Bounded quantum lattice pressure on cubes and van Hove sets

Write ${\mathcal{S}}_{\mathrm{\Phi }}$ for the finite anchored-shape list and $J_{\mathrm{\Phi }}={\sum }_{S\in {\mathcal{S}}_{\mathrm{\Phi }}}\mathrm{\parallel }\mathrm{\Phi }(S)\mathrm{\parallel }$. For each shape there are at most $\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }$ translates contained in $\mathrm{\Lambda }$, so $\mathrm{\parallel }{H}_{\mathrm{\Lambda }}\mathrm{\parallel }\le J_{\mathrm{\Phi }}\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }$. By [F3], $H_{\mathrm{\Lambda }}$ has real eigenvalues in $[-J_{\mathrm{\Phi }}\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid },J_{\mathrm{\Phi }}\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }]$. Therefore $q^{\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}{e}^{-\beta J_{\mathrm{\Phi }}\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}\le Z_{\mathrm{\Lambda }}(\beta )\le q^{\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}{e}^{\beta J_{\mathrm{\Phi }}\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}$, and $p_{\mathrm{\Lambda }}(\beta )$ is real and uniformly bounded.

For self-adjoint matrices $K,L$ on the same finite spin space, put $M_s=L+s(K-L)$. Apply [F2] to obtain $\frac{d}{ds}\log \mathrm{Tr}(e^{M_s})=\mathrm{Tr}({\rho }_s(K-L))$. The matrix ${\rho }_s$ is positive with trace one by [F3]; diagonalizing ${\rho }_s$ gives $\mathrm{\mid }\mathrm{Tr}({\rho }_s(K-L))\mathrm{\mid }\le \mathrm{\parallel }K-L\mathrm{\parallel }$, since each unit eigenvector $v$ obeys $\mathrm{\mid }⟨(K-L)v,v⟩\mathrm{\mid }\le \mathrm{\parallel }K-L\mathrm{\parallel }$. Integrating over $0\le s\le 1$ proves $\mathrm{\mid }\log \mathrm{Tr}(e^K)-\log \mathrm{Tr}(e^L)\mathrm{\mid }\le \mathrm{\parallel }K-L\mathrm{\parallel }$. In particular, for self-adjoint Hamiltonians $H,H^{\mathrm{\prime }}$ on that same space, $\mathrm{\mid }\log \mathrm{Tr}(e^{-\beta H})-\log \mathrm{Tr}(e^{-\beta H^{\mathrm{\prime }}})\mathrm{\mid }\le \beta \mathrm{\parallel }H-H^{\mathrm{\prime }}\mathrm{\parallel }$.

Fix an integer block side $\mathrm{\ell }\ge 1$ and tile $Q_L$ by $N=\lfloor L\mathrm{/}\mathrm{\ell }{\rfloor }^d$ disjoint full $\mathrm{\ell }$-cubes, leaving $E_L$ as the residual sites. The strips in the $d$ coordinate directions give $\mathrm{\mid }{E}_L\mathrm{\mid }\le d\mathrm{\ell }{L}^{d-1}$ when $L\ge \mathrm{\ell }$. Let $H_L^{\mathrm{c}\mathrm{u}\mathrm{t}}$ be the sum of the free-boundary Hamiltonians of the full blocks, embedded into ${\mathcal{A}}_{Q_L}$, and zero on residual sites. An interaction term omitted from this cut either meets $E_L$ or crosses an internal grid plane. For each anchored shape, at most $M\mathrm{\mid }{E}_L\mathrm{\mid }$ translates meet $E_L$, because choosing a site of the shape and a residual site determines its anchor. For a fixed coordinate grid plane, at most $(r+1)L^{d-1}$ anchors can have a support crossing it, where $r=\lceil R\rceil$: the support's coordinate span is at most $r$, the anchor must lie in an interval of at most $r+1$ lattice sites, and each of the other $d-1$ anchor coordinates has at most $L$ choices. There are at most $dL\mathrm{/}\mathrm{\ell }$ internal planes. Thus $\mathrm{\parallel }{H}_{Q_L}-H_L^{\mathrm{c}\mathrm{u}\mathrm{t}}\mathrm{\parallel }\le J_{\mathrm{\Phi }}(M\mathrm{\mid }{E}_L\mathrm{\mid }+d(r+1)L^d\mathrm{/}\mathrm{\ell })$.

