\nspinful case were indeed observed in lateral semiconductor\nquantum dots~\\cite{vanderWiel02,Granger05} and in carbon nanotube\nquantum dots~\\cite{HerreroSU4}. Still, there remains a\ndistinct gap between the idealized models that have been\nemployed, in which simplified symmetries are often imposed\non the tunnelling term, and the actual experimental systems\nthat obviously lack these symmetries. Our mapping should\nprovide a much needed bridge between the idealized models\nand the actual experimental systems. Similar to the present\nstudy, one may expect a single unified description\nencompassing all coupling regimes in terms of just\na few basic low-energy scales. This may provide valuable\nguidance for analyzing future experiments on such\ndevices.\n\n\\begin{acknowledgments}\nThe authors are thankful to V. Meden for kindly providing the\nnumerical data for the inset in Fig.~\\ref{fig:CIR}. VK is\ngrateful to Z.~A.~N\\'{e}meth for stimulating discussions of\nperturbative calculations. This research was supported by a\nCenter of Excellence of the Israel Science Foundation, and by a\ngrant from the German Federal Ministry of Education and\nResearch (BMBF) within the framework of the German-Israeli\nProject Cooperation (DIP). We have recently become aware of a\nrelated study by Silvestrov and Imry,~\\cite{SI-06} which\nindependently develops some of the ideas presented in this\nwork.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}