Uniqueness of states satisfying Kubo-Martin-Schwinger boundary conditions in the case of one-dimensional quantum spin systems with finite-range potential
Abstract We discuss the regularized boundary state e−τ0HB⟩a e − τ 0 H | B a on two aspects in both 2D CFT and higher dimensional free field theory. One is its entanglement and correlation properties, which exhibit exponential decay in 2D CFT, the parameter 1/τ 0 works as a mass scale. The other concerns with its time evolution, i.e., e−itHe−τ0HB⟩a e − i t H e − τ 0 H | B a . We investigate the Kubo-Martin-Schwinger (KMS) condition on correlation function of local operators to detect the thermal properties. Interestingly we find the correlation functions in the initial state e−τ0HB⟩a e − τ 0 H | B a also partially satisfy the KMS condition. In the limit t → ∞, the correlators will exactly satisfy the KMS condition. We generally analyse quantum quench by a pure state and obtain some constraints on the possible form of 2-point correlation function in the initial state if assuming they satisfies KMS condition in the final state. As a byproduct we find in an large τ 0 limit the thermal property of 2-point function in e−τ0HB⟩a e − τ 0 H | B a also appears.
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) with real spectrum, the biorthogonal Gibbs functional ωbi(A)=Zbi−1∑ne−βEn⟨φn∣A∣ψn⟩ satisfies ωbi(A†A)≥0 for all A if and only if H is quasi-Hermitian. The proof reconstructs the metric η directly from the eigenprojectors of ωbi 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 ωη(A)=Zη−1Tr[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, 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.