Knowlpedia

KMS condition in finite quantum systems

An analytic boundary condition on equilibrium correlation functions at inverse temperature beta.

Let A=B(H)\mathcal A=\mathcal B(\mathcal H) for a finite-dimensional Hilbert space, let H=H∗H=H^*, and use units with ℏ=1\hbar=1. Write

τt(A)=eitHAe−itH.\tau_t(A)=e^{itH}Ae^{-itH}.

A state ω\omega satisfies the β\beta-KMS condition if, for every A,B∈AA,B\in\mathcal A, there is a function FA,BF_{A,B} continuous on 0≤Im⁡z≤β0\le\operatorname{Im}z\le\beta, analytic in its interior, and satisfying

FA,B(t)=ω(Aτt(B)),FA,B(t+iβ)=ω(τt(B)A)F_{A,B}(t)=\omega(A\tau_t(B)), \qquad F_{A,B}(t+i\beta)=\omega(\tau_t(B)A)

for every t∈Rt\in\mathbb R.

Finite-dimensional characterization

The

ωβ(A)=Tr⁡(e−βHA)Tr⁡(e−βH)\omega_\beta(A)=\frac{\operatorname{Tr}(e^{-\beta H}A)}{\operatorname{Tr}(e^{-\beta H})}

satisfies the β\beta-KMS condition. Conversely, on the full matrix algebra it is the unique β\beta-KMS state for τ\tau.

Remarks

Every KMS state is invariant under the dynamics. The condition is formulated in terms of the observable algebra and time evolution, so it extends to infinite systems where a trace-class Gibbs density operator need not exist.