Kubo-Martin-Schwinger conditions for non-Hermitian systems

Abstract

We investigate the extension of the Kubo--Martin--Schwinger (KMS) thermal equilibrium condition to non-Hermitian Hamiltonians with real spectra and biorthogonal eigensystems, providing a systematic analysis through three complementary routes. Our central result is a thermodynamic characterisation of quasi-Hermiticity: for $H \in M_d(\mathbb{C})$ diagonalisable 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 $ω_{\rm{bi}}(A^†A) \geq 0$ for all $A$ if and only if $H$ is quasi-Hermitian. The proof constructs the metric $η$ directly from the eigenprojectors of $ω_{\rm{bi}}$ via the Riesz representation theorem, with no prior choice of $η$, providing a metric-free certificate of quasi-Hermiticity outside the Mostafazadeh--Scholtz framework. Under the full quasi-Hermitian hypothesis, we prove that the $η$-Gibbs state $ω_η(A) = Z_η^{-1}\, \rm{Tr}[ηe^{-βH}A]$ satisfies all three analytic KMS conditions, using the Hadamard three-line theorem and Bari's theorem on Riesz bases. The result is non-trivial: the transported state $\hatω(X) = \rm{Tr}[e^{-βh}Xη]/Z_η$ differs from the Gibbs state of the isospectral Hermitian partner $h = η^{1/2}Hη^{-1/2}$ whenever $[η,h]\neq 0$, so the KMS property cannot be deduced from the Hermitian theory by similarity. The gap between this result and the full Haag--Hugenholtz--Winnink $C^*$-algebraic framework is identified. Failure modes at exceptional points and for complex spectra are analysed, and the relation to the Fagnola--Umanità quantum detailed balance condition for open systems is discussed.