Accurate calculation of Green functions on the d-dimensional hypercubic lattice

TL;DR

This paper presents a new method for accurately computing Green functions on d-dimensional hypercubic lattices using piecewise integrals with Bessel functions, enabling fast calculations across dimensions and frequencies.

Contribution

It introduces a piecewise integral representation of the Green function that is efficient and accurate for any dimension and real frequency.

Findings

Fast computation of Green functions for any dimension

High accuracy across the entire frequency spectrum

Computational time scales linearly with dimension

Abstract

We write the Green function of the $d$-dimensional hypercubic lattice in a piecewise form covering the entire real frequency axis. Each piece is a single integral involving modified Bessel functions of the first and second kinds. The smoothness of the integrand allows both real and imaginary parts of the Green function to be computed quickly and accurately for any dimension $d$ and any real frequency, and the computational time scales only linearly with $d$.