Matsubara formalism: Difference between revisions

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The zero-temperature formalism of many-body perturbation theory breaks down for metals (system with zero band gap).
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.{{cite|KohnLuttinger:PR:1960}} This conundrum is lifted by considering diagrammatic perturbation theory at finite temperature <math>T>0</math>, which may be understood by an analytical continuation of the real-time <math>t</math> to the imaginary time axis <math>-i\tau</math>. Matsubara has shown that this Wick rotation in time <math>t\to-i\tau</math> reveals an intriguing connection to the inverse temperature <math>\beta=1/T</math> of the system.{{cite|Matsubara:PTP:1955}} 
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 <math>\chi(-i\tau)</math>) over the fundamental interval <math>-\beta\le\tau\le\beta</math>.  


[[Category:Theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]] [[Category:Many-Body Perturbation Theory]]
As a consequence, one decomposes imaginary time quantities into a Fourier series with period <math>\beta</math>
that determines the spacing of the Fourier modes. For instance the imaginary polarizability can be written as
 
<math>
\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
</math>
 
and the corresponding random-phase approximation of the correlation energy at finite temperature becomes a series over (in this case, bosonic) Matsubara frequencies
 
<math>
\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
</math>
 
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 <math>\epsilon_{n{\bf k}}\approx \mu</math>, 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.{{cite|Kaltak:PRB:2020}} This approach converges exponentially with the number of considered frequency points.
 
[[Category:Theory]][[Category:Low-scaling GW and RPA]][[Category:Many-body perturbation theory]]

Latest revision as of 06:32, 21 February 2024

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 Fourier 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 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.[3] This approach converges exponentially with the number of considered frequency points.