Two critical localization lengths in the Anderson transition on random graphs
Résumé
We present a full description of the nonergodic properties of wave functions on random graphs withoutboundary in the localized and critical regimes of the Anderson transition. We find that they are characterizedby two critical localization lengths: the largest one describes localization along rare branches and divergeswith a critical exponent ν = 1 at the transition. The second length, which describes localization along typicalbranches, reaches at the transition a finite universal value (which depends only on the connectivity of thegraph), with a singularity controlled by a new critical exponent ν ⊥ = 1/2. We show numerically that thesetwo localization lengths control the finite-size scaling properties of key observables: wave-function moments,correlation functions, and spectral statistics. Our results are identical to the theoretical predictions for the typicallocalization length in the many-body localization transition, with the same critical exponent. This stronglysuggests that the two transitions are in the same universality class and that our techniques could be directlyapplied in this context.