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Unique onsite product Gibbs KMS state

Statement

Assume Countable Choice ${A C}_{ω}$ and the exact finite-matrix C*-identity and positive-functional/density interfaces in [A2] below. Let $Γ = Z^{d}$ , $d \geq 1$ , and let $q \geq 1$ be a finite on-site dimension as in Local spin matrix algebras and their factor inclusions. Let $h = h^{*}$ be a $q \times q$ on-site Hamiltonian, with energy unit J, and set $Z : = Tr (e^{- β h}), r : = Z^{- 1} e^{- β h},$ where $h$ has unit J and $β > 0$ has unit J $^{- 1}$ . Let $ℏ > 0$ have unit J s. The consistent local densities $r^{\otimes Λ}$ have a unique state extension $ω_{r}$ to the norm-completed quasi-local algebra $A$ by Consistent local densities extend to an algebraic state, with local restrictions $ω_{r} (A) = {Tr}_{Λ} (r^{\otimes Λ} A) (A \in A_{Λ}) .$ Define the onsite interaction by $Φ ({x}) = h_{x}$ , the copy of $h$ at site $x$ , and $Φ (Z) = 0$ for all other finite nonempty $Z$ . Let $α_{t}$ be its thermodynamic dynamics under Strongly continuous thermodynamic spin automorphism group. Then $ω_{r}$ is the unique $β$ -KMS state for $α$ in the physical-time strip convention of Physical-time KMS strip equilibrium condition, whose width is $β ℏ$ seconds. The conclusion is mathematical; it does not assert relaxation or select a physical preparation.

Facts & Assumptions

Given: The finite spin system, ${A C}_{ω}$ , the exact finite-matrix interfaces [A2], $h = h^{*}$ , $β > 0$ , and $ℏ > 0$ .

[A1]

Countable Choice ${A C}_{ω}$ is the stated assumption of the quasi-local completion and of consistent-state extension (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Norm-completed quasi-local quantum spin algebra, Consistent local densities extend to an algebraic state).

[A2]

Assume the exact finite-matrix C*-identity and density/state norm interface supplied by Finite matrix C-star algebra and density-state duality. Its current proof supplies these finite-dimensional clauses and has an owner-recorded Step-1 ready record; that authoring-readiness record is not independent proof acceptance. The proof below is conditional on the stated interface.

[F1]

Local spin matrix algebras and their factor inclusions: for every finite $Λ \subset Γ$ , $A_{Λ}$ is the matrix algebra on the finite tensor product of the $q$ -dimensional site spaces; nested regions use tensor-with-identity inclusions.

[F2]

Norm-completed quasi-local quantum spin algebra and Quasi-local algebra completion and locality: $A_{l o c} = ⋃_{Λ} A_{Λ}$ is norm dense in its norm completion $A$ ; the conditional C*-structure uses [A1]–[A2].

[F3]

Uniformly bounded finite-range quantum spin interaction: $Φ$ consists of finite-region self-adjoint matrices with finite range and bounded per-site norm; $Φ ({x}) = h_{x}$ , $Φ (Z) = 0$ otherwise, is such an interaction, and its finite-volume Hamiltonian is $H_{Λ} = \sum_{x \in Λ} h_{x}$ .

[F4]

Strongly continuous thermodynamic spin automorphism group: under Countable Choice and the finite-matrix C*-identity condition, the finite-volume onsite dynamics have a thermodynamic limit that is a strongly continuous group of isometric *-automorphisms. Its current proof retains that finite-matrix condition explicitly.

[F5]

Consistent local densities extend to an algebraic state: positive trace-one densities on all finite regions that agree under partial traces extend uniquely to a bounded normalized positive functional on $A$ , under its stated Countable Choice and finite-matrix density/norm hypotheses.

[F6]

Physical-time KMS strip equilibrium condition: for each $A, B \in A$ , a $β$ -KMS state has a bounded function continuous on $S_{β ℏ} = {z : 0 \leq Im z \leq β ℏ}$ , holomorphic inside, with lower edge $ω (A α_{t} (B))$ and upper edge $ω (α_{t} (B) A)$ .

