Anderson localization transition in disordered hyperbolic lattices

TL;DR

This paper investigates Anderson localization in disordered hyperbolic lattices, revealing a transition at finite disorder strengths and characterizing critical behavior, bridging the gap between Euclidean and Bethe lattice models.

Contribution

It demonstrates the existence of an Anderson localization transition in hyperbolic lattices using advanced computational methods, providing new insights into their critical properties.

Findings

Existence of an Anderson transition on hyperbolic lattices

Large critical disorder strengths observed

Strong finite-size effects in level statistics

Abstract

We study Anderson localization in disordered tight-binding models on hyperbolic lattices. Such lattices are geometries intermediate between ordinary two-dimensional crystalline lattices, which localize at infinitesimal disorder, and Bethe lattices, which localize at strong disorder. Using state-of-the-art computational group theory methods to create large systems, we approximate the thermodynamic limit through appropriate periodic boundary conditions and numerically demonstrate the existence of an Anderson localization transition on the $\{8,3\}$ and $\{8,8\}$ lattices. We find unusually large critical disorder strengths, determine critical exponents, and observe a strong finite-size effect in the level statistics.