Accurate Calculation of Off-Diagonal Green Functions on Anisotropic Hypercubic Lattices

TL;DR

This paper introduces a precise method for calculating Green functions on anisotropic hypercubic lattices by deforming integrals into the complex plane, enabling accurate evaluations at various frequencies and lattice points.

Contribution

The paper presents a novel integral deformation technique and an alternative Levin collocation method for accurate Green function calculations on anisotropic lattices.

Findings

Accurate Green function values at selected frequencies and lattice vectors.

Effective integral deformation approach for rapid convergence.

Validation of methods through computed Green function data.

Abstract

We present a method for accurate evaluation of the Green function $G(\omega,r_1,...,r_d)$ at any real frequency $\omega$ and any lattice vector $(r_1,...,r_d)$ for a $d$-dimensional hypercubic lattice that may have anisotropic couplings $(\Omega_1,...,\Omega_d)$. In this method, we start with an integral representation of $G$, split the oscillatory integrand into combinations of Hankel functions, and deform the integration paths into the complex plane to obtain rapidly convergent integrals. We also discuss an alternative approach using the Levin collocation method. We report values of the Green function at selected frequencies on the branch cut and selected lattice vectors.