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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 ${\displaystyle T>0}$, which may be understood by an analytical continuation of the real-time ${\displaystyle t}$ to the imaginary time axis ${\displaystyle -i\tau }$. Matsubara has shown that this Wick-rotation in time ${\displaystyle t\to -i\tau }$ reveals an intriguing connection to the inverse temperature ${\displaystyle \beta =1/T}$ 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 ${\displaystyle \chi (-i\tau )}$) over the fundamental interval ${\displaystyle -\beta \leq \tau \leq \beta }$.

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

${\displaystyle \chi (-i\tau )={\frac {1}{\beta }}\sum _{m=-\infty }^{\infty }{\tilde {\chi }}(i\nu _{m})e^{-i\nu _{m}\tau },\quad \nu _{m}={\frac {2m}{\beta }}\pi }$

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

${\displaystyle \Omega _{c}^{\rm {RPA}}={\frac {1}{2}}{\frac {1}{\beta }}\sum _{m=-\infty }^{\infty }{\rm {Tr}}\left\lbrace \ln \left[1-{\tilde {\chi }}(i\nu _{m})V\right]-{\tilde {\chi }}(i\nu _{m})V\right\rbrace ,\quad \nu _{m}={\frac {2m}{\beta }}\pi }$

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 ${\displaystyle \epsilon _{n{\bf {k}}}\approx \mu }$, 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.