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

\[G(z) = ∫_{-∞}^{∞} dt G(t) \exp(izt)\]

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 and tt2z_lin which approximate the integral,

• tt2z_pade and tt2z_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:

\[G(iω_n) = 0.5 ∫_{-β}^{β}dτ G(τ) \exp(iω_n τ)\]

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:

• tau2iw for fermionic Green’s functions

• tau2iv for bosonic Green’s functions

Matsubara frequencies → imaginary time

The Fourier sum for the imaginary time Green’s function is defined as:

\[G(τ) = 1/β \sum_{n=-∞}^{∞} G(iω_n) \exp(-iω_n τ).\]

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:

• iv

• iw

• tau

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_herm2(tt, gf_t, z[, herm2, quad])	Square Fourier-Padé 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_lpz(tt, gf_t, z[, order, quad])	Linear prediction Z-transform of the real-time Green's function gf_t.
tt2z_pade(tt, gf_t, z[, degree, pade, quad])	Fourier-Padé transform of the real-time Green's function gf_t.
tt2z_simps(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.