title: Nonergodic Phases in Strongly Disordered Random Regular Graphs.
abstract: We combine numerical diagonalization with semianalytical calculations to prove the existence of the intermediate nonergodic 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→∞ we obtain the fractal dimensions D_{1}(W) and D_{2}(W) as well as the population dynamics exponent D(W) with the accuracy sufficient to claim that they are nontrivial 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 clear evidence for the first order transition between the two delocalized phases on RRG at W_{E}≈10.0. We discuss the implications of these results for quantum and classical nonintegrable and many-body systems.
authors: Physics Department, Columbia University, 538 West 120th Street, New York, New York 10027, USA.
authors: B L
authors: BL
authors: Altshuler
authors: Departamento de Física, Universidad de Murcia, E30071 Murcia, Spain.
authors: E
authors: E
authors: Cuevas
authors: CNRS and Universite Paris Sud, UMR 8626, LPTMS, Orsay Cedex F-91405, France.|L. D. Landau Institute for Theoretical Physics, Chernogolovka 142432, Moscow region, Russia.
authors: L B
authors: LB
authors: Ioffe
authors: L. D. Landau Institute for Theoretical Physics, Chernogolovka 142432, Moscow region, Russia.|Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy.
authors: V E
authors: VE
authors: Kravtsov