Conditional interacting spin dynamics and equilibrium predictions

Let $\mathrm{\Gamma }={\mathbb{Z}}^d$, $d\ge 1$, have finite on-site spin dimension $q\ge 1$, local algebras ${\mathcal{A}}_{\mathrm{\Lambda }}$, and quasi-local algebra $\mathcal{A}$ as in Local spin matrix algebras and their factor inclusions, Norm-completed quasi-local quantum spin algebra. Adopt the physical spin-model interpretation and equilibrium criterion of KMS equilibrium and local Born quantum spin model. Let $\mathrm{\Phi }$ be a uniformly bounded finite-range interaction as in Uniformly bounded finite-range quantum spin interaction, with free-boundary $H_{\mathrm{\Lambda }}$ and finite-volume dynamics ${\alpha }_t^{\mathrm{\Lambda }}(A)=e^{it{H}_{\mathrm{\Lambda }}\mathrm{/}\mathrm{\hslash }}A{e}^{-it{H}_{\mathrm{\Lambda }}\mathrm{/}\mathrm{\hslash }}$. Assume Countable Choice and the finite-matrix C${}^{\ast }$-identity and state/density-norm interface of Finite matrix C-star algebra and density-state duality, exactly at the stated conditional scope.

Then the thermodynamic dynamics ${\alpha }_t$ exists as a strongly continuous one-parameter group of isometric $\ast$-automorphisms of $\mathcal{A}$. For every fixed $\beta =(k_{B}T{)}^{-1}>0$, at least one $\beta$-KMS state for ${\alpha }_t$ exists. This existence conclusion is a model deduction from the stated mathematical hypotheses; the physical postulate adopts KMS as an equilibrium criterion but the theorem does not imply dynamical relaxation, uniqueness, or a global infinite-volume Gibbs density matrix.

For $x,y\in {\mathbb{Z}}^d$, write $d_{\mathrm{\infty }}(x,y):={\max }_{1\le j\le d}\mathrm{\mid }{x}_j-y_j\mathrm{\mid }$. For finite $\mathrm{\Lambda }$ and $x\in \mathrm{\Lambda }$, use ${\mathrm{dist}}_{\mathrm{\infty }}(x,{\mathbb{Z}}^d\setminus \mathrm{\Lambda }):={\min }_{y\in {\mathbb{Z}}^d\setminus \mathrm{\Lambda }}{d}_{\mathrm{\infty }}(x,y)$, exactly as in Bounded quantum lattice pressure on cubes and van Hove sets; its minimum exists because the complement is nonempty and the distances are positive integers.

If, in addition, $\mathrm{\Phi }$ is translation invariant in the sense of Translation-invariant finite-range spin interaction, then for every sequence of finite nonempty regions $({\mathrm{\Lambda }}_n)$ satisfying $\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }\to \mathrm{\infty }$ and, for each fixed integer $r\ge 0$, $\mathrm{\mid }{\mathrm{\partial }}_r{\mathrm{\Lambda }}_n\mathrm{\mid }\mathrm{/}\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }\to 0$, where ${\mathrm{\partial }}_r{\mathrm{\Lambda }}_n:=\{x\in {\mathrm{\Lambda }}_n:{\mathrm{dist}}_{\mathrm{\infty }}(x,{\mathbb{Z}}^d\setminus {\mathrm{\Lambda }}_n)\le r\}$, the dimensionless finite-volume log-partition per site $p_{{\mathrm{\Lambda }}_n}(\beta ):=\mathrm{\mid }{\mathrm{\Lambda }}_n{\mathrm{\mid }}^{-1}\log {\mathrm{Tr}}_{{\mathrm{\Lambda }}_n}(e^{-\beta H_{{\mathrm{\Lambda }}_n}})$ converges to the same finite $p_{\mathrm{\Phi }}(\beta )$. This is the van Hove condition stated in Bounded quantum lattice pressure on cubes and van Hove sets. The corresponding model free energy per site $f_{\mathrm{\Phi }}(\beta ):=-{\beta }^{-1}{p}_{\mathrm{\Phi }}(\beta )=-k_{B}T{p}_{\mathrm{\Phi }}(\beta )$ is in joules per site when interaction terms are assigned energy units. The theorem makes no claim about a physical pressure in pascals or the accuracy of a material fit.

The physical-quantities definition also records a translation-invariant entropy-density construction under its explicit Countable Choice and full Axiom of Choice hypotheses. This theorem uses only its local model and unit conventions, not that entropy-density clause; the KMS-existence argument retains exactly Countable Choice and the finite-matrix interfaces listed above.