Infinite quantum spin quantities and physical adoption

Fix an integer dimension $d\ge 1$, the lattice $\mathrm{\Gamma }={\mathbb{Z}}^d$, a finite on-site dimension $q\ge 1$, a lattice spacing $a>0$ metres, and finite-region Hilbert spaces ${\mathcal{H}}_{\mathrm{\Lambda }}$ and matrix algebras ${\mathcal{A}}_{\mathrm{\Lambda }}$ as in Local spin matrix algebras and their factor inclusions. The observable algebra is the norm completion $\mathcal{A}$ of the directed union of the local algebras, as in Norm-completed quasi-local quantum spin algebra. An infinite spin state is a bounded positive normalized functional $\omega :\mathcal{A}\to \mathbb{C}$ with local density matrices ${\rho }_{\mathrm{\Lambda }}$ specified by $\omega (A)={\mathrm{Tr}}_{\mathrm{\Lambda }}({\rho }_{\mathrm{\Lambda }}A)$ for $A\in {\mathcal{A}}_{\mathrm{\Lambda }}$, as in Infinite algebraic state and finite-region restrictions. A bounded local observable is an element of some ${\mathcal{A}}_{\mathrm{\Lambda }}$; a measurement observable is self-adjoint. Its model expectation is $\omega (A)$, real for self-adjoint $A$. For ordered bounded local observables $A_1,\dots ,A_n$, define the ordered correlation $C_{\omega }(A_1,\dots ,A_n):=\omega (A_1\cdots A_n)$. It may be complex when the product is not self-adjoint, depends on the order when observables do not commute, and has the product of their assigned units. The state functional is dimensionless on dimensionless operators; if $A$ has an assigned unit, $\omega (A)$ has that same unit.

Within this formulation, primitive model inputs are the dimensionless integers $d,q$, the lattice spacing $a$, the positive constants $\mathrm{\hslash },k_B$, the temperature $T$, and the interaction coefficients $\mathrm{\Phi }$ (or $J_{ij},h_i$ in the diagonal Ising submodel). The inverse temperature, finite Hamiltonians, partition functions, pressure/free-energy/entropy/energy densities, and state correlations are derived constructions under their stated hypotheses. The Born probability interpretation and the identification of KMS states with physical equilibrium are adopted separately; neither follows from this mathematical definition.

In SI bookkeeping, $a$ is measured in metres, $\mathrm{\hslash }>0$ in joule-seconds, $k_B>0$ in joules per kelvin, and $T>0$ in kelvins. The finite-range interaction coefficients are specified inputs; a term $\mathrm{\Phi }(Z)$ has units joules when interpreted as energy. The dimensionless local spin matrices (including the Pauli matrices in the spin-$\frac{1}{2}$ submodel) do not by themselves identify a magnetic moment or an applied field. In a term $-h_i{Z}_i$, $h_i$ is an energy coefficient in joules; any identification with a laboratory field or magnetic moment requires a separate coupling and calibration model. No numerical parameter value is specified here.

The inverse-temperature parameter is the derived quantity $\beta :=(k_{B}T{)}^{-1}>0$, in joules${}^{-1}$. For finite $\mathrm{\Lambda }$, the free-boundary Hamiltonian $H_{\mathrm{\Lambda }}={\sum }_{\mathrm{\varnothing }\mathrm{\ne }Z\subseteq \mathrm{\Lambda }}\mathrm{\Phi }(Z)$ and the finite partition function $Z_{\mathrm{\Lambda }}(\beta ):={\mathrm{Tr}}_{\mathrm{\Lambda }}(e^{-\beta H_{\mathrm{\Lambda }}})$ are defined by Uniformly bounded finite-range quantum spin interaction and Local spin matrix algebras and their factor inclusions. When $\mathrm{\Phi }$ is an energy interaction, $H_{\mathrm{\Lambda }}$ is in joules, $\beta H_{\mathrm{\Lambda }}$ is dimensionless, and $Z_{\mathrm{\Lambda }}$ is positive and dimensionless. The finite-volume Gibbs density ${\rho }_{\mathrm{\Lambda },\beta }=e^{-\beta H_{\mathrm{\Lambda }}}\mathrm{/}{Z}_{\mathrm{\Lambda }}$ is a finite-region density matrix; it is not an asserted density matrix on an infinite tensor-product Hilbert space.

