Stationary quantum spin entropy and energy densities

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.