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# Matsubara Formalism

The zero-temperature formalism of many-body perturbation theory breaks down for metals (systems with zero energy band-gap) as pointed out by Kohn and Luttinger.^{[1]} This conundrum is lifted by considering diagrammatic perturbation theory at finite temperature , which may be understood by an analytical continuation of the real-time to the imaginary time axis . Matsubara has shown that this Wick-rotation in time reveals an intriguing connection to the inverse temperature of the system.^{[2]}
More precisely, Matsubara has shown that all terms in perturbation theory at finite temperature can be expressed as integrals of imaginary time quantities (such as the polarizability ) over the fundamental interval .

As a consequence, one decomposes imaginary time quantities into a Fourier series with period that determines the spacing of the Fouier modes. For instance the imaginary polarizability can be written as

and the corresponding random phase approximation of the correlation energy at finite temperature becomes a series over (in this case bosonic) Matsubara frequencies

The Matsubara formalism has the advantage that all contributions to the Green's function and the polarizability are mathematically well-defined, including contributions from states close to the chemical potential , such that Matsubara series converge also for metallic systems.

Although formally convenient, Matsubara series converge poorly with the number of considered terms in practice. The VASP code, therefore, uses a compressed representation of the Fourier modes by employing the Minimax Isometry method.^{[3]} This approach converges exponentially with the number of considered frequency points.