Spectral and Combinatorial Aspects of Cayley-Crystals

TL;DR

This paper develops a numerical method to analyze quantum dynamics on Cayley graphs of finitely generated groups, revealing spectral properties and combinatorial insights, with applications to hyperbolic and fractal lattices.

Contribution

It introduces a convergent numerical approach for bulk spectrum computation and derives new combinatorial results from resolvent matrix elements.

Findings

Validated the numerical method against known spectral properties.

Derived new combinatorial statements from Cayley graph resolvent analysis.

Enabled systematic study of quantum dynamics on diverse Cayley graphs.

Abstract

Owing to their interesting spectral properties, the synthetic crystals over lattices other than regular Euclidean lattices, such as hyperbolic and fractal ones, have attracted renewed attention, especially from materials and meta-materials research communities. They can be studied under the umbrella of quantum dynamics over Cayley graphs of finitely generated groups. In this work, we investigate numerical aspects related to the quantum dynamics over such Cayley graphs. Using an algebraic formulation of the "periodic boundary condition" due to Lueck [Geom. Funct. Anal. 4, 455-481 (1994)], we devise a practical and converging numerical method that resolves the true bulk spectrum of the Hamiltonians. Exact results on the matrix elements of the resolvent, derived from the combinatorics of the Cayley graphs, give us the means to validate our algorithms and also to obtain new combinatorial statements. Our results open the systematic research of quantum dynamics over Cayley graphs of a very large family of finitely generated groups, which includes the free and Fuchsian groups.