The block Hamiltonians act on separate tensor factors. Diagonalize each block Hamiltonian using [F3]; tensor products of their eigenvectors and residual-site basis vectors form an orthonormal eigenbasis of $H_L^{\mathrm{c}\mathrm{u}\mathrm{t}}$. Hence $\mathrm{Tr}(e^{-\beta H_L^{\mathrm{c}\mathrm{u}\mathrm{t}}})=Z_{Q_{\mathrm{\ell }}}(\beta {)}^N{q}^{\mathrm{\mid }{E}_L\mathrm{\mid }}$. Step 1.2 and the residual-volume bound imply ${\mathrm{lim\hspace{0.17em}sup}}_{L\to \mathrm{\infty }}\mathrm{\mid }{p}_{Q_L}(\beta )-p_{Q_{\mathrm{\ell }}}(\beta )\mathrm{\mid }\le \beta J_{\mathrm{\Phi }}d(r+1)\mathrm{/}\mathrm{\ell }$.

For any $ϵ>0$, choose $\mathrm{\ell }$ so large that $2\beta J_{\mathrm{\Phi }}d(r+1)\mathrm{/}\mathrm{\ell }<ϵ\mathrm{/}2$. The estimate in step 3.1 then makes $\mathrm{\mid }{p}_{Q_L}(\beta )-p_{Q_{L^{\mathrm{\prime }}}}(\beta )\mathrm{\mid }<ϵ$ for all sufficiently large $L,L^{\mathrm{\prime }}$. The sequence is Cauchy in $\mathbb{R}$ and converges to a finite $p_{\mathrm{\Phi }}(\beta )$ by step 1.1. The estimate also gives $\mathrm{\mid }{p}_{\mathrm{\Phi }}(\beta )-p_{Q_{\mathrm{\ell }}}(\beta )\mathrm{\mid }\le \beta J_{\mathrm{\Phi }}d(r+1)\mathrm{/}\mathrm{\ell }$.

