Chebyshev polynomial representation of imaginary time response functions

TL;DR

This paper introduces a Chebyshev polynomial series method for representing imaginary-time Green's functions and self-energies in finite-temperature quantum mechanics, improving accuracy and computational efficiency over traditional grid methods.

Contribution

It demonstrates that Chebyshev polynomial expansions enable precise, efficient operations on Green's functions, including convolutions and Dyson equation solutions, for realistic quantum systems.

Findings

Chebyshev representation offers higher accuracy than uniform-power grid methods.

Operations like convolutions and Fourier transforms are efficiently expressed in Chebyshev coefficients.

The method improves the precision of finite-temperature quantum calculations.

Abstract

Problems of finite-temperature quantum statistical mechanics can be formulated in terms of imaginary (Euclidean) -time Green's functions and self-energies. In the context of realistic Hamiltonians, the large energy scale of the Hamiltonian (as compared to temperature) necessitates a very precise representation of these functions. In this paper, we explore the representation of Green's functions and self-energies in terms of a series of Chebyshev polynomials. We show that many operations, including convolutions, Fourier transforms, and the solution of the Dyson equation, can straightforwardly be expressed in terms of the series expansion coefficients. We then compare the accuracy of the Chebyshev representation for realistic systems with the uniform-power grid representation, which is most commonly used in this context.