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

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

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

${\mathrm{\Omega }}_c^{\mathrm{R}\mathrm{P}\mathrm{A}}=\frac{1}{2}\frac{1}{\beta }\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{T}\mathrm{r}\{\ln [1-\tilde{\chi }(i{\nu }_m)V]-\tilde{\chi }(i{\nu }_m)V\},{\nu }_m=\frac{2m}{\beta }\pi$

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 ${ϵ}_{n\mathbf{𝐤}}\approx \mu$, such that Matsubara series also converge for metallic systems.

Although formally convenient, the Matsubara series converges poorly with the number of considered terms in practice. VASP, 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.