Let $({\mathrm{\Lambda }}_n)$ be van Hove and fix $\mathrm{\ell }\ge 1$. Use any fixed offset to tile ${\mathbb{Z}}^d$ by $\mathrm{\ell }$-cubes, retain only cubes fully contained in ${\mathrm{\Lambda }}_n$, and let $N_n$ be their number. Every residual site $x\in E_n:={\mathrm{\Lambda }}_n\setminus \bigcup B$ lies in a grid cube that also contains some $y\mathrm{\notin }{\mathrm{\Lambda }}_n$, so ${\mathrm{dist}}_{\mathrm{\infty }}(x,{\mathbb{Z}}^d\setminus {\mathrm{\Lambda }}_n)\le \mathrm{\ell }-1$ and $E_n\subseteq {\mathrm{\partial }}_{\mathrm{\ell }}{\mathrm{\Lambda }}_n$. Thus $\mathrm{\mid }{E}_n\mathrm{\mid }\mathrm{/}\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }\to 0$ and $N_n{\mathrm{\ell }}^d=\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }-\mathrm{\mid }{E}_n\mathrm{\mid }$. Define $H_n^{\mathrm{c}\mathrm{u}\mathrm{t}}$ to be the sum of the free-boundary Hamiltonians on the retained full cubes, embedded in ${\mathcal{A}}_{{\mathrm{\Lambda }}_n}$, and zero on the residual sites. Any omitted interaction term either meets $E_n$, in which case its anchor lies in ${\mathrm{\partial }}_{\mathrm{\ell }+r}{\mathrm{\Lambda }}_n$, or crosses a face between two retained cubes. For each retained cube, at most $2d(r+1){\mathrm{\ell }}^{d-1}$ anchor sites lie in its face layers wide enough for a support of range $r=\lceil R\rceil$ to cross a face; hence all internal-cut anchors number at most $2d(r+1){\mathrm{\ell }}^{d-1}{N}_n\le 2d(r+1)\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }\mathrm{/}\mathrm{\ell }$. Since the norm sum over shapes at one anchor is $J_{\mathrm{\Phi }}$, $\mathrm{\parallel }{H}_{{\mathrm{\Lambda }}_n}-H_n^{\mathrm{c}\mathrm{u}\mathrm{t}}\mathrm{\parallel }\le J_{\mathrm{\Phi }}(\mathrm{\mid }{\mathrm{\partial }}_{\mathrm{\ell }+r}{\mathrm{\Lambda }}_n\mathrm{\mid }+2d(r+1)\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }\mathrm{/}\mathrm{\ell })$. Translation invariance makes every full-block partition function equal to $Z_{Q_{\mathrm{\ell }}}$, and the zero residual Hamiltonian contributes $q^{\mathrm{\mid }{E}_n\mathrm{\mid }}$, so ${\mathrm{Tr}}_{{\mathrm{\Lambda }}_n}(e^{-\beta H_n^{\mathrm{c}\mathrm{u}\mathrm{t}}})=Z_{Q_{\mathrm{\ell }}}^{N_n}{q}^{\mathrm{\mid }{E}_n\mathrm{\mid }}$. Divide the comparison from step 1.2 by $\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }$; the van Hove condition sends the fixed-width boundary term and $\mathrm{\mid }{E}_n\mathrm{\mid }\mathrm{/}\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }$ to zero, while $N_n\mathrm{/}\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }\to {\mathrm{\ell }}^{-d}$. Therefore ${\mathrm{lim\hspace{0.17em}sup}}_n\mathrm{\mid }{p}_{{\mathrm{\Lambda }}_n}(\beta )-p_{Q_{\mathrm{\ell }}}(\beta )\mathrm{\mid }\le 2\beta J_{\mathrm{\Phi }}d(r+1)\mathrm{/}\mathrm{\ell }$. Letting $\mathrm{\ell }\to \mathrm{\infty }$ and using step 4.1 proves $p_{{\mathrm{\Lambda }}_n}(\beta )\to p_{\mathrm{\Phi }}(\beta )$.

Put ${\tilde{p}}_n:=\mathrm{\mid }{\mathrm{\Lambda }}_n{\mathrm{\mid }}^{-1}\log {\mathrm{Tr}}_{{\mathrm{\Lambda }}_n}(e^{-\beta (H_{{\mathrm{\Lambda }}_n}+B_n)})$. Applying step 1.2 to the self-adjoint matrices $-\beta (H_{{\mathrm{\Lambda }}_n}+B_n)$ and $-\beta H_{{\mathrm{\Lambda }}_n}$ shows that $\mathrm{\mid }{\tilde{p}}_n-p_{{\mathrm{\Lambda }}_n}(\beta )\mathrm{\mid }\le \beta \mathrm{\parallel }{B}_n\mathrm{\parallel }\mathrm{/}\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }$. This tends to zero by hypothesis, so step 5.1 gives ${\tilde{p}}_n\to p_{\mathrm{\Phi }}(\beta )$.

