Nonergodic Phases in Strongly Disordered Random Regular Graphs

2016/05/31 by B. L. Altshuler, E. Cuevas, L. B. Ioffe +1 · 4 citations
Mathematics · Physics and Astronomy · #Anderson impurity model #Anderson localization #Delocalized electron #Electron #Ergodic theory #Ergodicity #Extrapolation #Mathematical analysis #Mathematical physics #Mathematics #Opinion Dynamics and Social Influence #Physics #Population #Pure mathematics #Quantum many-body systems #Quantum mechanics #Singularity #Statistical physics #Statistics #Theoretical and Computational Physics #cond-mat.dis-nn #cond-mat.stat-mech

paper · pdf · doi:10.1103/physrevlett.117.156601

published as Phys. Rev. Lett. 117, 156601 (2016) · 4 pages paper with 5 figures + Supplementary Material with 5 figures

arxiv created 2016/06/29 · openalex publication_date 2016/10/06 · arxiv updated 2016/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

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 D₁(W) and D₂(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>10⁵ 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.

Citations

• Random Graphs
• Nonergodic metallic and insulating phases of Josephson junction chains
• A selfconsistent theory of localization
• Quasiparticle Lifetime in a Finite System: A Nonperturbative Approach
• Localization and fluctuations of local spectral density on treelike structures with large connectivity: Application to the quasiparticle line shape in quantum dots
• Localization of interacting fermions at high temperature
• PROOF OF A THEOREM OF A. N. KOLMOGOROV ON THE INVARIANCE OF QUASI-PERIODIC MOTIONS UNDER SMALL PERTURBATIONS OF THE HAMILTONIAN

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