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
Currently, to 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:
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
iw2tau (gf_iw, beta[, moments, fourier, n_fit]) |
Discrete Fourier transform of the Hermitian Green’s function gf_iw. |
iw2tau_dft (gf_iw, beta) |
Discrete Fourier transform of the Hermitian Green’s function gf_iw. |
iw2tau_dft_soft (gf_iw, beta) |
Discrete Fourier transform of the Hermitian Green’s function gf_iw. |
izp2tau (izp, gf_izp, tau, beta[, moments]) |
Fourier transform of the Hermitian Green’s function gf_izp to tau. |
tau2iv (gf_tau, beta[, fourier]) |
Fourier transformation of the real Green’s function gf_tau. |
tau2iv_dft (gf_tau, beta) |
Discrete Fourier transform of the real Green’s function gf_tau. |
tau2iv_ft_lin (gf_tau, beta) |
Fourier integration of the real Green’s function gf_tau. |
tau2iw (gf_tau, beta[, n_pole, moments, fourier]) |
Fourier transform of the real Green’s function gf_tau. |
tau2iw_dft (gf_tau, beta) |
Discrete Fourier transform of the real Green’s function gf_tau. |
tau2iw_ft_lin (gf_tau, beta) |
Fourier integration of the real Green’s function gf_tau. |
tau2izp (gf_tau, beta, izp[, moments, occ, …]) |
Fourier transform of the real Green’s function gf_tau to izp. |
tt2z (tt, gf_t, z[, laplace]) |
Laplace transform of the real-time Green’s function gf_t. |
tt2z_lin (tt, gf_t, z) |
Laplace transform of the real-time Green’s function gf_t. |
tt2z_trapz (tt, gf_t, z) |
Laplace transform of the real-time Green’s function gf_t. |