Bosonic Matsubara frequencies are . They are the Fourier modes on the periodic imaginary-time circle.
For , contour summation over bosonic Matsubara frequencies givesIt separates into the zero-point term and a thermal Bose--Einstein occupation term.
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Matsubara frequency is a concept commonly used in condensed matter physics and statistical mechanics, specifically in the context of finite-temperature field theory and many-body quantum systems. It arises in the formalism known as Matsubara techniques, which are used to evaluate correlations and Green's functions in systems at finite temperature. Matsubara frequencies are defined as discrete frequencies that appear in the solution of the equations describing quantum systems at finite temperature.