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Unique one-dimensional Ising KMS state at every positive temperature

Statement

Assume Countable Choice, the finite-matrix C*-identity, $d = 1$ , finite $β > 0$ , $ℏ > 0$ , a uniform field $h_{i} = h \in R$ , and nearest-neighbor couplings $J_{i, i + 1} = J_{i + 1, i} = J \in R$ with all other pair couplings zero, in the diagonal model of Diagonal quantum Ising interaction and classical conditional energies. Then the full quasi-local matrix algebra has exactly one $β$ -KMS state. The conclusion concerns the longitudinal diagonal Ising chain; no transverse-field term is included.

Facts & Assumptions

Given: Countable Choice, the finite-matrix C*-identity, the one-dimensional nearest-neighbor Ising model, finite $J, h$ , $β > 0$ , and $ℏ > 0$ .

[A1]

Countable Choice is assumed for the CSM cylinder-measure extension and DLR uniqueness suppliers (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]

All-positive-temperature one-dimensional Ising DLR uniqueness states that for this nearest-neighbor chain and every finite $β > 0$ , $J, h \in R$ , there is exactly one DLR state under Countable Choice. Its current batch-27 carrier is draft and authoring-ready; this is not independent acceptance.

[F2]

Full-algebra quantum Ising KMS–DLR correspondence gives an affine bijection between DLR measures and full-algebra $β$ -KMS states for the same diagonal model, under Countable Choice, [A2], and its stated finite-range assumptions. Its current batch-42 proof is conditional on the exact draft suppliers listed in its Facts & Assumptions.

Proof

technique · Transfer the exact classical DLR uniqueness result through the full-algebra KMS–DLR bijection

Given: [A1], [F1], [F2], and the stated one-dimensional Ising parameters.

1.1givenA1F1F2algebra

Under these parameters, the conditional energy in Diagonal quantum Ising interaction and classical conditional energies is the one-dimensional nearest-neighbor Ising energy used by [F1]: each unordered edge is counted once and the same constant field term is included at every site. Thus the DLR sets in [F1] and [F2] are identical.

2.1A1A2F1F2step 1.1∎

By [F1], that DLR set has exactly one measure. The bijection [F2] is both injective and surjective, so it sends this singleton to exactly one full-algebra $β$ -KMS state. This is a model deduction conditional on [A1]–[A2], [F1], and [F2].

Remarks

The finite-volume transfer-matrix result and the infinite-volume uniqueness conclusion are distinct: the latter follows from the CSM DLR uniqueness theorem plus the proved quantum correspondence. The conclusion does not describe a transverse-field chain or an empirical material.

Unique one-dimensional Ising KMS state at every positive temperature

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources