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

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.