Non-ergodic delocalized phase in Anderson model on Bethe lattice and regular graph

Abstract

We develop a novel analytical approach to the problem of single particle localization in infinite dimensional spaces such as Bethe lattice and random regular graph models. The key ingredient of the approach is the notion of the inverted order thermodynamic limit (IOTL) in which the coupling to the environment goes to zero before the system size goes to infinity. Using IOTL and Replica Symmetry Breaking (RSB) formalism we derive analytical expressions for the fractal dimension D1 that distinguishes between the extended ergodic, D1=1, and extended non-ergodic (multifractal), 0<D1<1 states on the Bethe lattice and random regular graphs with the branching number K. We also employ RSB formalism to derive the analytical expression lnStyp−1=−〈lnS〉∼(Wc−W)−1 for the typical imaginary part of self-energy Styp in the non-ergodic phase close to the Anderson transition in the conventional thermodynamic limit. We prove the existence of an extended non-ergodic phase in a broad range of disorder strength and energy and establish the phase diagrams of the models as a function of disorder and energy. The results of the analytical theory are compared with large-scale population dynamics and with the exact diagonalization of Anderson model on random regular graphs. We discuss the consequences of these results for the many body localization.