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

Abstract

We present a method for calculating the complex Green function Gij(ω) at any real frequency ω 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 Gij(ω). The convergence of the Chebyshev series can be accelerated by constructing functions f (ω) that mimic the van Hove singularities in Gij(ω) 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.Consider the quantum mechanical tight-binding HamiltonianĤwhere t is the hopping amplitude between nearestneighbor sites i and j on a lattice. Define the "Greenian" operatorĜ(ω) = (ω −Ĥ) −1 where ω −Ĥ ≡ ω1 −Ĥ and 1 is the identity operator. The matrix elements of the Greenian

On the fully anisotropic honeycomb lattice Green function, Bessel function integrals and Pearson random walks

“…For the isotropic case α 1 = α 2 = α 3 = 1 the formula (6.7) gives the well-known connection between the number of random walks on the simple cubic and honeycomb lattices which return to their starting point after a walk consisting of 2n nearest-neighbour steps (see Joyce 1994, Guttmann 2010and Loh 2017. The geometrical argument used by Loh (2017) is particularly instructive because the honeycomb lattice is considered to be a puckered subset of a simple cubic lattice.…”