Non-ergodic phases in strongly disordered random regular graphs

Understand

We combine numerical diagonalization with a semi-analytical calculations to prove the existence of the intermediate non-ergodic but delocalized phase in the Anderson model on disordered hierarchical lattices.

• We suggest a new generalized population dynamics that is able to detect the violation of ergodicity of the delocalized states within the Abou-Chakra, Anderson and Thouless recursive scheme.
• This result is supplemented by statistics of random wave functions extracted from exact diagonalization of the Anderson model on ensemble of disordered Random Regular Graphs (RRG) of N sites with the connectivity K=2.
• By extrapolation of the results of both approaches to N->infinity we obtain the fractal dimensions D_{1}(W) and D_{2}(W) as well as the population dynamic exponent D(W) with the accuracy sufficient to claim that they are non-trivial in the broad interval of disorder strength W_{E}<W<W_{c}.