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Two-dimensional diagonal quantum Ising KMS coexistence

Statement

Assume Countable Choice, the finite-matrix C*-identity, $d = 2$ , uniform nearest-neighbor coupling $J > 0$ , zero field, $β J \geq 2$ , and $ℏ > 0$ , in the diagonal spin- $\frac{1}{2}$ model of Diagonal quantum Ising interaction and classical conditional energies. Let $τ_{k}$ be the spatial translation automorphisms of Translation-invariant finite-range spin interaction; a state is translation-invariant when $ω (τ_{k} (A)) = ω (A)$ for every local $A$ and $k \in Z^{2}$ . Then there are two distinct translation-invariant full-algebra $β$ -KMS states $ω_{+}$ and $ω_{-}$ with $ω_{+} (Z_{i}) \geq m_{*} > 0, ω_{-} (Z_{i}) \leq - m_{*} < 0 (i \in Z^{2}),$ where one may take $m_{*} = 1 - 2 δ (β J)$ , $δ (K) = \frac{4}{3} \frac{r^{4} (5 - 4 r)}{(1 - r)^{2}}$ , and $r = 3 e^{- 2 K}$ . This conservative bound does not assert extremality or that $β J = 2$ is critical.

For the same uniform model with field parameter $h_{i} = h$ , let $p_{Q} (h) : = \lim_{n \to \infty} \frac{1}{∣ Λ_{n} ∣} \log {Tr}_{Λ_{n}} ⁣ (e^{- β H_{Λ_{n}} (h)}), Λ_{n} = [- n, n]^{2},$ the dimensionless pressure per site along cubic boxes. This limit is the corresponding classical CSM pressure. It is not differentiable at $h = 0$ ; for every $h > 0$ its secant slope from zero is at least $β m_{*}$ , and for every $h < 0$ it is at most $- β m_{*}$ . These are exact consequences of the stated diagonal model and the CSM suppliers, not empirical material claims.

Facts & Assumptions

Given: Countable Choice, the finite-matrix C*-identity, the two-dimensional nearest-neighbor Ising model, $J > 0$ , zero field for coexistence, $β J \geq 2$ , $ℏ > 0$ , and the finite-range pressure and KMS–DLR suppliers.

[A1]

Countable Choice is the exact assumption of the CSM DLR construction (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[A2]

Finite matrix C-star algebra and density-state duality supplies the local finite-matrix C*-identity used by the QSM KMS dynamics. Its current batch-36 carrier is draft and is used conditionally.

[F1]

Two-dimensional Ising Gibbs coexistence at low temperature states that for this two-dimensional model and $β J \geq 2$ there are translation-stationary DLR measures with one-site magnetizations at least $m_{*} > 0$ and at most $- m_{*}$ . Its current batch-27 carrier is draft and authoring-ready; independent proof acceptance is not asserted.

[F2]

Full-algebra quantum Ising KMS–DLR correspondence is the affine bijection for the same diagonal model, including its full matrix algebra, under Countable Choice and [A2]. Its current proof is conditional on the inspected draft dependencies in its own Facts & Assumptions.

[F3]

Diagonal quantum Ising interaction and classical conditional energies defines $H_{Λ} (h)$ as the free-boundary diagonal matrix with one term per internal unordered pair and one field term per site.

[F4]

Local spin matrix algebras and their factor inclusions defines the finite spin Hilbert space, local matrix algebra, product basis, and ordinary finite trace used in $p_{Q} (h)$ .

[F5]

Finite-range lattice interaction and conditional energy uses the free-boundary energy from translated finite shapes, each translated term counted once.

[F6]

Bounded lattice pressure on van Hove sets states that the free-boundary dimensionless pressure along cubes converges to the finite-range CSM pressure per site.

[F7]

Coexistence implies an Ising pressure cusp states that for the CSM model at $β J \geq 2$ , the pressure $p (h)$ is nondifferentiable at zero and has positive/negative secant slopes bounded below by $β m_{*}$ and above by $- β m_{*}$ . Its current batch-27 carrier is draft and authoring-ready.

