Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase

We develop a novel analytical approach to the problem of single particle localization in infinite dimensional spaces such as Bethe lattice and random regular graph models. The key ingredient of the approach is the notion of the inverted order thermodynamic limit (IOTL) in which the coupling to the environment goes to zero before the system size goes to infinity. Using IOTL and Replica Symmetry Breaking (RSB) formalism we derive analytical expressions for the fractal dimension D1 that distinguishes between the extended ergodic, D1 = 1, and extended non-ergodic (multifractal), 0 < D1 < 1 states on the Bethe lattice and random regular graphs with the branching number K. We also employ RSB formalism to derive the analytical expression ln Sfor the typical imaginary part of self-energy Styp in the non-ergodic phase close to the Anderson transition in the conventional thermodynamic limit. We prove the existence of an extended nonergodic phase in a broad range of disorder strength and energy and establish the phase diagrams of the models as a function of disorder and energy. The results of the analytical theory are compared with large-scale population dynamics and with the exact diagonalization of Anderson model on random regular graphs. We discuss the consequences of these results for the many body localization. Contents

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“…The absence of non-ergodic extended state on RRG in work 45 is based on an analysis of the level compressibility χ…”

Section: Symmetry Of the Correlation Volume Dependence On W − Wc In Dmentioning

confidence: 99%

Non-ergodic delocalized phase in Anderson model on Bethe lattice and regular graph

2018

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“…53 and 54. These works motivated an intensive numerical research on properties of the delocalized phase in the RRG and SRM models [55][56][57][58] . A detailed numerical investigation of level and eigenfunctions statistics on the delocalized side of the Anderson transition on RRG carried out in Ref.…”

Section: Introductionmentioning

confidence: 99%

Multifractality of wave functions on a Cayley tree: From root to leaves

2017

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We explore the evolution of wave-function statistics on a finite Bethe lattice (Cayley tree) from the central site ("root") to the boundary ("leaves"). We show that the eigenfunction moments Pq = N |ψ| 2q (i) exhibit a multifractal scaling Pq ∝ N −τq with the volume (number of sites) N at N → ∞. The multifractality spectrum τq depends on the strength of disorder and on the parameter s characterizing the position of the observation point i on the lattice. Specifically, s = r/R, where r is the distance from the observation point to the root, and R is the "radius" of the lattice. We demonstrate that the exponents τq depend linearly on s and determine the evolution of the spectrum with increasing disorder, from delocalized to the localized phase. Analytical results are obtained for the n-orbital model with n 1 that can be mapped onto a supersymmetric σ model. These results are supported by numerical simulations (exact diagonalization) of the conventional (n = 1) Anderson tight-binding model.

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“…In many-body systems, this can be considered as a proxy for the non-equilibrium dynamics of local operators after quench [28,60,61] and also as a direct measure for entanglement propagation [60,61]. Our finding gives evidence of the existence of sub-diffusive dynamics over an entire range of parameters, even in a part of the phase diagram where most of the works [29,[37][38][39][40][41][42][45][46][47][48][49][50][51][52][53][54][55][56] agree that eigenstates are ergodic according to standard multifractal analysis of wave functions.…”

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confidence: 89%

Subdiffusion in the Anderson model on the random regular graph

et al. 2020

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We study the finite-time dynamics of an initially localized wave-packet in the Anderson model on the random regular graph (RRG). Considering the full probability distribution Π(x, t) of a particle to be at some distance x from the initial state at time t, we give evidence that Π(x, t) spreads subdiffusively over a range of disorder strengths, wider than a putative non-ergodic phase. We provide a detailed analysis of the propagation of Π(x, t) in space-time (x, t) domain, identifying four different regimes. These regimes in (x, t) are determined by the position of a wave-front X front (t), which moves sub-diffusively to the most distant sites X front (t) ∼ t β with an exponent β < 1. We support our numerical results by a self-consistent semiclassical picture of wavepacket propagation relating the exponent β with the relaxation rate of the return probability Π(0, t) ∼ e −Γt β . Importantly, the Anderson model on the RRG can be considered as proxy of the many-body localization transition (MBL) on the Fock space of a generic interacting system. In the final discussion, we outline possible implications of our findings for MBL. arXiv:1908.11388v1 [cond-mat.dis-nn]

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Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase

Cited by 38 publications

References 43 publications

Non-ergodic delocalized phase in Anderson model on Bethe lattice and regular graph

Non-ergodic delocalized phase in Anderson model on Bethe lattice and regular graph

Multifractality of wave functions on a Cayley tree: From root to leaves

Subdiffusion in the Anderson model on the random regular graph

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