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

Understand

We develop a novel analytical approach to the problem of single particle localization in infinite dimensional spaces such as Bethe lattice and random regular graphs.

• 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 D_{1} that distinguishes between the extended ergodic, D_{1}=1, and extended non-ergodic (multifractal), 0<D_{1}<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 ln(1/S_{typ})~(W_{c}-W)^{-1} for the typical imaginary part of self-energy S_{typ} in the non-ergodic phase close to the Anderson transition in the conventional thermodynamic limit.