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Finite-range interacting quantum spin KMS existence

Statement

Let $Γ = Z^{d}$ , $d \geq 1$ , have finite on-site spin dimension $q \geq 1$ and quasi-local spin algebra $A$ . Let $Φ$ be a uniformly bounded finite-range interaction with free-boundary finite-volume Hamiltonians, and let $α_{t}$ be the strongly continuous thermodynamic dynamics of Strongly continuous thermodynamic spin automorphism group. Fix $β > 0$ and $ℏ > 0$ . Assume Countable Choice, the finite-matrix C $^{*}$ -identity, and the finite-matrix state/density-norm interface exactly as stated in Facts [A1]–[A2].

For any increasing finite exhaustion $Λ_{n} ↑ Γ$ , extend each finite Gibbs density on $A_{Λ_{n}}$ by normalized trace densities on the outside sites, as in Uniform local Gibbs strip limits. Every pointwise convergent subsequence of these algebraic states has a limit $ω$ that is a $β$ -KMS state for $α$ in physical time, with strip width $β ℏ$ . At least one such subsequential limit exists under [A1]–[A2].

The conclusion is a conditional mathematical existence result for bounded finite-dimensional spin interactions. It does not assert uniqueness, clustering, phase classification, a global infinite-volume Gibbs density matrix, or dynamical relaxation to equilibrium.

Facts & Assumptions

Given: $d \geq 1$ , finite $q \geq 1$ , $Γ = Z^{d}$ , $β > 0$ , $ℏ > 0$ , a uniformly bounded finite-range interaction $Φ$ , and the free-boundary thermodynamic dynamics.

[A1]

Countable Choice and the finite-matrix C $^{*}$ -identity $∥ T^{*} T ∥ = ∥ T ∥^{2}$ are the exact assumptions of Strongly continuous thermodynamic spin automorphism group. The current Finite matrix C-star algebra and density-state duality carrier proves that identity for every nonzero finite local matrix algebra, but remains a draft pending independent review; the dynamics supplier still states the identity as a hypothesis. This theorem therefore records the exact conditional input and does not claim independent acceptance of either draft.

[A2]

The finite-matrix state/density correspondence and norm-one trace-functional clauses in Finite matrix C-star algebra and density-state duality are assumed exactly as used by Consistent local densities extend to an algebraic state and Sequential compactness of the quasi-local spin state space. They are present in a complete current draft argument, but this theorem carries them as explicit conditional inputs and makes no independent acceptance claim.

[F1]

Infinite algebraic state and finite-region restrictions: an algebraic state is a bounded normalized positive functional on the quasi-local algebra; it is not assumed to be represented by a global trace-class density.

[F2]

Norm-completed quasi-local quantum spin algebra and Quasi-local algebra completion and locality: the strictly local algebra is norm dense in $A$ ; the completion proof uses the finite-matrix C $^{*}$ -identity of [A1] through Finite matrix C-star algebra and density-state duality.

[F3]

Strongly continuous thermodynamic spin automorphism group: under [A1], $α_{t}$ is a strongly continuous group of isometric $*$ -automorphisms and finite-volume dynamics converge in norm on local observables uniformly on compact real-time intervals.

[F4]

Uniform local Gibbs strip limits: under [A1]–[A2], the product-extended finite Gibbs states have pointwise convergent subsequences, and every such limit has bounded analytic strip functions with the required lower and upper boundaries for each local pair.

[F5]

Bounded KMS strip norm and zero-edge uniqueness: bounded strip functions with boundary suprema at most $C$ have the same bound throughout, and their locally uniform limits are unique when their lower boundary values are fixed.

[F6]

Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly: a locally uniform limit of holomorphic functions on an open set is holomorphic.

[F7]

Finite unitary dynamics and Gibbs KMS identity: each finite Gibbs density has the KMS strip identity for its finite-volume Heisenberg group. Its current Step 1 record is ready against the current Finite matrix spectral and state calculus supplier; it is a finite-volume identity and supplies no infinite-volume result.

Proof

technique · extend the local-pair strip functions by norm approximation and the three-lines bound

Given: [A1]–[A2], [F1]–[F7], the finite-range interaction, and a pointwise limit $ω$ supplied by [F4].

