Selberg trace formula in hyperbolic band theory

TL;DR

This paper extends the Selberg trace formula to hyperbolic band theory, enabling the analysis of band structures on hyperbolic lattices and relating spectral properties to geometric symmetries.

Contribution

It introduces a method to incorporate higher-dimensional crystal momentum into the trace formula for hyperbolic lattices, specifically on the Bolza surface, and relates spectral features to lattice symmetries.

Findings

Computed partition functions on the Bolza surface.

Proposed an approximate relation between the lowest bands of the Bolza surface and the {8,3} hyperbolic lattice.

Analyzed the role of automorphism symmetry in the trace formula.

Abstract

We apply Selberg's trace formula to solve problems in hyperbolic band theory, a recently developed extension of Bloch theory to model band structures on experimentally realized hyperbolic lattices. For this purpose we incorporate the higher-dimensional crystal momentum into the trace formula and evaluate the summation for periodic orbits on the Bolza surface of genus two. We apply the technique to compute partition functions on the Bolza surface and propose an approximate relation between the lowest bands on the Bolza surface and on the $\{8,3\}$ hyperbolic lattice. We discuss the role of automorphism symmetry and its manifestation in the trace formula.