Stationary quantum variational pressure maximum

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.