Density of states of tight-binding models in the hyperbolic plane

TL;DR

This paper computes the density of states for tight-binding models on hyperbolic tilings using a continued-fraction Green function approach, revealing significant differences from hyperbolic band theory predictions.

Contribution

It introduces a method to accurately compute the density of states for large hyperbolic systems and challenges existing hyperbolic band theory assumptions.

Findings

Density of states differs from hyperbolic band theory predictions

The hyperbolic Bloch-like wave eigenfunctions occupy a vanishing fraction of the spectrum

The continued-fraction expansion converges rapidly, enabling large-scale analysis

Abstract

We study the energy spectrum of tight-binding Hamiltonian for regular hyperbolic tilings. More specifically, we compute the density of states using the continued-fraction expansion of the Green function on finite-size systems with more than $10^9$ sites and open boundary conditions. The coefficients of this expansion are found to quickly converge so that the thermodynamical limit can be inferred quite accurately. This density of states is in stark contrast with the prediction stemming from the recently proposed hyperbolic band theory. Thus, we conclude that the fraction of the energy spectrum described by the hyperbolic Bloch-like wave eigenfunctions vanishes in the thermodynamical limit.