Matsubara formalism: Difference between revisions

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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 converges for metallic systems.
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. The VASP code, 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.  
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:VASP6]][[Category:Low-scaling GW and RPA]][[Category:Many-body perturbation theory]]
[[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.