[F8]

Translation-invariant finite-range spin interaction defines the spatial translation $τ_{k}$ on local operators by unitary relabeling of product-basis vectors, proves it is independent of localization and isometric, and extends it to an automorphism of the quasi-local completion. Its current batch-39 carrier is used at that exact interface.

[F9]

Finite-range inhomogeneous Ising kernels form a specification states that in the translation-invariant pair-coupling subcase, one singleton field shape and one pair shape for each displacement pair ${v, - v}$ reproduce the same conditional and finite-volume energies under the CSM translated-shape convention. Its current batch-42 draft proof is used conditionally.

Proof

technique · Transfer the CSM coexistence measures through the KMS–DLR bijection and identify the diagonal quantum trace with the classical partition sum

Given: [A1]–[A2], [F1]–[F9], $J > 0$ , $h = 0$ , $β J \geq 2$ , $ℏ > 0$ , and the stated finite-range conventions.

1.1A1A2F1F2F8givenalgebra

Let $μ_{+}, μ_{-}$ be the translation-stationary DLR measures from [F1]. By the bijection [F2], each has a unique associated full-algebra $β$ -KMS state $ω_{μ_{\pm}}$ . Since $Z_{i} e_{σ} = σ_{i} e_{σ}$ , the diagonal lift gives $ω_{μ_{\pm}} (Z_{i}) = \int σ_{i} d μ_{\pm} = μ_{\pm} (σ_{i})$ , so the magnetization bounds hold and the states are distinct. For local $A \in A_{Λ}$ , the basis relabeling in [F8] gives $diag (τ_{k} (A)) (σ) = diag (A) (T_{k} σ)$ , where $(T_{k} σ)_{i} = σ_{i + k}$ . Stationarity means $(T_{k})_{*} μ_{\pm} = μ_{\pm}$ , so the lift formula gives $ω_{μ_{\pm}} (τ_{k} (A)) = ω_{μ_{\pm}} (A)$ . Density of local observables and continuity of both the state and $τ_{k}$ extend the equality to all $A \in A$ .

1.2A2F3F4F5F6F9algebra

For a uniform field parameter $h$ , the finite matrix $H_{Λ_{n}} (h)$ is diagonal in the joint $Z_{i}$ eigenbasis. Its eigenvalue on each spin configuration $σ_{Λ_{n}}$ is exactly the CSM free-boundary energy $H_{Λ_{n}}^{f r e e} (σ_{Λ_{n}})$ : [F9] identifies the translation-invariant pair energy with one CSM shape per unordered displacement pair, and [F3] counts each internal pair and site field once, matching [F5]. The ordinary trace is the sum over this product basis by [F4]. Therefore ${Tr}_{Λ_{n}} (e^{- β H_{Λ_{n}} (h)}) = \sum_{σ_{Λ_{n}} \in {- 1, + 1}^{Λ_{n}}} e^{- β H_{Λ_{n}}^{f r e e} (σ_{Λ_{n}})}$ for every $n$ and real $h$ . By [F6], the limit defining $p_{Q} (h)$ exists and equals the CSM pressure $p (h)$ .

2.1A1A2F1F7step 1.2algebra∎

Applying [F7] to the pressure equality from step 1.2 gives the asserted secant-slope bounds and nondifferentiability at $h = 0$ . The coexistence threshold remains the sufficient $β J \geq 2$ from [F1]; no critical-point or extremality claim is used.

Remarks

The infinite-volume phase conclusion depends on the full quantum-to-classical DLR correspondence, not on uniqueness of each finite Gibbs matrix. The proof is conditional on the current draft CSM and QSM carriers listed in Facts & Assumptions. It gives no measured transition temperature, material identification, or experimental evidence.

Two-dimensional diagonal quantum Ising KMS coexistence

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources