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 ${\tilde{p}}_n\to p_{\mathrm{\Phi }}(\beta )$.