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. This conundrum is lifted by considering diagrammatic perturbation theory at finite temperature $T>0$, which may be understood by an analytical continuation of the real-time $t$ to the imaginary time axis $-i\tau$. Matsubara has shown that this Wick rotation in time $t\to-i\tau$ reveals an intriguing connection to the inverse temperature $\beta=1/T$ of the system. 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 $\chi(-i\tau)$) over the fundamental interval $-\beta\le\tau\le\beta$.

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

$\chi(-i\tau)=\frac1\beta\sum_{m=-\infty}^\infty \tilde \chi(i\nu_m)e^{-i\nu_m\tau},\quad \nu_m=\frac{2m}\beta\pi$

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

$\Omega_c^{\rm RPA}=\frac12\frac1\beta \sum_{m=-\infty}^\infty {\rm Tr}\left\lbrace \ln\left[ 1 -\tilde \chi(i\nu_m) V \right] -\tilde \chi(i\nu_m) V \right\rbrace,\quad \nu_m=\frac{2m}\beta\pi$

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 $\epsilon_{n{\bf k}}\approx \mu$, such that Matsubara series also converges for metallic systems.

Although formally convenient, the Matsubara series converges 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. This approach converges exponentially with the number of considered frequency points.