Fourier transformation and response functions

TL;DR

This paper presents improved Fourier transform techniques for quantum cluster methods, enhancing the accuracy of Green's functions and response functions through boundary conditions and efficient splitting methods.

Contribution

It introduces a novel approach to Fourier transforms that handles singularities and asymptotic behaviors more effectively in quantum many-body calculations.

Findings

Enhanced Green's function asymptotic behavior with sumrule boundary conditions

Efficient two-dimensional Fourier transform method for singular functions

Improved accuracy in response function calculations

Abstract

We improve on Fourier transforms (FT) between imaginary time $\tau$ and imaginary frequency $\omega_n$ used in certain quantum cluster approaches using the Hirsch-Fye method. The asymptotic behavior of the electron Green's function can be improved by using a "sumrule" boundary condition for a spline. For response functions a two-dimensional FT of a singular function is required. We show how this can be done efficiently by splitting off a one-dimensional part containing the singularity and by performing a semi-analytical FT for the remaining more innocent two-dimensional part.