Walking on Archimedean Lattices: Insights from Bloch Band Theory

TL;DR

This paper derives analytical formulas for returning walks on Archimedean lattices using Bloch band theory, providing insights into lattice models and their spectral properties, with potential for broader applications.

Contribution

It introduces a novel connection between returning walk enumeration and Bloch energy band theory for Archimedean lattices, enabling analytical computation of walk numbers and density of states.

Findings

Derived explicit formulas for returning walk numbers on 11 Archimedean lattices.

Validated results through an alternative adjacency matrix moment method.

Computed density of states and asymptotic return probabilities for tight-binding models.

Abstract

Returning walks on a lattice are sequences of moves that start at a given lattice site and return to the same site after $n$ steps. Determining the total number of returning walks of a given length $n$ is a typical graph-theoretical problem with connections to lattice models in statistical and condensed matter physics. We derive analytical expressions for the returning walk numbers on the eleven two-dimensional Archimedean lattices by developing a connection to the theory of Bloch energy bands. We benchmark our results through an alternative method that relies on computing the moments of adjacency matrices of large graphs, whose construction we explain explicitly. As condensed matter physics applications, we use our formulas to compute the density of states of tight-binding models on the Archimedean lattices and analytically determine the asymptotics of the return probability. While the Archimedean lattices provide a sufficiently rich structure and are chosen here for concreteness, our techniques can be generalized straightforwardly to other two- or higher-dimensional Euclidean lattices.