# Two critical localization lengths in the Anderson transition on random graphs

## TL;DR

This paper characterizes two distinct critical localization lengths in the Anderson transition on random graphs, revealing their roles in nonergodic wavefunction properties and linking them to many-body localization universality.

## Contribution

It introduces and analyzes two critical localization lengths in the Anderson transition on random graphs, connecting these findings to many-body localization universality classes.

## Key findings

- Largest localization length diverges with exponent 1 at transition.
- Finite universal value for typical localization length at transition.
- Two localization lengths govern key observable scaling behaviors.

## Abstract

We present a full description of the nonergodic properties of wavefunctions on random graphs without boundary in the localized and critical regimes of the Anderson transition. We find that they are characterized by two critical localization lengths: the largest one describes localization along rare branches and diverges with a critical exponent $\nu_\parallel=1$ at the transition. The second length, which describes localization along typical branches, reaches at the transition a finite universal value (which depends only on the connectivity of the graph), with a singularity controlled by a new critical exponent $\nu_\perp=1/2$. We show numerically that these two localization lengths control the finite-size scaling properties of key observables: wavefunction moments, correlation functions and spectral statistics. Our results are identical to the theoretical predictions for the typical localization length in the many-body localization transition, with the same critical exponent. This strongly suggests that the two transitions are in the same universality class and that our techniques could be directly applied in this context.