# Anderson Localization on the Bethe Lattice using Cages and the Wegner Flow

## TL;DR

This paper investigates Anderson localization on the Bethe lattice using cages and Wegner flow techniques, providing insights into ergodic, non-ergodic extended, and localized phases through a novel combination of graph theory and numerical methods.

## Contribution

It introduces a new approach combining cages and Wegner flows to better understand localization phenomena on the Bethe lattice.

## Key findings

- Identifies regimes of ergodic, non-ergodic extended, and localized behavior.
- Provides an intuitive picture of phase transitions in Anderson localization.
- Extrapolates finite structures to the infinite Bethe lattice context.

## Abstract

Anderson localization on tree-like graphs such as the Bethe lattice, Cayley tree, or random regular graphs has attracted attention due to its apparent mathematical tractability, hypothesized connections to many-body localization, and the possibility of non-ergodic extended regimes. This behavior has been conjectured to also appear in many-body localization as a "bad metal" phase, and constitutes an intermediate possibility between the extremes of ergodic quantum chaos and integrable localization. Despite decades of research, a complete consensus understanding of this model remains elusive. Here, we use cages, maximally tree-like structures from extremal graph theory; and numerical continuous unitary Wegner flows of the Anderson Hamiltonian to develop an intuitive picture which, after extrapolating to the infinite Bethe lattice, appears to capture ergodic, non-ergodic extended, and fully localized behavior.