For nonempty finite $\mathrm{\Lambda }$, diagonalize $H_{\mathrm{\Lambda }}$ using [F3], with eigenvalues $E_j$ and Gibbs weights $w_j=e^{-\beta E_j}\mathrm{/}{Z}_{\mathrm{\Lambda }}(\beta )$. Then $p_{\mathrm{\Lambda }}(\beta )=\mathrm{\mid }\mathrm{\Lambda }{\mathrm{\mid }}^{-1}\log {\sum }_j{e}^{-\beta E_j}$ is convex because $p_{\mathrm{\Lambda }}^{\mathrm{\prime }\mathrm{\prime }}(\beta )=\mathrm{\mid }\mathrm{\Lambda }{\mathrm{\mid }}^{-1}({\sum }_j{w}_j{E}_j^2-({\sum }_j{w}_j{E}_j{)}^2)=\mathrm{\mid }\mathrm{\Lambda }{\mathrm{\mid }}^{-1}{\sum }_j{w}_j(E_j-{\sum }_k{w}_k{E}_k{)}^2\ge 0$. Applying [F2] to $M(\beta )=-\beta H_{\mathrm{\Lambda }}$ also gives $p_{\mathrm{\Lambda }}^{\mathrm{\prime }}(\beta )=-\mathrm{Tr}({\rho }_{\mathrm{\Lambda },\beta }{H}_{\mathrm{\Lambda }})\mathrm{/}\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }$.

For nonempty finite $\mathrm{\Lambda }$, the measure ${\mu }_{\mathrm{\Lambda },\beta }$ assigns weight $w_j$ to each eigenvalue, combining weights when eigenvalues coincide, and is a probability measure because ${\sum }_j{w}_j=1$. For $t\in (0,\beta )$ and $e=e_{\mathrm{\Phi }}(\beta )$, each eigenvalue $E\ge \mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }(e+ϵ)$ satisfies $e^{tE}\ge e^{t\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }(e+ϵ)}$. Summing the corresponding weights gives ${\mu }_{\mathrm{\Lambda },\beta }\{E\mathrm{/}\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }\ge e+ϵ\}\le \exp \text{\hspace{0.17em}⁣}(\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }[p_{\mathrm{\Lambda }}(\beta -t)-p_{\mathrm{\Lambda }}(\beta )-t(e+ϵ)])\mathrm{.}$ The identity ${\sum }_j{w}_j{e}^{t{E}_j}=Z_{\mathrm{\Lambda }}(\beta -t)\mathrm{/}{Z}_{\mathrm{\Lambda }}(\beta )$ gives the displayed exponent.

Fix $h\in (0,\beta )$. Convexity in step 1.1 gives $(p_{\mathrm{\Lambda }}(\beta )-p_{\mathrm{\Lambda }}(\beta -h))\mathrm{/}h\le p_{\mathrm{\Lambda }}^{\mathrm{\prime }}(\beta )\le (p_{\mathrm{\Lambda }}(\beta +h)-p_{\mathrm{\Lambda }}(\beta ))\mathrm{/}h$. Along ${\mathrm{\Lambda }}_n$, [F1] makes the two secants converge for fixed $h$ to the corresponding secants of $p_{\mathrm{\Phi }}$. Letting $h\downarrow 0$ and using differentiability at $\beta$ squeezes $p_{{\mathrm{\Lambda }}_n}^{\mathrm{\prime }}(\beta )\to p_{\mathrm{\Phi }}^{\mathrm{\prime }}(\beta )$; step 1.1 therefore proves $\mathrm{Tr}({\rho }_{{\mathrm{\Lambda }}_n,\beta }{H}_{{\mathrm{\Lambda }}_n})\mathrm{/}\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }\to e_{\mathrm{\Phi }}(\beta )$.

Differentiability and $e=-p_{\mathrm{\Phi }}^{\mathrm{\prime }}(\beta )$ give $p_{\mathrm{\Phi }}(\beta -t)-p_{\mathrm{\Phi }}(\beta )=te+o(t)$ as $t\downarrow 0$. Choose $t\in (0,\beta )$ so the remainder has magnitude at most $tϵ\mathrm{/}2$. For all sufficiently large $n$, convergence in [F1] at $\beta -t$ and $\beta$ makes the exponent bracket in step 1.2 at most $-tϵ\mathrm{/}4$. Thus the upper-tail probability is at most $e^{-tϵ\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }\mathrm{/}4}$.

For the lower tail, if $E\le \mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }(e-ϵ)$ then $e^{-tE}\ge e^{-t\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }(e-ϵ)}$. The same finite spectral sum gives ${\mu }_{\mathrm{\Lambda },\beta }\{E\mathrm{/}\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }\le e-ϵ\}\le \exp \text{\hspace{0.17em}⁣}(\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }[p_{\mathrm{\Lambda }}(\beta +t)-p_{\mathrm{\Lambda }}(\beta )+t(e-ϵ)])\mathrm{.}$ Since $p_{\mathrm{\Phi }}(\beta +t)-p_{\mathrm{\Phi }}(\beta )=-te+o(t)$, choose a possibly smaller fixed $t>0$ so the limiting bracket is at most $-tϵ\mathrm{/}2$; for large $n$ the bracket is at most $-tϵ\mathrm{/}4$. Let $t_{+}$ be the choice in step 2.2, let $t_{-}$ be this lower-tail choice, and set $c_0=\min (t_{+},t_{-})ϵ\mathrm{/}4>0$. Once $\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }\ge 4\log 2\mathrm{/}{c}_0$, the sum of the two tail bounds $2e^{-c_0\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }}$ is at most $e^{-c_0\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }\mathrm{/}2}$; hence $c_{ϵ}=c_0\mathrm{/}2$ works after increasing $n_0$.

Diagonalize a local density matrix $\rho$ on a region of $m$ sites using [F3], with eigenvalues ${\lambda }_j\ge 0$ summing to one and matrix dimension $D=q^m$, as supplied by [F4]. Then $S(\rho )=-{\sum }_{{\lambda }_j>0}{\lambda }_j\log {\lambda }_j\ge 0$. For each positive ${\lambda }_j$, the scalar inequality $-\log x\ge 1-x$ at $x=(D{\lambda }_j{)}^{-1}$ gives ${\lambda }_j\log (D{\lambda }_j)\ge {\lambda }_j-D^{-1}$. Summing over the positive eigenvalues yields $\log D-S(\rho )\ge 1-r\mathrm{/}D\ge 0$, where $r$ is the rank. Thus $0\le S(\rho )\le m\log q$.

For each $S\in {\mathcal{S}}_{\mathrm{\Phi }}$, let $N_S(L)$ count translations $x$ with $S+x\subseteq Q_L$. With $r=\lceil R\rceil$, $(L-2r{)}^d\le N_S(L)\le L^d$ for $L>2r$: every anchor at least $r$ sites from each face contains the whole support, and the anchor of every included term lies in $Q_L$. Translation invariance gives $\omega (\mathrm{\Phi }(S+x))=\omega (\mathrm{\Phi }(S))$. Therefore $\omega (H_{Q_L})\mathrm{/}{L}^d={\sum }_{S\in {\mathcal{S}}_{\mathrm{\Phi }}}(N_S(L)\mathrm{/}{L}^d)\omega (\mathrm{\Phi }(S))\to {\sum }_{S\in {\mathcal{S}}_{\mathrm{\Phi }}}\omega (\mathrm{\Phi }(S))=u_{\mathrm{\Phi }}(\omega )$.

