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Communications in Mathematical Physics

On a class of equilibrium states under the Kubo-Martin-Schwinger condition

F. Rocca, M. Sirugue, D. Testard

Published Jun 1, 1970• 42 citations
Publisher
DOI
10.1007/bf01646630

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We investigate the extension of the Kubo--Martin--Schwinger (KMS) thermal equilibrium condition to bounded non-Hermitian Hamiltonians with real spectra and biorthogonal eigensystems, providing a unified framework through three complementary constructions: a complete KMS theorem under quasi-Hermiticity, a biorthogonal KMS-type identity whose positivity characterises quasi-Hermiticity, and a quantum-detailed-balance condition for the associated open-system dynamics. Our main result is a thermodynamic characterisation of quasi-Hermiticity: for any diagonalisable $H\in M_d(\mathbb C)$ with real spectrum, the biorthogonal Gibbs functional $ω_{\rm bi}(A)=Z_{\rm bi}^{-1}\sum_n e^{-βE_n}\langleφ_n|A|ψ_n\rangle$ satisfies $ω_{\rm bi}(A^\dagger A)\ge0$ for all $A$ if and only if $H$ is quasi-Hermitian. The proof reconstructs the metric $η$ directly from the eigenprojectors of $ω_{\rm bi}$ via the Riesz representation theorem, yielding a metric-free criterion for quasi-Hermiticity. Under the quasi-Hermitian hypothesis, we prove that the $η$-Gibbs state $ω_η(A)=Z_η^{-1}{\rm Tr}[ηe^{-βH}A]$ satisfies the full analytic KMS condition using the Hadamard three-line theorem and Bari's theorem on Riesz bases. The transported state generally differs from the Gibbs state of the isospectral Hermitian partner whenever $[η,h]\neq0$, so the KMS property cannot be obtained by similarity transformation alone. Finally, within the Haag--Hugenholtz--Winnink programme, we establish the Tomita--Takesaki modular structure of the $η$-Gibbs state in finite dimensions, while the construction of a compatible $C^*$-norm and the proof of $σ$-weak continuity remain open.

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On the approach to equilibrium in quantum systems

Pierre M. V. Résibois

Physica•Jun 1, 1961•92 citations
Abstract We show that the long-time limit of the two-point correlation function obtained via the standard quantum regression theorem (QRT), a standard tool to compute correlation functions in open quantum systems, does not respect the Kubo–Martin–Schwinger equilibrium condition to the non-zero order of the system-bath coupling. We then follow the recently developed Heisenberg operator method for open quantum systems and by applying a ‘weak’ Markov approximation, derive a new modified version of the QRT that not only respects the KMS condition but further predicts exact answers for certain paradigmatic models in specific limits. We also show that in cases where the modified QRT does not match with exact answers, it always performs better than the standard QRT.
We investigate the extension of the Kubo--Martin--Schwinger (KMS) thermal equilibrium condition to bounded non-Hermitian Hamiltonians with real spectra and biorthogonal eigensystems, providing a unified framework through three complementary constructions: a complete KMS theorem under quasi-Hermiticity, a biorthogonal KMS-type identity whose positivity characterises quasi-Hermiticity, and a quantum-detailed-balance condition for the associated open-system dynamics. Our main result is a thermodynamic characterisation of quasi-Hermiticity: for any diagonalisable H∈Md(C)H\in M_d(\mathbb C)H∈Md​(C) with real spectrum, the biorthogonal Gibbs functional ωbi(A)=Zbi−1∑ne−βEn⟨φn∣A∣ψn⟩ω_{\rm bi}(A)=Z_{\rm bi}^{-1}\sum_n e^{-βE_n}\langleφ_n|A|ψ_n\rangleωbi​(A)=Zbi−1​∑n​e−βEn​⟨φn​∣A∣ψn​⟩ satisfies ωbi(A†A)≥0ω_{\rm bi}(A^\dagger A)\ge0ωbi​(A†A)≥0 for all AAA if and only if HHH is quasi-Hermitian. The proof reconstructs the metric ηηη directly from the eigenprojectors of ωbiω_{\rm bi}ωbi​ via the Riesz representation theorem, yielding a metric-free criterion for quasi-Hermiticity. Under the quasi-Hermitian hypothesis, we prove that the ηηη-Gibbs state ωη(A)=Zη−1Tr[ηe−βHA]ω_η(A)=Z_η^{-1}{\rm Tr}[ηe^{-βH}A]ωη​(A)=Zη−1​Tr[ηe−βHA] satisfies the full analytic KMS condition using the Hadamard three-line theorem and Bari's theorem on Riesz bases. The transported state generally differs from the Gibbs state of the isospectral Hermitian partner whenever [η,h]≠0[η,h]\neq0[η,h]=0, so the KMS property cannot be obtained by similarity transformation alone. Finally, within the Haag--Hugenholtz--Winnink programme, we establish the Tomita--Takesaki modular structure of the ηηη-Gibbs state in finite dimensions, while the construction of a compatible C∗C^*C∗-norm and the proof of σσσ-weak continuity remain open.