# A general method for calculating lattice Green functions on the branch cut

## TL;DR

This paper introduces a versatile Chebyshev expansion method for accurately computing lattice Green functions at any real frequency, applicable to various lattice types, and improves convergence by modeling van Hove singularities.

## Contribution

The authors develop a general Chebyshev expansion approach for calculating lattice Green functions, including a technique to accelerate convergence by subtracting modeled singularities.

## Key findings

- Achieves 6-9 significant figure accuracy with 1000 series terms.
- Applicable to multiple lattice types including square, cubic, and diamond.
- Method effectively captures van Hove singularities in Green functions.

## Abstract

We present a method for calculating the complex Green function $G_{ij} (\omega)$ at any real frequency $\omega$ between any two sites $i$ and $j$ on a lattice. Starting from numbers of walks on square, cubic, honeycomb, triangular, bcc, fcc, and diamond lattices, we derive Chebyshev expansion coefficients for $G_{ij} (\omega)$. The convergence of the Chebyshev series can be accelerated by constructing functions $f(\omega)$ that mimic the van Hove singularities in $G_{ij} (\omega)$ and subtracting their Chebyshev coefficients from the original coefficients. We demonstrate this explicitly for the square lattice and bcc lattice. Our algorithm achieves typical accuracies of 6--9 significant figures using 1000 series terms.