Fix $\mathrm{\ell }\ge 1$ and tile $Q_L$ by $N=\lfloor L\mathrm{/}\mathrm{\ell }{\rfloor }^d$ disjoint $\mathrm{\ell }$-cubes and residual sites $E_L$, where $\mathrm{\mid }{E}_L\mathrm{\mid }\le d\mathrm{\ell }{L}^{d-1}$. Repeated use of [L1] subadditivity gives $S_{Q_L}(\omega )\le N{S}_{Q_{\mathrm{\ell }}}(\omega )+S_{E_L}(\omega )$. By step 1.1 the last term is at most $\mathrm{\mid }{E}_L\mathrm{\mid }\log q$; if $E_L=\mathrm{\varnothing }$, its scalar local density has entropy zero. Dividing by $L^d$ and taking the upper limit gives ${\mathrm{lim\hspace{0.17em}sup}}_L{S}_{Q_L}(\omega )\mathrm{/}{L}^d\le S_{Q_{\mathrm{\ell }}}(\omega )\mathrm{/}{\mathrm{\ell }}^d$. Every ratio is at least their infimum over $\mathrm{\ell }$, so the limit exists and equals that infimum.

On a fixed finite region, pointwise convergence of algebraic states gives convergence of all matrix coefficients of their density matrices by [F2] and [F4], hence convergence in operator norm. Ordered eigenvalues are continuous in that norm: the finite-dimensional min-max formula follows from an ordered orthonormal eigenbasis, since any $k$-dimensional subspace meets the span of the last $D-k+1$ eigenvectors, while the span of the first $k$ realizes the reverse bound; each Rayleigh quotient changes by at most the operator norm of the matrix difference. The function $-x\log x$, extended by $0$ at $x=0$, is continuous on $[0,1]$, so each $S_{Q_{\mathrm{\ell }}}(\omega )$ is continuous under pointwise convergence. By step 2.1, $s(\omega )$ is the infimum of these continuous normalized local entropies and is upper semicontinuous. The energy density is a finite sum of local state evaluations by [F1], so it is continuous.

If a sequence of translation-invariant algebraic states converges pointwise to $\omega$, then for each fixed $x,A$ the equalities ${\omega }_n({\tau }_x(A))={\omega }_n(A)$ pass to the limit, so the invariant state set is closed. By [L2] every sequence in the full state space has a pointwise-convergent subsequence; the closed invariant subset is therefore sequentially compact.

For any $\omega \in {\mathcal{S}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$, [F5] represents its restriction to $Q_L$ by a trace-one density matrix, and [L1] applies to that density. Writing $S_{Q_L}=-\mathrm{Tr}({\rho }_{Q_L}\log {\rho }_{Q_L})$ (dimensionless), the supplier's physical entropy is $k_{\mathrm{B}}{S}_{Q_L}$ and $T=(k_{\mathrm{B}}\beta {)}^{-1}$, so its inequality rearranges to $S_{Q_L}-\beta \omega (H_{Q_L})\le \log Z_{Q_L}(\beta )$. Divide by $L^d$ and use [F1]–[F2] to obtain $s(\omega )-\beta u_{\mathrm{\Phi }}(\omega )\le p_{\mathrm{\Phi }}(\beta )$. This is conditional on the stated finite-dimensional inequality [L1].

For each positive integer $\mathrm{\ell }$, let $Q_{\mathrm{\ell }}$ be one block, $H_{\mathrm{\ell }}=H_{Q_{\mathrm{\ell }}}$, and ${\rho }_{\mathrm{\ell }}=e^{-\beta H_{\mathrm{\ell }}}\mathrm{/}{Z}_{Q_{\mathrm{\ell }}}(\beta )$. Tile ${\mathbb{Z}}^d$ by blocks $B_k=\mathrm{\ell }k+\{0,\dots ,\mathrm{\ell }-1{\}}^d$, $k\in {\mathbb{Z}}^d$. For a finite region $\mathrm{\Lambda }$, set ${\rho }_{\mathrm{\Lambda }}={⨂}_{k:B_k\cap \mathrm{\Lambda }\mathrm{\ne }\mathrm{\varnothing }}{\mathrm{Tr}}_{B_k\setminus \mathrm{\Lambda }}({\rho }_{\mathrm{\ell }}^{(k)})$, where ${\rho }_{\mathrm{\ell }}^{(k)}$ is the translated copy on $B_k$. This is a finite product; for $\mathrm{\Lambda }=\mathrm{\varnothing }$ the empty tensor product is the scalar unit density. Each nonempty block marginal is positive trace-one by [F3]–[F5].

