KMS imaginary-time periodicity

For a β-KMS equilibrium state, real-time correlation functions extend analytically to a complex-time strip and satisfy a β-periodicity (KMS boundary) relation in imaginary time.

KMS imaginary-time periodicity

Statement

Let $(\mathcal A,\tau_t)$ be a quantum dynamical system (a $C^*$ -algebra $\mathcal A$ with a strongly continuous one-parameter group of $*$ -automorphisms $t\mapsto \tau_t$ ). Let $\omega$ be a $\beta$ -KMS state at inverse temperature $\beta>0$ (see KMS condition ).

For any $\tau$ -analytic observables $A,B\in\mathcal A$ , the real-time two-point function $F_{A,B}(t):=\omega\!\big(A\,\tau_t(B)\big)$ , $t\in\mathbb R$ , admits an analytic continuation to the strip

S_\beta:=\{z\in\mathbb C: 0<\operatorname{Im}z<\beta\},

continuous on the closed strip $\overline{S_\beta}$ , with boundary values

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) \quad (t\in\mathbb R).

Equivalently, whenever $\tau_{t+i\beta}(B)$ is defined (e.g. for $\tau$ -analytic $B$ ),

\omega\!\big(A\,\tau_{t+i\beta}(B)\big)=\omega\!\big(\tau_t(B)\,A\big), \quad t\in\mathbb R.

In particular (take $t=0$ ),

\omega\!\big(A\,\tau_{i\beta}(B)\big)=\omega(BA),

which is the imaginary-time “ $\beta$ -periodicity” relation (a shift by $i\beta$ in time corresponds to moving the second observable past the first).

Key hypotheses

$\beta>0$ .
$(\mathcal A,\tau_t)$ is a $C^*$ -dynamical system.
$\omega$ is a $\beta$ -KMS state for $\tau_t$ (KMS condition ).
$A,B$ belong to the $\tau$ -analytic subalgebra (so $\tau_z(B)$ is meaningful for complex $z$ in the strip).

Main conclusions

Analyticity in complex time: $t\mapsto \omega(A\tau_t(B))$ extends to an analytic function on the strip $0<\operatorname{Im}z<\beta$ .
Imaginary-time boundary relation: the values on the upper boundary $\operatorname{Im}z=\beta$ match the lower boundary $\operatorname{Im}z=0$ after a cyclic permutation of operators:
$\omega(A\tau_{t+i\beta}(B))=\omega(\tau_t(B)A).$
Endpoint condition at imaginary time $\beta$ : $\omega(A\tau_{i\beta}(B))=\omega(BA)$ .

Cross-links to definitions and context

The equilibrium notion here is the KMS condition .
In finite quantum systems, the standard equilibrium state is the quantum Gibbs state written via a density operator and the quantum partition function .
The identification “Gibbs state $\Rightarrow$ KMS” is commonly packaged as Gibbs–KMS theorem (and converses as KMS–Gibbs converse in settings where it holds).

Proof idea / significance

Finite-dimensional (Gibbs) case.
Let $H$ be the Hamiltonian and $\tau_t(B)=e^{itH}Be^{-itH}$ . Let $\rho_\beta=e^{-\beta H}/Z_\beta$ be the Gibbs density operator (Gibbs state ), and define

F_{A,B}(z)=\operatorname{Tr}\!\big(\rho_\beta\,A\,e^{izH}Be^{-izH}\big).

Because $z\mapsto e^{izH}$ is entire, $F_{A,B}(z)$ is analytic in $z$ . Evaluating at $z=t+i\beta$ and using $e^{i(t+i\beta)H}=e^{itH}e^{-\beta H}$ together with cyclicity of the trace gives the boundary identity

F_{A,B}(t+i\beta)=\operatorname{Tr}\!\big(\rho_\beta\,e^{itH}Be^{-itH}\,A\big)=\omega(\tau_t(B)A),

which is exactly the KMS imaginary-time periodicity.

Why it matters.
This boundary relation is the conceptual basis of the imaginary-time (Matsubara) formalism: equilibrium correlation functions can be studied as analytic functions on a strip and, after Wick rotation, as functions on an imaginary-time circle of circumference $\beta$ (with the KMS cyclic permutation at the “cut”).