[F7]

Uniform local Gibbs strip limits: for any pointwise-convergent sequence of the specified finite-volume Gibbs state extensions, each fixed pair of local observables has a bounded continuous strip function, holomorphic inside, with the two limiting KMS boundary values. This is a local-pair statement only. Its proof explicitly assumes the finite-matrix interfaces in [A2]; the current supplier and its transitive inputs have current authoring-readiness records documented in batch-43 dependency evidence, not independent proof acceptance.

[F8]

Bounded KMS strip norm and zero-edge uniqueness: bounded continuous functions holomorphic inside the same strip obey the three-lines boundary norm bound; equality on the lower edge determines the full function, including the upper edge.

[F9]

Finite KMS condition exactly determines the Gibbs density: on a finite-dimensional matrix algebra with self-adjoint Hamiltonian $H$ , physical-time dynamics $e^{i t H / ℏ} (\cdot) e^{- i t H / ℏ}$ has exactly the Gibbs state $A \mapsto Tr (e^{- β H} A) / Tr (e^{- β H})$ as its $β$ -KMS state. Its proof uses the finite Hermitian spectral-calculus clause [F11] and the finite-matrix interface [A2].

[F10]

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

[F11]

Finite matrix spectral and state calculus: every finite-dimensional Hermitian matrix has an orthonormal eigenbasis with real eigenvalues, and the finite matrix trace is cyclic and basis-independent. Its current proof correctly handles normalized rational-coordinate approximants; the owner-recorded Step-1 ready record is authoring readiness only. The finite C*-identity and converse density/state correspondence are supplied separately by [A2].

Proof

technique · construct the consistent product state, apply the inspected local Gibbs-strip limit conditionally, extend the strip functions by norm density, and use finite-volume uniqueness on each local algebra

Given: [A1]–[A2], [F1]–[F11], $h$ , $β$ , $ℏ$ , and the onsite interaction $Φ$ .

1.1A1A2F1F5F11construct

The finite Hermitian spectral calculus [F11] diagonalizes $h$ with real eigenvalues; exponentiating them gives $Z > 0$ and a positive trace-one matrix $r$ . For nested finite regions $Λ \subseteq Λ^{'}$ , the product-basis trace factorization and $Tr r = 1$ give ${Tr}_{Λ^{'} ∖ Λ} (r^{\otimes Λ^{'}}) = r^{\otimes Λ}$ . Thus the family is consistent, and [F5] defines a unique state $ω_{r}$ with the displayed local restrictions.

2.1A1A2F1F2F3F5F11step 1.1algebra

Let $Λ_{n} = [- n, n]^{d} \cap Γ$ . In a finite-volume product eigenbasis, $H_{Λ_{n}} = \sum_{x \in Λ_{n}} h_{x}$ has eigenvalues equal to sums of the corresponding one-site eigenvalues by [F11]. Therefore $e^{- β H_{Λ_{n}}} = (e^{- β h})^{\otimes Λ_{n}}$ and its partition function is $Z^{∣ Λ_{n} ∣}$ , so its normalized Gibbs density is $r^{\otimes Λ_{n}}$ . For every finite $X \subseteq Γ$ , define $ρ_{X}^{(n)} : = {Tr}_{(X \cup Λ_{n}) ∖ X} ⁣ (r^{\otimes Λ_{n}} \otimes (q^{- 1} I)^{\otimes (X ∖ Λ_{n})}),$ placing the tensor factors on $Λ_{n}$ and $X ∖ Λ_{n}$ respectively. These densities are consistent: removing a site outside $Λ_{n}$ traces a factor $q^{- 1} I$ to $1$ , while tracing sites in $Λ_{n}$ commutes with the displayed partial trace because both are finite tensor-basis sums. Therefore [F5] gives an algebraic state $ω_{n}$ on $A$ . If $A$ is local in $A_{X}$ , then for all sufficiently large $n$ , $X \subseteq Λ_{n}$ and $ω_{n} (A) = ω_{r} (A)$ . For arbitrary $A \in A$ , choose local $C$ with $∥ A - C ∥ < ε / 4$ using [F2]; since both states have norm one, $∣ ω_{n} (A) - ω_{r} (A) ∣ \leq 2 ∥ A - C ∥ + ∣ ω_{n} (C) - ω_{r} (C) ∣ < ε$ for all sufficiently large $n$ . Hence $ω_{n} \to ω_{r}$ pointwise on $A$ .