If $\mathrm{\Lambda }\subseteq {\mathrm{\Lambda }}^{\mathrm{\prime }}$, then within each block the partial trace of ${\mathrm{Tr}}_{B_k\setminus {\mathrm{\Lambda }}^{\mathrm{\prime }}}{\rho }_{\mathrm{\ell }}^{(k)}$ over $(B_k\cap {\mathrm{\Lambda }}^{\mathrm{\prime }})\setminus \mathrm{\Lambda }$ equals ${\mathrm{Tr}}_{B_k\setminus \mathrm{\Lambda }}{\rho }_{\mathrm{\ell }}^{(k)}$: the coordinate formula in [F3] sums the same finite indices in either order. A block meeting ${\mathrm{\Lambda }}^{\mathrm{\prime }}$ but not $\mathrm{\Lambda }$ is traced over completely and contributes $\mathrm{Tr}({\rho }_{\mathrm{\ell }}^{(k)})=1$. The partial trace of a tensor product factors by the same finite sums, so ${\mathrm{Tr}}_{{\mathrm{\Lambda }}^{\mathrm{\prime }}\setminus \mathrm{\Lambda }}({\rho }_{{\mathrm{\Lambda }}^{\mathrm{\prime }}})={\rho }_{\mathrm{\Lambda }}$. Thus the local density family is consistent, and [L3] extends it to a state ${\eta }_{\mathrm{\ell }}$ on $\mathcal{A}$.

The state ${\eta }_{\mathrm{\ell }}$ is invariant under shifts in $\mathrm{\ell }{\mathbb{Z}}^d$ because these shifts permute identical block factors, using the covariance and action law [F6]–[F7]. Average its ${\mathrm{\ell }}^d$ translates over residues $r\in \{0,\dots ,\mathrm{\ell }-1{\}}^d$ to obtain ${\omega }_{\mathrm{\ell }}={\mathrm{\ell }}^{-d}{\sum }_r{\eta }_{\mathrm{\ell }}\circ {\tau }_r$. The group law in [F6] makes each lattice translation permute the residue classes modulo $\mathrm{\ell }$, so ${\omega }_{\mathrm{\ell }}\in {\mathcal{S}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$.

For each fixed phase $r$, the restriction of ${\eta }_{\mathrm{\ell }}\circ {\tau }_r$ to $Q_L$ is a tensor product of full block densities and partial-block marginals. In each coordinate, $Q_L$ contains $L\mathrm{/}\mathrm{\ell }+O(1)$ full intervals of length $\mathrm{\ell }$, so the number of full blocks is $L^d\mathrm{/}{\mathrm{\ell }}^d+O_{\mathrm{\ell }}(L^{d-1})$. Entropy additivity in [L2] gives the full-block contribution, while every partial marginal has nonnegative entropy by the opening finite-density spectral calculation in the proof of [F2]; hence ${\mathrm{lim\hspace{0.17em}inf}}_{L\to \mathrm{\infty }}{S}_{Q_L}({\eta }_{\mathrm{\ell }}\circ {\tau }_r)\mathrm{/}{L}^d\ge s({\rho }_{\mathrm{\ell }})\mathrm{/}{\mathrm{\ell }}^d$. At each finite $L$, concavity in [L2] gives $S_{Q_L}({\omega }_{\mathrm{\ell }})\ge {\mathrm{\ell }}^{-d}{\sum }_r{S}_{Q_L}({\eta }_{\mathrm{\ell }}\circ {\tau }_r)$. Divide by $L^d$, take the limit on the invariant left side using [F2], and use the finite number of phases to obtain $s({\omega }_{\mathrm{\ell }})\ge s({\rho }_{\mathrm{\ell }})\mathrm{/}{\mathrm{\ell }}^d$.