For $t$ in seconds, the finite-volume Heisenberg evolution is ${\alpha }_t^{\mathrm{\Lambda }}(A)=e^{it{H}_{\mathrm{\Lambda }}\mathrm{/}\mathrm{\hslash }}A{e}^{-it{H}_{\mathrm{\Lambda }}\mathrm{/}\mathrm{\hslash }}$. The thermodynamic dynamics ${\alpha }_t$ exists under the exact hypotheses of Strongly continuous thermodynamic spin automorphism group. The $\beta$-KMS criterion for this dynamics uses a bounded continuous strip function with lower edge $\omega (A{\alpha }_t(B))$, upper edge $\omega ({\alpha }_t(B)A)$, and imaginary-time width $\beta \mathrm{\hslash }$ seconds, exactly as in Physical-time KMS strip equilibrium condition. This is a mathematical state criterion; no global infinite-volume Gibbs trace is part of the definition.

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 a finite region $\mathrm{\Lambda }\subseteq {\mathbb{Z}}^d$ and $x\in \mathrm{\Lambda }$, define ${\mathrm{dist}}_{\mathrm{\infty }}(x,{\mathbb{Z}}^d\setminus \mathrm{\Lambda }):={\min }_{y\in {\mathbb{Z}}^d\setminus \mathrm{\Lambda }}{d}_{\mathrm{\infty }}(x,y)$, using the convention of Bounded quantum lattice pressure on cubes and van Hove sets. The minimum exists because the complement is nonempty and its distances from $x$ are positive integers. For a translation-invariant interaction $\mathrm{\Phi }$ of Translation-invariant finite-range spin interaction and finite nonempty $\mathrm{\Lambda }$, define the finite-volume dimensionless log-partition per site by $p_{\mathrm{\Lambda }}(\beta ):=\mathrm{\mid }\mathrm{\Lambda }{\mathrm{\mid }}^{-1}\log Z_{\mathrm{\Lambda }}(\beta )$. A sequence $({\mathrm{\Lambda }}_n)$ is van Hove here when $\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }\to \mathrm{\infty }$ and for each fixed integer $r\ge 0$, the inner boundary ${\mathrm{\partial }}_r{\mathrm{\Lambda }}_n=\{x\in {\mathrm{\Lambda }}_n:{\mathrm{dist}}_{\mathrm{\infty }}(x,{\mathbb{Z}}^d\setminus {\mathrm{\Lambda }}_n)\le r\}$ satisfies $\mathrm{\mid }{\mathrm{\partial }}_r{\mathrm{\Lambda }}_n\mathrm{\mid }\mathrm{/}\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }\to 0$. Under the precise finite-range and translation-invariance assumptions of Bounded quantum lattice pressure on cubes and van Hove sets, the common finite limit is $p_{\mathrm{\Phi }}(\beta ):={\lim }_n{p}_{{\mathrm{\Lambda }}_n}(\beta )$. It is dimensionless per lattice site, not a pressure in pascals. Define the finite-volume free energy per site $f_{\mathrm{\Lambda }}(\beta ):=-{\beta }^{-1}{p}_{\mathrm{\Lambda }}(\beta )$ and the thermodynamic free energy per site $f_{\mathrm{\Phi }}(\beta ):=-{\beta }^{-1}{p}_{\mathrm{\Phi }}(\beta )=-k_{B}T{p}_{\mathrm{\Phi }}(\beta )$, in joules per site. Dividing by the model cell volume $a^d$ gives a free-energy density in joules per cubic metre when $d=3$; this fixed-lattice quantity does not by itself define a mechanical equation of state or pressure in pascals.

A state is spatially translation invariant when $\omega ({\tau }_x(A))=\omega (A)$ for every $x\in {\mathbb{Z}}^d$ and local $A$, with ${\tau }_x$ as defined in Translation-invariant finite-range spin interaction. For such a state, let $Q_L=\{0,\dots ,L-1{\}}^d$ and define the dimensionless cube entropy per site $s(\omega ):={\lim }_{L\to \mathrm{\infty }}\frac{-{\mathrm{Tr}}_{Q_L}({\rho }_{Q_L}\log {\rho }_{Q_L})}{L^d},0\log 0:=0.$ Under Countable Choice and the full Axiom of Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice, the exact entropy-density supplier Stationary quantum spin entropy and energy densities establishes that this limit exists for every such state and equals the infimum over positive integer cube sides. The corresponding physical entropy density is $k_{B}s(\omega )\mathrm{/}{a}^d$, in joules per kelvin per cubic metre for $d=3$. For the translation-invariant finite-range energy interaction, the same supplier defines the energy density per site $u_{\mathrm{\Phi }}(\omega )={\sum }_{S\in {\mathcal{S}}_{\mathrm{\Phi }}}\omega (\mathrm{\Phi }(S))$; it is in joules per site when $\mathrm{\Phi }$ is an energy interaction, and $u_{\mathrm{\Phi }}(\omega )\mathrm{/}{a}^d$ is the energy density per physical volume in $d=3$. These thermodynamic limits are restricted to the supplier's stated assumptions and do not assert that a prepared physical material realizes a translation-invariant state.

Adopt, for this bounded finite-dimensional quantum spin model, the following physical interpretation and equilibrium rule. A prepared state is a positive normalized functional $\omega$ on the quasi-local observable algebra of Infinite algebraic state and finite-region restrictions. The model assigns the Born probability $\omega (P)$ to a local projective-measurement outcome represented by a projection $P\in {\mathcal{A}}_{\mathrm{\Lambda }}$; the probabilities of the mutually orthogonal spectral projections of a finite-region self-adjoint observable sum to one. For a physical equilibrium preparation at temperature $T>0$, adopt as the model's equilibrium criterion that $\omega$ is a $\beta$-KMS state for the thermodynamic dynamics ${\alpha }_t$, with $\beta =(k_{B}T{)}^{-1}$ and the physical-time strip of width $\beta \mathrm{\hslash }$ seconds as in Physical-time KMS strip equilibrium condition. This is an adopted model postulate, not a deduction from the algebraic state definition or from the finite-volume Gibbs identity.

The postulate does not claim that an actual material has the specified interaction parameters, that a system dynamically relaxes to a KMS state, that equilibrium is unique, or that an apparatus realizes a specified projective measurement. It reports no experiment or measured value. Finite-volume approximations, when used, remain distinct from the infinite-volume state and its local restrictions.

The entropy-density construction in the physical-quantities definition is a separate conditional quantity under its stated Countable Choice and full Axiom of Choice hypotheses. This postulate uses only the model state, local measurement and KMS/unit conventions; it does not assume an entropy limit or assert that a material realizes the model.

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.

Adopt the model and units of Infinite quantum spin physical quantities and the equilibrium criterion of KMS equilibrium and local Born quantum spin model. Let $T>0$, $k_B>0$, and $\mathrm{\hslash }>0$. Let $Z_i$ be the dimensionless Pauli matrix with eigenvalues $\pm 1$ at site $i$, and let the diagonal spin-$\frac{1}{2}$ Ising interaction of Diagonal quantum Ising interaction and classical conditional energies have symmetric energy coefficients $J_{ij}=J_{ji}$ and fields $h_i$ in joules, finite range, and a uniform per-site bound. Thus $H_{\mathrm{\Lambda }}=-{\sum }_{i\in \mathrm{\Lambda }}{h}_i{Z}_i-{\sum }_{\{i,j\}\subseteq \mathrm{\Lambda }}{J}_{ij}{Z}_i{Z}_j,$ with each unordered internal pair counted once. Here $\mathrm{\mid }i-j{\mathrm{\mid }}_1:={\sum }_{k=1}^d\mathrm{\mid }{i}_k-j_k\mathrm{\mid }$ is the lattice ${\mathrm{\ell }}^1$ distance, and $\beta =(k_{B}T{)}^{-1}$ in ${\mathrm{J}}^{-1}$.

1. Sufficient high-temperature uniqueness. For translation-invariant symmetric pair couplings, a constant finite field $h_i=h$, and ${\sup }_{i\in {\mathbb{Z}}^d}{\sum }_{j\mathrm{\ne }i}\tanh \text{\hspace{0.17em}⁣}\left(\frac{\mathrm{\mid }{J}_{ij}\mathrm{\mid }}{k_{B}T}\right)<1,$ the full quasi-local algebra has exactly one $\beta$-KMS state. In the homogeneous nearest-neighbor model this sufficient condition is $2d\tanh (\mathrm{\mid }J\mathrm{\mid }\mathrm{/}(k_{B}T))<1$.

2. One-dimensional uniqueness. For $d=1$, uniform nearest-neighbor coupling $J\in \mathbb{R}$ and constant finite field $h$, every $T>0$ gives exactly one full-algebra $\beta$-KMS state.

3. Two-dimensional sufficient coexistence. For $d=2$, uniform nearest-neighbor ferromagnetic coupling $J>0$, zero field, and $T\le J\mathrm{/}(2k_B)$, there are at least two distinct spatially translation-invariant full-algebra $\beta$-KMS states ${\omega }_{+}$ and ${\omega }_{-}$ with opposite nonzero dimensionless magnetizations ${\omega }_{+}(Z_i)\ge m_{\ast }>0$ and ${\omega }_{-}(Z_i)\le -m_{\ast }<0$. Here spatial translation invariance means $\omega ({\tau }_x(A))=\omega (A)$ for every local $A$ and $x\in {\mathbb{Z}}^2$, with ${\tau }_x$ from Translation-invariant finite-range spin interaction. Put $K=\beta J$, $r=3e^{-2K}$, $\delta (K)=\frac{4}{3}\frac{r^4(5-4r)}{(1-r{)}^2}$, and $m_{\ast }=1-2\delta (K)$. This is a sufficient coexistence bound; $T=J\mathrm{/}(2k_B)$ is not asserted to be the critical temperature, the states are not asserted to be extremal, and no transverse-field result follows.

For the same two-dimensional model with uniform coupling $J$ and variable uniform longitudinal energy coefficient $h$, put $H_{\mathrm{\Lambda }}(h)=-J{\sum }_{\begin{array}{c}\{i,j\}\subseteq \mathrm{\Lambda }\\ \mathrm{\mid }i-j{\mathrm{\mid }}_1=1\end{array}}{Z}_i{Z}_j-h{\sum }_{i\in \mathrm{\Lambda }}{Z}_i,$ where each internal unordered nearest-neighbor pair is counted once. At fixed $\beta$, let ${\mathrm{\Lambda }}_n=[-n,n{]}^2$ and define the dimensionless log-partition pressure per site $p_{\beta }(h):={\lim }_{n\to \mathrm{\infty }}\frac{1}{\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }}\log {\mathrm{Tr}}_{{\mathrm{\Lambda }}_n}\text{\hspace{0.17em}⁣}\left(e^{-\beta H_{{\mathrm{\Lambda }}_n}(h)}\right)\mathrm{.}$ Under the stated coexistence condition this limit exists, is not differentiable at $h=0$, and satisfies $(p_{\beta }(h)-p_{\beta }(0))\mathrm{/}h\ge \beta m_{\ast }$ for every $h>0$ and $(p_{\beta }(h)-p_{\beta }(0))\mathrm{/}h\le -\beta m_{\ast }$ for every $h<0$. The $p_{\beta }$ here is dimensionless per site, not a mechanical pressure in pascals.

These statements are conditional deductions from the adopted spin model and current mathematical supplier interfaces. High-temperature uniqueness is a sufficient influence bound, one-dimensional uniqueness holds at every positive model temperature, and the two-dimensional low-temperature bound only guarantees coexistence in the stated range. No interval of actual material parameters, experimental uncertainty, transverse field, or observed transition is asserted.

The conditional Ising conclusions use the units and interaction definitions from the physical-quantities item, but do not use its separate AC-dependent entropy-density clause. The phase suppliers carry Countable Choice and the stated finite-matrix interfaces; the physical definition records full AC only for the entropy-density construction. No empirical premise is used.