Diagonal quantum Ising interaction and classical conditional energies

Fix an integer $d\ge 1$, the lattice $\mathrm{\Gamma }={\mathbb{Z}}^d$, and at each site the two-dimensional complex Hilbert space ${\mathcal{H}}_i={\mathbb{C}}^2$ with an orthonormal basis $e_{+1},e_{-1}$. Define the dimensionless local spin matrix $Z_i$ by $Z_i{e}_s=s{e}_s$ for $s\in \{-1,+1\}$. It is self-adjoint, diagonal in this basis, and has operator norm one. The local matrix algebras and factor inclusions are those of Local spin matrix algebras and their factor inclusions, and the interaction/Hamiltonian convention is that of Uniformly bounded finite-range quantum spin interaction.

Let $J_{ij}=J_{ji}\in \mathbb{R}$ be specified for distinct sites, and let $h_i\in \mathbb{R}$. Assume there is a finite $R\ge 0$ such that $J_{ij}=0$ when $\mathrm{\parallel }i-j{\mathrm{\parallel }}_{\mathrm{\infty }}>R$, and ${\sup }_{i\in \mathrm{\Gamma }}(\mathrm{\mid }{h}_i\mathrm{\mid }+{\sum }_{j\mathrm{\ne }i}\mathrm{\mid }{J}_{ij}\mathrm{\mid })<\mathrm{\infty }\mathrm{.}$ The associated quantum interaction in the convention of Uniformly bounded finite-range quantum spin interaction is $\mathrm{\Phi }(\{i\})=-h_i{Z}_i,\mathrm{\Phi }(\{i,j\})=-J_{ij}{Z}_i{Z}_j(i\mathrm{\ne }j),\mathrm{\Phi }(S)=0\text{ for all other finite nonempty }S\mathrm{.}$ Its per-site norm sum is finite by the displayed bound, and all its terms are diagonal and commute. For a finite $\mathrm{\Lambda }\subset \mathrm{\Gamma }$, its free-boundary Hamiltonian is $H_{\mathrm{\Lambda }}=-{\sum }_{i\in \mathrm{\Lambda }}{h}_i{Z}_i-{\sum }_{\{i,j\}\subseteq \mathrm{\Lambda }}{J}_{ij}{Z}_i{Z}_j,$ where every unordered pair is counted once.

For a classical configuration $\sigma \in X:=\{-1,+1{\}}^{\mathrm{\Gamma }}$, a finite region $\mathrm{\Delta }\subset \mathrm{\Gamma }$, and an exterior assignment $\eta \in \{-1,+1{\}}^{{\mathrm{\Delta }}^c}$, write ${\widehat{\sigma }}_i={\sigma }_i$ for $i\in \mathrm{\Delta }$ and ${\widehat{\sigma }}_i={\eta }_i$ otherwise. The classical conditional energy is $E_{\mathrm{\Delta }}({\sigma }_{\mathrm{\Delta }}\mid \eta )=-{\sum }_{i\in \mathrm{\Delta }}{h}_i{\sigma }_i-{\sum }_{\{i,j\}:\{i,j\}\cap \mathrm{\Delta }\mathrm{\ne }\mathrm{\varnothing }}{J}_{ij}{\widehat{\sigma }}_i{\widehat{\sigma }}_j\mathrm{.}$ This is the finite-range conditional-energy convention of Finite-range lattice interaction and conditional energy: each interaction term meeting $\mathrm{\Delta }$ is included once, including terms crossing its boundary. The sum is finite because $\mathrm{\Delta }$ is finite and the interaction range is finite. Define the finite exterior boundary $C_{\mathrm{\Delta }}:=\{j\mathrm{\notin }\mathrm{\Delta }:\text{there is }i\in \mathrm{\Delta }\text{ with }{J}_{ij}\mathrm{\ne }0\}\mathrm{.}$ It is finite, and $E_{\mathrm{\Delta }}({\sigma }_{\mathrm{\Delta }}\mid \eta )$ depends on $\eta$ only through $\eta {\mathrm{\mid }}_{C_{\mathrm{\Delta }}}$.

For $\beta >0$ in inverse-energy units, set $Z_{\mathrm{\Delta }}^{\eta }:={\sum }_{{\sigma }_{\mathrm{\Delta }}\in \{-1,+1{\}}^{\mathrm{\Delta }}}{e}^{-\beta E_{\mathrm{\Delta }}({\sigma }_{\mathrm{\Delta }}\mid \eta )},{\gamma }_{\mathrm{\Delta }}({\sigma }_{\mathrm{\Delta }}\mid \eta ):=\frac{e^{-\beta E_{\mathrm{\Delta }}({\sigma }_{\mathrm{\Delta }}\mid \eta )}}{Z_{\mathrm{\Delta }}^{\eta }}\mathrm{.}$ The kernel sends an exterior configuration to a probability distribution on the spins in $\mathrm{\Delta }$. A DLR measure for this model means a Borel probability measure $\mu$ on $X$ fixed by these kernels for every finite $\mathrm{\Delta }$, in the specification terminology of Lattice Gibbs kernels and DLR states. The finite kernel is defined without a choice principle; Countable Choice is stated separately when used to extend consistent cylinder probabilities to a measure.

The model is mathematical and supplies no measured coefficients. If interpreted with SI energy units, $J_{ij}$, $h_i$, $H_{\mathrm{\Lambda }}$, and $E_{\mathrm{\Delta }}$ are in joules, $\beta$ is in joules${}^{-1}$, and $\beta E_{\mathrm{\Delta }}$ is dimensionless.

The finite kernel requires no choice principle. Borel-measure extension from consistent cylinder probabilities uses The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).