Put $r=\lceil R\rceil$. By the phase average and translation covariance [F7], $u_{\mathrm{\Phi }}({\omega }_{\mathrm{\ell }})={\mathrm{\ell }}^{-d}{\sum }_{x\in Q_{\mathrm{\ell }}}{\sum }_{S\in {\mathcal{S}}_{\mathrm{\Phi }}}{\eta }_{\mathrm{\ell }}(\mathrm{\Phi }(S+x))$. Because each anchored shape contains $0$, the terms with $S+x\subseteq Q_{\mathrm{\ell }}$ sum exactly to $\mathrm{Tr}({\rho }_{\mathrm{\ell }}{H}_{\mathrm{\ell }})$. Every remaining pair $(S,x)$ has $x\in D_{\mathrm{\ell },r}:=\{y\in Q_{\mathrm{\ell }}:{\mathrm{dist}}_{\mathrm{\infty }}(y,{\mathbb{Z}}^d\setminus Q_{\mathrm{\ell }})\le r\}$, since the support contains $x$ and has diameter at most $R$. If $r=0$, no support crosses a block face and $D_{\mathrm{\ell },0}=\mathrm{\varnothing }$. If $r\ge 1$, a union bound over the $2d$ faces gives $\mathrm{\mid }{D}_{\mathrm{\ell },r}\mathrm{\mid }\le 2dr{\mathrm{\ell }}^{d-1}$. Terms anchored in these layers may use partial block marginals of ${\eta }_{\mathrm{\ell }}$; those marginals are positive trace-one by steps 1.2–3.1, and [F5] gives $\mathrm{\mid }{\eta }_{\mathrm{\ell }}(A)\mathrm{\mid }\le \mathrm{\parallel }A\mathrm{\parallel }$ for each local term. At each anchor the sum of term norms is at most $J_{\mathrm{\Phi }}\le \kappa$ by [F8]. Hence $\mathrm{\mid }{u}_{\mathrm{\Phi }}({\omega }_{\mathrm{\ell }})-\mathrm{Tr}({\rho }_{\mathrm{\ell }}{H}_{\mathrm{\ell }})\mathrm{/}{\mathrm{\ell }}^d\mathrm{\mid }\le 2dr\kappa \mathrm{/}\mathrm{\ell }$. By the equality case of [L1], $s({\rho }_{\mathrm{\ell }})-\beta \mathrm{Tr}({\rho }_{\mathrm{\ell }}{H}_{\mathrm{\ell }})=\log Z_{Q_{\mathrm{\ell }}}(\beta )$. Hence $s({\omega }_{\mathrm{\ell }})-\beta u_{\mathrm{\Phi }}({\omega }_{\mathrm{\ell }})\ge p_{Q_{\mathrm{\ell }}}(\beta )-2\beta dr\kappa \mathrm{/}\mathrm{\ell }$.

Take any sequence of positive integers ${\mathrm{\ell }}_j\to \mathrm{\infty }$. By [F2], a subsequence of ${\omega }_{{\mathrm{\ell }}_j}$ converges pointwise to some invariant algebraic state ${\omega }_{\ast }$. The right side of step 5.1 tends to $p_{\mathrm{\Phi }}(\beta )$ by [F1], while step 1.1 bounds every variational value above by $p_{\mathrm{\Phi }}(\beta )$. Thus the sequence of variational values converges to $p_{\mathrm{\Phi }}(\beta )$. Upper semicontinuity of $s$ and continuity of $u_{\mathrm{\Phi }}$ from [F2] imply $s({\omega }_{\ast })-\beta u_{\mathrm{\Phi }}({\omega }_{\ast })\ge p_{\mathrm{\Phi }}(\beta )$; step 1.1 gives the reverse inequality. Therefore equality is attained, proving the maximum formula.