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arXiv:1904.08869 · Physical Review Research

Two critical localization lengths in the Anderson transition on random graphs

Ignacio García-Mata, John Martin, R. Dubertrand, O. Giraud, Bertrand Georgeot, Gabriel Lemarié

Published Jan 21, 2020• 86 citations
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Abstract

This paper shows that the Anderson transition on random graphs has two critical localization lengths, which control the critical behavior of specific observables, and are associated with two different critical exponents (the known ${\ensuremath{\nu}}_{\ensuremath{\parallel}}=1$ for the average localization length ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ and the new ${\ensuremath{\nu}}_{\ensuremath{\perp}}=0.5$ for the typical localization length ${\ensuremath{\xi}}_{\ensuremath{\perp}}$). The behavior we find for ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ is identical to the recent predictions for the many-body localization transition, strongly suggesting that both transitions belong to the same universality class.

DOI
10.1103/physrevresearch.2.012020

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This paper shows that the Anderson transition on random graphs has two critical localization lengths, which control the critical behavior of specific observables, and are associated with two different critical exponents (the known \ensuremathν\ensuremath∥=1{\ensuremath{\nu}}_{\ensuremath{\parallel}}=1\ensuremathν\ensuremath∥​=1 for the average localization length \ensuremathξ\ensuremath∥{\ensuremath{\xi}}_{\ensuremath{\parallel}}\ensuremathξ\ensuremath∥​ and the new \ensuremathν\ensuremath⊥=0.5{\ensuremath{\nu}}_{\ensuremath{\perp}}=0.5\ensuremathν\ensuremath⊥​=0.5 for the typical localization length \ensuremathξ\ensuremath⊥{\ensuremath{\xi}}_{\ensuremath{\perp}}\ensuremathξ\ensuremath⊥​). The behavior we find for \ensuremathξ\ensuremath⊥{\ensuremath{\xi}}_{\ensuremath{\perp}}\ensuremathξ\ensuremath⊥​ is identical to the recent predictions for the many-body localization transition, strongly suggesting that both transitions belong to the same universality class.
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.
We perform a thorough and complete analysis of the Anderson localization transition on several models of random graphs with regular and random connectivity. The unprecedented precision and abundance of our exact diagonalization data (both spectra and eigenstates), together with new finite size scaling and statistical analysis of the graph ensembles, unveils a universal behavior which is described by two simple, integer, scaling exponents. A by-product of such analysis is a reconciliation of the tension between the results of perturbation theory coming from strong disorder and earlier numerical works, which seemed to suggest that there should be a non-ergodic region above a given value of disorder WEW_{E}WE​ which is strictly less than the Anderson localization critical disorder WCW_CWC​, and that of other works which suggest that there is no such region. We find that, although no separate WEW_{E}WE​ exists from WCW_CWC​, the length scale at which fully developed ergodicity is found diverges like ∣W−WC∣−1|W-W_C|^{-1}∣W−WC​∣−1, while the critical length over which delocalization develops is ∼∣W−WC∣−1/2\sim |W-W_C|^{-1/2}∼∣W−WC​∣−1/2. The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. In addition, by looking at eigenstates and studying leading and sub-leading terms in system size-dependence of participation entropies, we show that the former contain information about the non-ergodicity volume which becomes non-trivial already deep in the delocalized regime. We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition.
We perform a thorough and complete analysis of the Anderson localization transition on several models of random graphs with regular and random connectivity. The unprecedented precision and abundance of our exact diagonalization data (both spectra and eigenstates), together with new finite size scaling and statistical analysis of the graph ensembles, unveils a universal behavior which is described by two simple, integer, scaling exponents. A by-product of such analysis is a reconciliation of the tension between the results of perturbation theory coming from strong disorder and earlier numerical works, which seemed to suggest that there should be a non-ergodic region above a given value of disorder W_{E} W E which is strictly less than the Anderson localization critical disorder W_C W C , and that of other works which suggest that there is no such region. We find that, although no separate W_{E} W E exists from W_C W C , the length scale at which fully developed ergodicity is found diverges like |W-W_C|^{-1} | W − W C | − 1 , while the critical length over which delocalization develops is \sim |W-W_C|^{-1/2} ∼ | W − W C | − 1 / 2 . The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. In addition, by looking at eigenstates and studying leading and sub-leading terms in system size-dependence of participation entropies, we show that the former contain information about the non-ergodicity volume which becomes non-trivial already deep in the delocalized regime. We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition.
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.
We investigate the relation between two complementary descriptions of the Anderson transition on small-world random graphs: finite-size scaling of eigenfunction moments and renormalization-group (RG) flows of multifractal dimensions DqD_qDq​. The RG flows recover the qualitative structure previously reported for the information dimension D1D_1D1​, while their extension to other DqD_qDq​ with smaller moment orders q<1/2q<1/2q<1/2 confirms the strongly multifractal nature of the localized phase characterized by a clear separation between multifractal and localized behaviors at strong disorder. On the other hand, we show that we can construct another beta function characterizing the flow of eigenfunction moments, whose collapse onto a single function across different disorder strengths provides a clear confirmation of single-parameter scaling property. These results indicate that the family of RG trajectories observed for the information dimension D1D_1D1​ need not imply two-parameter scaling and Kosterlitz-Thouless like critical behavior. To characterize these properties, one should consider different observables, such as large q>1/2q>1/2q>1/2 and small q<1/2q<1/2q<1/2 eigenfunction moments, which are associated with distinct critical behaviors, in particular different critical exponents.
The Anderson transition on random graphs is a paradigm for understanding high-dimensional quantum phase transitions driven by disorder and closely mirrors several features of many-body localization. In this Letter, we introduce a unitary Anderson model on small-world graphs, enabling large-scale, long-time simulations of wave-packet dynamics. This allows us to uncover logarithmically slow critical dynamics, two distinct localization times, and a crossover to ergodic diffusion. Finite-time scaling reveals distinct critical exponents, establishing a dynamical universality class for this exotic transition. Our model opens the way to an experimental realization of the Anderson transition on random graphs in quantum simulators. By combining disorder, high dimensionality, and dynamics, our approach provides a general framework for probing universal features of out-of-equilibrium quantum phase transitions, including those relevant to many-body localization.