Anderson localization and ergodicity on random regular graphs

K. S. Tikhonov, A. D. Mirlin, M. A. Skvortsov

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    82 Citations (Scopus)


    A numerical study of Anderson transition on random regular graphs (RRGs) with diagonal disorder is performed. The problem can be described as a tight-binding model on a lattice with N sites that is locally a tree with constant connectivity. In a certain sense, the RRG ensemble can be seen as an infinite-dimensional (d→) cousin of the Anderson model in d dimensions. We focus on the delocalized side of the transition and stress the importance of finite-size effects. We show that the data can be interpreted in terms of the finite-size crossover from a small (NNc) to a large (NNc) system, where Nc is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wave-function statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly nontrivial. Our results support an analytical prediction that states in the delocalized phase (and at NNc) are ergodic in the sense that their inverse participation ratio scales as 1/N.

    Original languageEnglish
    Article number220203
    JournalPhysical Review B
    Issue number22
    Publication statusPublished - 22 Dec 2016


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