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Finite Gibbs uniqueness does not imply infinite KMS uniqueness

Statement refuted

For every bounded finite-range quantum spin model, uniqueness of the finite-volume Gibbs state for each finite region implies uniqueness of the infinite-volume KMS state.

Facts & Assumptions

Given: The two-dimensional diagonal nearest-neighbor quantum Ising model with $J > 0$ , zero field, $β J \geq 2$ , $ℏ > 0$ , Countable Choice, the finite-matrix C*-identity, and its finite- and infinite-volume equilibrium suppliers.

[F1]

Diagonal quantum Ising interaction and classical conditional energies defines the finite-volume Hamiltonians as finite self-adjoint matrices on the tensor-product spin spaces and the infinite quasi-local model; Local spin matrix algebras and their factor inclusions defines the finite spin Hilbert spaces, local matrix algebras, product bases, and ordinary finite trace.

[A1]

Countable Choice and the finite-matrix C*-identity are inherited as exact hypotheses from the infinite-volume theorem (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Finite matrix C-star algebra and density-state duality).

[A2]

Finite matrix spectral and state calculus supplies the finite Hermitian spectral calculus and trace cyclicity required by the finite KMS characterization; this current batch-36 result is used conditionally.

[F2]

Finite KMS condition exactly determines the Gibbs density states that under [A1]–[A2], on every nonzero finite-dimensional Hilbert space and for every finite self-adjoint Hamiltonian, the normalized Gibbs trace is the unique $β$ -KMS state.

[F3]

Two-dimensional diagonal quantum Ising KMS coexistence proves that for the stipulated infinite diagonal Ising model, under Countable Choice and the finite-matrix C*-identity, there are at least two distinct full-algebra $β$ -KMS states.

Counterexample

Given: [A1], [F1]–[F3], and the parameter conditions above.

1.1A1A2F1F2given

Fix any finite $Λ \subset Z^{2}$ . Its Hilbert space $H_{Λ} = (C^{2})^{\otimes ∣ Λ ∣}$ is nonzero and finite-dimensional, and $H_{Λ} = H_{Λ}^{*}$ by [F1]. The finite characterization [F2] gives exactly one finite-volume $β$ -KMS state, namely $ρ_{Λ} = e^{- β H_{Λ}} / Tr (e^{- β H_{Λ}})$ . This holds for every finite $Λ$ , regardless of finite-volume energy degeneracies.

2.1A1F3step 1.1∎

The same coefficients and $β J \geq 2$ satisfy the hypotheses of [F3], so the infinite quasi-local system has at least two distinct $β$ -KMS states. Together, the finite-volume conclusion and this infinite-volume conclusion provide a counterexample to the refuted implication: every finite-volume Gibbs state is unique, while the infinite-volume KMS state is not unique. The distinction is between a fixed finite matrix Hamiltonian and the thermodynamic-limit KMS problem.

Remarks

This is an exact mathematical counterexample for the stated diagonal interaction. It does not say that finite-volume Gibbs states are ill-defined, nor does it identify a real material or report an observation.

Finite Gibbs uniqueness does not imply infinite KMS uniqueness

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

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