3.1A1A2F3F6F7step 2.1

Applying the local-pair conclusion of [F7] to this particular pointwise-convergent finite Gibbs sequence gives, for every local $C, D$ , a bounded continuous strip function $F_{C, D}$ , holomorphic inside, with lower and upper edges $ω_{r} (C α_{t} (D))$ and $ω_{r} (α_{t} (D) C)$ . This is precisely the local conclusion of [F7]; no whole-algebra KMS conclusion is attributed to that supplier.

4.1F2F4F6F8F10step 3.1algebra

Let $A, B \in A$ and choose local sequences $A_{n} \to A$ , $B_{n} \to B$ . On either boundary, the correlation difference for local pairs is bounded by $∥ A_{n} - A_{m} ∥ ∥ B_{n} ∥ + ∥ A_{m} ∥ ∥ B_{n} - B_{m} ∥,$ because $ω_{r}$ has norm one and $α_{t}$ is isometric by [F4]. Decomposing the difference into the two local-pair differences $(A_{n} - A_{m}, B_{n})$ and $(A_{m}, B_{n} - B_{m})$ , [F8] gives the same bound throughout the strip; its zero-edge uniqueness makes this decomposition equal to $F_{A_{n}, B_{n}} - F_{A_{m}, B_{m}}$ . The right side tends to zero, since the convergent sequences have bounded norms. Thus $F_{A_{n}, B_{n}}$ is uniformly Cauchy on the closed strip. Its limit $F_{A, B}$ is bounded and continuous there and is holomorphic inside by [F10]. Norm continuity of $ω_{r}$ , $α_{t}$ , and multiplication gives the required lower and upper boundary values. Therefore $ω_{r}$ is KMS for all $A, B \in A$ .

5.1A2F1F2F4F6F9step 2.1step 4.1∎

Let $ν$ be any $β$ -KMS state on $A$ . For a finite region $Λ$ , every larger finite-volume Hamiltonian has the form $H_{Λ^{'}} = H_{Λ} + K_{Λ^{'} ∖ Λ}$ , where $K_{Λ^{'} ∖ Λ}$ is a sum of onsite terms supported outside $Λ$ and therefore commutes with every $A \in A_{Λ}$ . Hence $α_{t}^{Λ^{'}} (A) = e^{i t H_{Λ} / ℏ} A e^{- i t H_{Λ} / ℏ}$ whenever $Λ^{'} \supseteq Λ$ . Taking the thermodynamic limit in [F4] shows that $A_{Λ}$ is invariant under $α_{t}$ and its restriction is exactly this finite-dimensional conjugation group. Restricting the KMS functions to $A, B \in A_{Λ}$ shows that $ν ∣_{A_{Λ}}$ is a finite-dimensional $β$ -KMS state. By [F9], its density is $e^{- β H_{Λ}} / Z^{∣ Λ ∣} = r^{\otimes Λ}$ . Thus $ν$ and $ω_{r}$ agree on every finite local algebra; continuity and density of $A_{l o c}$ imply $ν = ω_{r}$ . This proves uniqueness.

Remarks

The current finite-matrix and KMS suppliers are used at their inspected conditional interfaces. The proof does not infer that a finite Gibbs trace is an infinite-volume density matrix; it constructs an algebraic product state and proves the full KMS condition by local-pair strip limits and norm approximation.

Unique onsite product Gibbs KMS state

Statement

Facts & Assumptions

Proof

Remarks

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