1.1F2construct

The local algebra $A_{l o c}$ is norm dense in $A$ by [F2]. Fix arbitrary $A, B \in A$ . Choose sequences $A_{j}, B_{j} \in A_{l o c}$ with $A_{j} \to A$ and $B_{j} \to B$ in norm. They are bounded sequences, so set $M_{A} : = \sup_{j} ∥ A_{j} ∥ < \infty$ and $M_{B} : = \sup_{j} ∥ B_{j} ∥ < \infty$ .

2.1A1A2F3F4F5F7step 1.1algebra

For each $j$ , [F4] gives a bounded continuous strip function $F_{j}$ holomorphic in $S_{b}^{\circ}$ , $b = β ℏ$ , with lower and upper boundaries $ω (A_{j} α_{t} (B_{j}))$ and $ω (α_{t} (B_{j}) A_{j})$ ; the finite-volume Gibbs boundary input used by [F4] is the exact identity in [F7]. On both boundaries, the state norm and the isometry of $α_{t}$ from [A2]–[F3] give modulus at most $∥ A_{j} ∥ ∥ B_{j} ∥$ . Since $F_{j}$ is already bounded by [F4], applying [F5] to $F_{j}$ gives $∣ F_{j} (z) ∣ \leq ∥ A_{j} ∥ ∥ B_{j} ∥ \leq M_{A} M_{B}$ throughout $S_{b}$ . For $j, k$ , each boundary difference is bounded uniformly in real $t$ by $η_{j k} : = ∥ A_{j} - A_{k} ∥ M_{B} + M_{A} ∥ B_{j} - B_{k} ∥$ , because $∥ ω ∥ = 1$ by [A2] and each $α_{t}$ is isometric by [F3]. Applying the three-lines estimate of [F5] to the bounded function $F_{j} - F_{k}$ gives $\sup_{z \in S_{b}} ∣ F_{j} (z) - F_{k} (z) ∣ \leq η_{j k}$ . Since $(A_{j})$ and $(B_{j})$ are norm-Cauchy, $η_{j k} \to 0$ as $j, k \to \infty$ . Thus $(F_{j})$ is uniformly Cauchy on the whole closed strip.

3.1step 1.1step 2.1F6

The uniform limit $F_{A, B} : = \lim_{j} F_{j}$ exists on $S_{b}$ because $C$ is complete. It is bounded by $M_{A} M_{B}$ and continuous on $S_{b}$ as a uniform limit of bounded continuous functions. The convergence is locally uniform in the open strip, so [F6] makes $F_{A, B}$ holomorphic on $S_{b}^{\circ}$ .

4.1A2F3step 1.1step 3.1

For fixed real $t$ , the lower boundaries converge to $ω (A α_{t} (B))$ : indeed, $∣ ω (A_{j} α_{t} (B_{j})) - ω (A α_{t} (B)) ∣ \leq ∥ A_{j} - A ∥ ∥ B_{j} ∥ + ∥ A ∥ ∥ B_{j} - B ∥$ , using $∥ ω ∥ = 1$ and isometry of $α_{t}$ . The right-hand side tends to zero. The same calculation with the order reversed shows the upper boundaries converge to $ω (α_{t} (B) A)$ . Since $F_{j} \to F_{A, B}$ uniformly on $S_{b}$ , its two edge values are therefore the required KMS boundary values.

5.1F1F4step 3.1step 4.1∎

Steps 3.1 and 4.1 show that for every $A, B \in A$ the bounded continuous function $F_{A, B}$ is holomorphic inside and has the two boundaries in Physical-time KMS strip equilibrium condition. Thus $ω$ is a $β$ -KMS state for $α$ in physical time. Existence of at least one such subsequential $ω$ is supplied conditionally by [F4].

Remarks

The theorem uses finite Gibbs states only as an existence construction. It does not identify all KMS states with limits of a particular boundary condition, infer that all stationary states are KMS, or turn the mathematical criterion into an empirical claim about thermalization.

Finite-range interacting quantum spin KMS existence

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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Sources