KMS condition in finite quantum systems

Let $\mathcal H$ be finite-dimensional and let $H$ be the Hamiltonian (quantum Hamiltonian ). The Heisenberg time evolution on the observable algebra (observable algebra ) is the one-parameter group

Fix $\beta>0$ (inverse temperature β ). A state $\omega$ (equivalently, a density operator state density-operator state ) is said to satisfy the $\beta$-KMS condition with respect to $\tau_t$ if for every pair of observables $A,B$ there exists a complex-valued function $F_{A,B}(z)$ such that:

• $F_{A,B}(z)$ is analytic in the open strip $0<\operatorname{Im} z<\beta$ and continuous on its closure,
• for all real $t$, $F_{A,B}(t)\;=\;\omega\!\big(A\,\tau_t(B)\big), \qquad F_{A,B}(t+i\beta)\;=\;\omega\!\big(\tau_t(B)\,A\big).$

In finite dimension, one can write $\tau_{t+i\beta}(B)$ explicitly as

so the boundary condition can equivalently be stated as

Physical interpretation

The KMS condition is a precise, representation-independent way to say that the state $\omega$ is in thermal equilibrium for the dynamics generated by $H$.

Two physical features are built in:

• Imaginary-time structure: equilibrium correlation functions extend to complex time and satisfy a specific shift relation by $i\beta$ (“imaginary-time periodicity”).
• Detailed balance / equilibrium symmetry: the swap $A\tau_t(B)\leftrightarrow \tau_t(B)A$ after an imaginary-time shift is the operator-algebraic form of thermal balance.

These features control the behavior of thermal two-point functions (quantum correlation functions ) and underlie fluctuation–dissipation relations in linear response.

Key properties

1. Gibbs states satisfy KMS.
   The Gibbs state (quantum Gibbs state )

   $$\rho_\beta\;=\;\frac{e^{-\beta H}}{\operatorname{Tr}(e^{-\beta H})}$$

   defines $\omega_\beta(A)=\operatorname{Tr}(\rho_\beta A)$, and $\omega_\beta$ is a $\beta$-KMS state for $\tau_t$.

2. Uniqueness for full matrix algebras.
   For the full observable algebra $\mathcal B(\mathcal H)$ in finite dimension, the $\beta$-KMS state for the Hamiltonian dynamics is unique and equals the Gibbs state.

3. Time-translation invariance.
   Any $\beta$-KMS state $\omega$ is stationary:

   $$\omega(\tau_t(A))\;=\;\omega(A)$$

   for all $A$ and $t$. This formalizes equilibrium as invariance under the system’s own time evolution.

4. Imaginary-time boundary condition for correlators.
   If one defines the thermal correlator $G_{A,B}(t)=\omega(A\tau_t(B))$, then KMS gives

   $$G_{A,B}(t+i\beta)\;=\;\omega(\tau_t(B)A),$$

   which is the fundamental identity behind the periodicity properties of Matsubara (imaginary-time) correlators.

5. Finite-dimensional regularity.
   In finite dimension, all observables are bounded and the map $z\mapsto \omega(A\tau_z(B))$ can be constructed explicitly using matrix exponentials, so the analytic continuation required by the KMS condition is technically straightforward.