gftool.fourier
Fourier transformations of Green’s functions.
Fourier transformation between imaginary time and Matsubara frequencies. The function in this module should be used after explicitly treating the high-frequency behavior, as this is not yet implemented. Typically, transformation from τ-space to Matsubara frequency are unproblematic.
The Fourier transforms are defined in the following way:
Definitions
real time → complex frequencies
The Laplace integral for the Green’s function is defined as
This integral is only well defined
in the upper complex half-plane z.imag>=0 for retarded Green’s function \(∝θ(t)\)
in the lower complex half-plane z.imag<=0 for advanced Green’s function \(∝θ(-t)\)
The recommended high-level function to perform this Laplace transform is:
tt2z
for both retarded and advanced Green’s function
Two different kind of algorithms are available
tt2z_trapz
andtt2z_lin
which approximate the integral,tt2z_pade
andtt2z_herm2
which are Padé-Fourier type transformations.
Currently, sub-functions can be used equivalently, the abstraction tt2z
is
mostly for consistency with the imaginary time ↔ Matsubara frequencies
Fourier transformations.
imaginary time → Matsubara frequencies
The Fourier integral for the Matsubara Green’s function is defined as:
with \(iw_n = iπn/β\). For fermionic Green’s functions only odd frequencies are non-vanishing, for bosonic Green’s functions only even.
The recommended high-level function to perform this Fourier transform is:
Matsubara frequencies → imaginary time
The Fourier sum for the imaginary time Green’s function is defined as:
The recommended high-level function to perform this Fourier transform is:
iw2tau
for fermionic Green’s functions
Glossary
- dft
<discrete Foruier transform>
- ft
<Fourier transformation> In contrast to dft, this is used for Fourier integration of continous variables without discretization.
Previously defined:
API
Functions
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Discrete Fourier transform of the Hermitian Green's function gf_iw. |
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Discrete Fourier transform of the Hermitian Green's function gf_iw. |
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Discrete Fourier transform of the Hermitian Green's function gf_iw. |
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Fourier transform of the Hermitian Green's function gf_izp to tau. |
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Fourier transformation of the real Green's function gf_tau. |
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Discrete Fourier transform of the real Green's function gf_tau. |
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Fourier integration of the real Green's function gf_tau. |
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Fourier transform of the real Green's function gf_tau. |
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Discrete Fourier transform of the real Green's function gf_tau. |
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Fourier integration of the real Green's function gf_tau. |
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Fourier transform of the real Green's function gf_tau to izp. |
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Laplace transform of the real-time Green's function gf_t. |
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Square Fourier-Padé transform of the real-time Green's function gf_t. |
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Laplace transform of the real-time Green's function gf_t. |
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Linear prediction Z-transform of the real-time Green's function gf_t. |
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Fourier-Padé transform of the real-time Green's function gf_t. |
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Laplace transform of the real-time Green's function gf_t. |
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Laplace transform of the real-time Green's function gf_t. |