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arXiv:1903.07877 · Physical review. B./Physical review. B

Critical behavior at the localization transition on random regular graphs

Konstantin S. Tikhonov, A. D. Mirlin

Published Jun 11, 2019• 79 citations
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Abstract

Anderson localization on random regular graphs (RRG) attracts much attention as a toy model of many-body localization. Here, the critical behavior at the localization transition on RRG, related to that on infinite Bethe lattices, is studied numerically by means of a population dynamics approach. Proceeding up to correlation volumes as large as ${10}^{19}$, the authors obtain the value of one half for the correlation-length critical exponent, in agreement with analytical predictions. They find very well pronounced corrections to scaling, similar to those in models with high spatial dimensionality and with many-body localization transitions.

DOI
10.1103/physrevb.99.214202

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Anderson localization on random regular graphs (RRG) attracts much attention as a toy model of many-body localization. Here, the critical behavior at the localization transition on RRG, related to that on infinite Bethe lattices, is studied numerically by means of a population dynamics approach. Proceeding up to correlation volumes as large as 1019{10}^{19}1019, the authors obtain the value of one half for the correlation-length critical exponent, in agreement with analytical predictions. They find very well pronounced corrections to scaling, similar to those in models with high spatial dimensionality and with many-body localization transitions.
Anderson localization on random regular graphs (RRG) serves as a toy-model of many-body localization (MBL). We explore the transition from ergodicity to localization on RRG with large connectivity mmm. In the analytical part, we focus on the inverse participation ratio of eigenstates and identify several regimes on the ergodic side of the transition---self-consistent golden rule, golden rule, precritical, and critical---that the system consecutively goes through when the disorder increases toward the point Wc{W}_{c}Wc​ of the localization transition. We also perform exact-diagonalization numerics as well as a population-dynamics analysis combining analytical and numerical techniques. Results of all the approaches are in excellent mutual agreement. We further explore the evolution from ergodicity to localization in two models of Fock-space MBL: fermionic and spin quantum dots. Large-connectivity RRG models serve as an approximation for this class of many-body problems; one of our central goals is to better understand the status of this approximation. Our numerical simulations support the conjecture that, in the limit of a large system, there is a sharp ergodicity-to-localization transition in the quantum-dot models. Furthermore, our results are consistent with the RRG-like scaling of the critical disorder, ${W}_{c}\ensuremath{\sim}mlnm$. While in the golden-rule range the behavior of quantum-dot models is in agreement with analytical predictions based on the RRG model, substantial deviations occur in the precritical regime. Our results indicate that the precritical and critical behavior (as well as the numerical coefficient in the formula for Wc{W}_{c}Wc​) in the quantum-dot models may be different from what one would expect from the RRG-like approximation.
We present a renormalization group (RG) analysis of the problem of Anderson localization on a random regular graph (RRG) which generalizes the RG of Abrahams, Anderson, Licciardello, and Ramakrishnan to infinite-dimensional graphs. The RG equations necessarily involve two parameters (one being the changing connectivity of subtrees), but we show that the one-parameter scaling hypothesis is recovered for sufficiently large system sizes for both eigenstates and spectrum observables. We also explain the nonmonotonic behavior of dynamical and spectral quantities as a function of the system size for values of disorder close to the transition, by identifying two terms in the beta function of the running fractal dimension of different signs and functional dependence. Our theory provides a simple and coherent explanation for the unusual scaling behavior observed in numerical data of the Anderson model on RRG and of many-body localization.
