Theory of Anderson localization on the hyperbolic plane

TL;DR

This paper develops a unified theoretical framework for Anderson localization on the hyperbolic plane, capturing the transition between metallic and insulating phases through a two-parameter flow.

Contribution

It introduces a two-parameter flow model that interpolates between low- and high-dimensional Anderson localization principles on the hyperbolic plane.

Findings

Derived a two-parameter flow in curvature and conductivity space.

Identified an extended critical line separating metallic and insulating phases.

Unified treatment of localization at different scales on hyperbolic geometry.

Abstract

The two-dimensional hyperbolic plane, $\mathbb{H}^2$, is an unusual system in that dimensionality changes with scale: locally two-dimensional and planar at short distances, but effectively infinite-dimensional at large scales, it provides an interesting paradigm for the study of (quantum) phase transitions, notably the disorder-driven Anderson transition. Generalizing previous work, which treated short and large distance scales separately, we develop a unified framework interpolating between the principles of low- and high-dimensional Anderson localization. As a main result, we derive a two-parameter flow in a plane spanned by scale-dependent curvature (setting the system's effective dimensionality) and conductivity, with an extended critical line separating metallic and insulating phases.