Imaginary-time-ordered correlation function

An imaginary-time-ordered correlation function orders operators along the Euclidean thermal circle. Bosonic correlators are periodic with period $\beta$.

The Kubo--Martin--Schwinger condition uses cyclicity of the thermal trace to relate correlation functions separated by imaginary time $\beta$.

Matsubara frequencies are the discrete Fourier frequencies on an imaginary time thermal interval. With $\hslash =k_B=1$ and $\beta =1/T$, periodic bosonic fields have ${\omega }_n=2\pi nT$, while antiperiodic fermionic fields have ${\omega }_n=(2n+1)\pi T$.

Their difference follows from requiring the Fourier phase to return with the appropriate sign after one thermal period. The bosonic zero mode can dominate long-distance finite-temperature fluctuations, whereas fermions have no zero Matsubara mode in the ordinary thermal trace. These are labels in an imaginary time calculation, not discrete frequencies of an externally applied real-time drive. Extracting a real-frequency response requires the corresponding analytic continuation and spectral information.

For $E>0$, contour summation over bosonic Matsubara frequencies givesIt separates into the zero-point term $1/(2E)$ and a thermal Bose--Einstein occupation term.