A disordered quantum system can be driven, e.g., by increasing disorder, through the Anderson transition between a delocalized and localized phase. Recently, Anderson localization on treelike graphs has attracted much attention in view of its connections with problems of many-body localization. One of the central questions in this context is that of ergodicity of eigenstates. The authors of this paper perform a numerical study of Anderson transition on random regular graphs (RRGs) with diagonal disorder. The problem can be described as a tight-binding model on a lattice with NNN sites that is locally a tree with constant connectivity. Focusing on the delocalized side of the transition, they show that the data can be interpreted in terms of the finite-size crossover from a small ($N\ensuremath{\ll}{N}_{c}$) to a large ($N\ensuremath{\gg}{N}_{c}$) system, where Nc{N}_{c}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. The results support an analytical prediction that states in the delocalized phase (and at $N\ensuremath{\gg}{N}_{c}$) are ergodic.
Anderson localization on random regular graphs (RRG) attracts much attention in view of its relation to many-body localization. Here, the authors study the statistics of eigenfunctions and energy levels on RRG, focussing on the delocalized (ergodic) phase and the critical regime. Results of exact diagonalization are in outstanding agreement with analytical theory based on the saddle-point analysis of the effective supersymmetric action supplemented by numerical solution of the corresponding self-consistency equation.
We discuss the dependence of the critical properties of the Anderson model on the dimension d in the language of β -function and renormalization group recently introduced in Vanoni et al. [C. Vanoni et al. , Proc. Natl. Acad. Sci. U.S.A. 121 , e2401955121 (2024)] in the context of Anderson transition on random regular graphs. We show how in the delocalized region, including the transition point, the one-parameter scaling part of the β -function for the fractal dimension D 1 evolves smoothly from its d = 2 form, in which β 2 ≤ 0 , to its β ∞ ≥ 0 form, which is represented by the random regular graph (RRG) result. We show how the ϵ = d − 2 expansion and the 1 / d expansion around the RRG result can be reconciled and how the initial part of a renormalization group trajectory governed by the irrelevant exponent y depends on dimensionality. We also show how the irrelevant exponent emerges out of the high-gradient terms of expansion in the nonlinear sigma model and put forward a conjecture about a lower bound for the fractal dimension. The framework introduced here may serve as a basis for investigations of disordered many-body systems and of more general nonequilibrium quantum systems.
The Anderson transition in random graphs has raised great interest, partly out of the hope that its analogy with the many-body localization (MBL) transition might lead to a better understanding of this hotly debated phenomenon. Unlike the latter, many results for random graphs are now well established, in particular, the existence and precise value of a critical disorder separating a localized from an ergodic delocalized phase. However, the renormalization group flow and the nature of the transition are not well understood. In turn, recent works on the MBL transition have made the remarkable prediction that the flow is of Kosterlitz-Thouless type. In this paper, we show that the Anderson transition on graphs displays the same type of flow. Our work attests to the importance of rare branches along which wave functions have a much larger localization length ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ than the one in the transverse direction ${\ensuremath{\xi}}_{\ensuremath{\perp}}$. Importantly, these two lengths have different critical behaviors: ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ diverges with a critical exponent ${\ensuremath{\nu}}_{\ensuremath{\parallel}}=1$, while ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ reaches a finite universal value ${\ensuremath{\xi}}_{\ensuremath{\perp}}^{c}$ at the transition point Wc{W}_{c}Wc​. Indeed, ${\ensuremath{\xi}}_{\ensuremath{\perp}}^{\ensuremath{-}1}\ensuremath{\approx}{{\ensuremath{\xi}}_{\ensuremath{\perp}}^{c}}^{\ensuremath{-}1}+{\ensuremath{\xi}}^{\ensuremath{-}1}$, with $\ensuremath{\xi}\ensuremath{\sim}{(W\ensuremath{-}{W}_{c})}^{\ensuremath{-}{\ensuremath{\nu}}_{\ensuremath{\perp}}}$ associated with a new critical exponent ${\ensuremath{\nu}}_{\ensuremath{\perp}}=\frac{1}{2}$, where $exp(\ensuremath{\xi})$ controls finite-size effects. The delocalized phase inherits the strongly nonergodic properties of the critical regime at short scales, but is ergodic at large scales, with a unique critical exponent $\ensuremath{\nu}=\frac{1}{2}$. This shows a very strong analogy with the MBL transition: the behavior of ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ is identical to that recently predicted for the typical localization length of MBL in a phenomenological renormalization group flow. We demonstrate these important properties for a small-world complex network model and show the universality of our results by considering different network parameters and different key observables of Anderson localization.