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Quantum Ising KMS states have diagonal local densities

Statement

Assume Countable Choice and the finite-matrix C*-identity for the local spin algebras. Let $ω$ be a $β$ -KMS state, with $β > 0$ , for the thermodynamic dynamics of Diagonal quantum Ising interaction and classical conditional energies in the full quasi-local matrix algebra. For every finite region $Λ$ and any two distinct product-basis configurations $σ, τ \in {- 1, + 1}^{Λ}$ , the local matrix unit $E_{σ, τ}$ satisfies $ω (E_{σ, τ}) = 0$ . Consequently the density matrix representing $ω$ on each finite local matrix algebra is diagonal in the joint eigenbasis of the matrices $Z_{i}$ .

Facts & Assumptions

Given: Countable Choice, the finite-matrix C*-identity, the diagonal Ising dynamics, a $β$ -KMS state for it in the convention of Physical-time KMS strip equilibrium condition, and the closed-strip hypotheses below.

[A1]

Countable Choice is the axiom that every sequence of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )); it is inherited through the thermodynamic dynamics used here.

[A2]

Finite matrix C-star algebra and density-state duality supplies the C*-identity for each finite local matrix algebra and the representation of its positive functionals by density matrices, under its exact finite-dimensional hypotheses.

[F1]

Exact local dynamics for commuting diagonal interactions states that every local diagonal projection is fixed by $α_{t}$ .

[F2]

Physical-time KMS strip equilibrium condition says the KMS correlation for any pair is bounded and continuous on the closed strip, holomorphic inside, and has lower and upper boundaries $ω (A α_{t} (B))$ and $ω (α_{t} (B) A)$ .

[F3]

Bounded KMS strip norm and zero-edge uniqueness says two bounded continuous functions holomorphic inside the same strip with the same lower boundary agree on the whole strip.

Proof

technique · Compare a KMS correlation with the constant function determined by a fixed diagonal projection

Given: [F1]–[F3], $ω$ , and the model's finite local matrix algebras.

1.1A1A2F1F2F3given

Let $P$ be any local rank-one diagonal projection and $A$ any local observable. By [F1], $α_{t} (P) = P$ , so the lower boundary of the KMS function for the pair $(A, P)$ is the constant $ω (A P)$ . The constant function with that value is bounded, continuous, holomorphic, and has the same lower boundary. By [F3] it equals the KMS function. Evaluating the upper boundary in [F2] at $t = 0$ gives $ω (P A) = ω (A P)$ .

2.1A2step 1.1algebra∎

Fix a finite $Λ$ and a product-basis matrix unit $E_{σ, τ}$ with $σ \neq τ$ . Take $P = P_{τ}$ , the diagonal rank-one projection onto the basis vector $e_{τ}$ . Matrix-unit multiplication gives $E_{σ, τ} P_{τ} = E_{σ, τ}$ and $P_{τ} E_{σ, τ} = 0$ . Applying step 1.1 with $A = E_{σ, τ}$ yields $ω (E_{σ, τ}) = 0$ . By [A2], the restricted positive functional has a unique density matrix whose off-diagonal entries are precisely these matrix-unit expectations, so the finite local density matrix is diagonal.

Quantum Ising KMS states have diagonal local densities

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources