Infinite quantum spin physical quantities

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.