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High-temperature full quantum Ising KMS uniqueness

Statement

Assume Countable Choice, the finite-matrix C*-identity, $d \geq 1$ , a translation-invariant symmetric pair coupling $J_{i + k, j + k} = J_{i j}$ with $J_{i j} = 0$ for $∥ i - j ∥_{\infty} > R$ for some finite $R$ , a constant finite field $h_{i} = h \in R$ , $β > 0$ , and a fixed physical-time scale $ℏ > 0$ , in the diagonal model of Diagonal quantum Ising interaction and classical conditional energies. Use the thermodynamic dynamics with finite-volume convention $e^{i t H_{Λ} / ℏ} (\cdot) e^{- i t H_{Λ} / ℏ}$ and the $β$ -KMS strip of width $β ℏ$ from Full-algebra quantum Ising KMS–DLR correspondence. If $\sup_{i \in Z^{d}} \sum_{j \neq i} \tanh (β ∣ J_{i j} ∣) < 1,$ then the full quasi-local matrix algebra has exactly one $β$ -KMS state. This strict influence bound is sufficient, not sharp. For homogeneous nearest-neighbor coupling $J$ , it reduces to $2 d \tanh (β ∣ J ∣) < 1$ . The result is for the longitudinal diagonal interaction and makes no transverse-field claim.

Facts & Assumptions

Given: Countable Choice, the finite-matrix C*-identity, the translation-invariant finite-range Ising pair interaction, constant field, $β > 0$ , and $ℏ > 0$ , with physical-time dynamics and KMS strip width $β ℏ$ .

[A1]

Countable Choice is assumed in the CSM DLR existence and extension framework (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[A2]

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

[F1]

Unique interacting Ising state at sufficient high temperature states that for this same pair-energy convention and strict bound $\sup_{i} \sum_{j} \tanh (β ∣ J_{i j} ∣) < 1$ , exactly one DLR state exists. Its current batch-27 carrier is draft and authoring-ready; no independent acceptance is inferred.

[F2]

Full-algebra quantum Ising KMS–DLR correspondence gives the affine bijection between DLR measures and full-algebra $β$ -KMS states under the exact diagonal finite-range assumptions, Countable Choice, [A2], $β > 0$ , and $ℏ > 0$ , using physical-time dynamics and strip width $β ℏ$ . Its current proof remains conditional on the draft suppliers recorded there.

[F3]

Diagonal quantum Ising interaction and classical conditional energies uses one term per unordered pair and the same constant field energy as the CSM high-temperature statement in [F1].

Proof

technique · Apply the classical sufficient uniqueness condition and transfer its unique DLR state through the bijection

Given: [A1]–[A2], [F1]–[F3], the stated coefficients, $β > 0$ , and $ℏ > 0$ with the physical-time convention in [F2].

1.1givenF1F2F3algebra

The energy formula in [F3] agrees term by term with the pair Hamiltonian in [F1], including the unordered-pair convention, finite range, translation symmetry, constant field, and the zero transverse component. Hence the DLR specifications addressed by [F1] and [F2] coincide.

2.1A1A2F1F2step 1.1algebra∎

By [F1], the common DLR set contains exactly one measure. The affine bijection [F2] transfers this singleton to exactly one full-algebra $β$ -KMS state. In the homogeneous nearest-neighbor case there are $2 d$ nonzero neighboring couplings at each site, so the displayed row sum is $2 d \tanh (β ∣ J ∣)$ ; the strict bound remains only a sufficient criterion.

High-temperature full quantum Ising KMS uniqueness

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources