Non-ergodic phases in strongly disordered random regular graphs

TL;DR

This paper demonstrates the existence of a non-ergodic yet delocalized phase in the Anderson model on disordered random regular graphs, combining numerical and semi-analytical methods to analyze phase transitions and fractal dimensions.

Contribution

It introduces a new generalized population dynamics method to detect ergodicity violation and provides evidence for a first-order transition between delocalized phases in RRG.

Findings

Identification of a non-ergodic delocalized phase in RRG.

Evidence of a first-order transition at W_{E}≈10.0.

Determination of fractal dimensions D_{1}(W), D_{2}(W) and dynamic exponent D(W).

Abstract

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}. The thorough analysis of the exact diagonalization results for RRG with N>10^{5} reveals a singularity in D_{1,2}(W)-dependencies which provides a clear evidence for the first order transition between the two delocalized phases on RRG at W_{E}\approx 10.0. We discuss the implications of these results for quantum and classical non-integrable and many-body systems.