Conditional quantum Ising uniqueness and phase predictions

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.