Home » Classical Statistical Mechanics

Gibbs–KMS theorem (finite quantum systems)

The finite-volume Gibbs state satisfies the KMS condition at inverse temperature β for the Heisenberg time evolution generated by the Hamiltonian.
Gibbs–KMS theorem (finite quantum systems)

Statement

Let H\mathcal{H} be a finite-dimensional Hilbert space and let H=H∗H=H^\ast be a Hamiltonian. Consider the algebra of observables A=B(H)\mathcal{A}=B(\mathcal{H}) and the Heisenberg time evolution

αt(A):=eitH A e−itH,t∈R. \alpha_t(A) := e^{itH}\,A\,e^{-itH}, \qquad t\in\mathbb{R}.

Fix β>0\beta>0 and define the (normalized) Gibbs state

ρβ:=e−βHTr(e−βH),ωβ(A):=Tr(ρβA), \rho_\beta := \frac{e^{-\beta H}}{\mathrm{Tr}(e^{-\beta H})}, \qquad \omega_\beta(A):=\mathrm{Tr}(\rho_\beta A),

with partition function Tr(e−βH)\mathrm{Tr}(e^{-\beta H}) (see and ).

Gibbs–KMS theorem.
The state ωβ\omega_\beta satisfies the β\beta-KMS condition with respect to αt\alpha_t (see ). Concretely: for all A,B∈AA,B\in\mathcal{A}, the function

FA,B(t):=ωβ ⁣(A αt(B)) F_{A,B}(t):=\omega_\beta\!\big(A\,\alpha_t(B)\big)

extends to a function analytic in the strip 0<Im(t)<β0<\mathrm{Im}(t)<\beta, continuous on the closed strip, and satisfies the boundary relation

FA,B(t+iβ)=ωβ ⁣(αt(B) A),t∈R. F_{A,B}(t+i\beta)=\omega_\beta\!\big(\alpha_t(B)\,A\big), \qquad t\in\mathbb{R}.

Key hypotheses and conclusions

Hypotheses

  • Finite-dimensional quantum system: A=B(H)\mathcal{A}=B(\mathcal{H}).
  • Hamiltonian H=H∗H=H^\ast generating αt(A)=eitHAe−itH\alpha_t(A)=e^{itH}Ae^{-itH}.
  • Inverse temperature β>0\beta>0 and Gibbs state ωβ\omega_\beta built from the ρβ\rho_\beta.

Conclusions

  • Equilibrium characterization: the Gibbs state is an equilibrium state in the KMS sense for the given dynamics.
  • Imaginary-time periodicity: the KMS boundary relation encodes the β\beta-periodicity in imaginary time (compare ).
  • Bridge to infinite volume: in algebraic quantum statistical mechanics, KMS states serve as the definition of thermal equilibrium even when Gibbs density matrices may fail to exist (finite-volume Gibbs ⇒\Rightarrow KMS is the finite-system prototype).

Proof idea / significance

In finite dimension, the proof is a direct computation using cyclicity of the trace and analytic continuation:

  • write FA,B(t)=Tr(ρβAeitHBe−itH)F_{A,B}(t)=\mathrm{Tr}(\rho_\beta A e^{itH}Be^{-itH}),
  • move factors around using Tr(XY)=Tr(YX)\mathrm{Tr}(XY)=\mathrm{Tr}(YX),
  • observe that inserting e−βHe^{-\beta H} corresponds to shifting t↦t+iβt\mapsto t+i\beta.

This theorem is one half of the “Gibbs ↔\leftrightarrow KMS” correspondence; the converse direction is typically formulated as a separate result (see under appropriate finiteness assumptions).