Quantum thermal correlation function (construction)

Equilibrium time and imaginary-time correlation functions computed from the thermal density matrix; they satisfy the KMS condition and control response.

Quantum thermal correlation function (construction)

A quantum thermal correlation function is a two-point (or multipoint) expectation value in a thermal equilibrium state, with operators evolved in time. These objects generalize the two-point correlation function of classical statistical mechanics and are central for linear response and spectroscopy.

Thermal expectation and time evolution

In the canonical ensemble at inverse temperature $\beta$ , the thermal state is

\rho_\beta = \frac{e^{-\beta H}}{Z(\beta)}, \qquad Z(\beta)=\operatorname{Tr}\big(e^{-\beta H}\big),

with $Z(\beta)$ the canonical partition function . The thermal expectation of an observable $X$ is the ensemble average

\langle X\rangle_\beta := \operatorname{Tr}(\rho_\beta X).

In the Heisenberg picture, the real-time evolution of an operator $A$ is

A(t) := e^{iHt/\hbar}\,A\,e^{-iHt/\hbar}.

Real-time and imaginary-time two-point functions

Given two observables $A$ and $B$ , the real-time thermal correlation function is

C_{AB}(t) := \langle A(t)\,B\rangle_\beta = \operatorname{Tr}\!\big(\rho_\beta\, A(t)\,B\big).

The imaginary-time (Matsubara) correlation function is defined for $\tau\in[0,\beta]$ by

G_{AB}(\tau) := \langle A(-i\tau)\,B\rangle_\beta = \operatorname{Tr}\!\big(\rho_\beta\, e^{\tau H}\,A\,e^{-\tau H}\,B\big).

Imaginary-time correlations are particularly natural in path-integral and transfer-matrix approaches (compare transfer matrix constructions in one dimension).

If $A$ and $B$ are supported in a finite region, the same correlations can be computed using the reduced density matrix for that region instead of the full $\rho_\beta$ .

Connected thermal correlations

The connected (or truncated) two-point function is obtained by subtracting the product of means:

\langle AB\rangle_{\beta,c} := \langle AB\rangle_\beta - \langle A\rangle_\beta\,\langle B\rangle_\beta.

This is the quantum counterpart of the connected correlation function and is the basic building block for cumulant expansions (see connected correlations via cumulants ).

KMS condition (thermal analyticity)

A defining equilibrium property of thermal correlations is the KMS condition, which can be stated (for bosonic observables) as the analyticity/periodicity relation

C_{AB}(t-i\beta) = \langle B\,A(t)\rangle_\beta,

whenever the analytic continuation is well-defined. This identity encodes equilibrium and replaces time-translation invariance plus detailed balance in the quantum setting.

Spectral (energy-eigenbasis) representation

Let $H|n\rangle=E_n|n\rangle$ and define matrix elements $A_{nm}=\langle n|A|m\rangle$ . Then

C_{AB}(t) ={} \frac{1}{Z(\beta)}\sum_{n,m} e^{-\beta E_n}\, e^{i(E_n-E_m)t/\hbar}\, A_{nm}\,B_{mn}.

This formula shows that thermal correlations probe energy differences and selection rules through operator matrix elements.

Physical interpretation and response

Fluctuations: $C_{AA}(t)$ quantifies how fluctuations of $A$ persist over time in equilibrium, complementing static quantities like variance in an ensemble .
Susceptibility: static and dynamical response coefficients (see susceptibility ) can be expressed in terms of time-integrated connected thermal correlations; in many settings, imaginary-time integrals of $G_{AA}(\tau)$ control static susceptibilities.
Clustering: in phases with finite correlation length , connected thermal correlations decay in space (and often in imaginary time), reflecting locality and the absence of long-range order.

These correlation functions are also the natural objects that appear when differentiating $\log Z$ with respect to sources, matching the general “derivatives of $\log Z$ produce cumulants” philosophy in cumulant generating constructions .