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arXiv:1707.05701 · Physical review. B./Physical review. B

Multifractal finite-size scaling at the Anderson transition in the unitary symmetry class

Jakob Lindinger, Alberto Rodríguez

Published Oct 5, 2017• 23 citations
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Abstract

We use multifractal finite-size scaling to perform a high-precision numerical study of the critical properties of the Anderson localization-delocalization transition in the unitary symmetry class, considering the Anderson model including a random magnetic flux. We demonstrate the scale invariance of the distribution of wave-function intensities at the critical point and study its behavior across the transition. Our analysis, involving more than $4\ifmmode\times\else\texttimes\fi{}{10}^{6}$ independently generated wave functions of system sizes up to ${L}^{3}={150}^{3}$, yields accurate estimates for the critical exponent of the localization length, $\ensuremath{\nu}=1.446(1.440,1.452)$, the critical value of the disorder strength, and the multifractal exponents.

DOI
10.1103/physrevb.96.134202

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We use multifractal finite-size scaling to perform a high-precision numerical study of the critical properties of the Anderson localization-delocalization transition in the unitary symmetry class, considering the Anderson model including a random magnetic flux. We demonstrate the scale invariance of the distribution of wave-function intensities at the critical point and study its behavior across the transition. Our analysis, involving more than 4\ifmmode×\else\texttimes\fi1064\ifmmode\times\else\texttimes\fi{}{10}^{6}4\ifmmode×\else\texttimes\fi106 independently generated wave functions of system sizes up to L3=1503{L}^{3}={150}^{3}L3=1503, yields accurate estimates for the critical exponent of the localization length, \ensuremathν=1.446(1.440,1.452)\ensuremath{\nu}=1.446(1.440,1.452)\ensuremathν=1.446(1.440,1.452), the critical value of the disorder strength, and the multifractal exponents.
We describe a new multifractal finite-size scaling (MFSS) procedure and its application to the Anderson localization-delocalization transition. MFSS permits the simultaneous estimation of the critical parameters and the multifractal exponents. Simulations of system sizes up to L3=1203{L}^{3}={120}^{3}L3=1203 and involving nearly 106{10}^{6}106 independent wave functions have yielded unprecedented precision for the critical disorder Wc=16.530(16.524,16.536){W}_{c}=16.530(16.524,16.536)Wc​=16.530(16.524,16.536) and the critical exponent $\ensuremath{\nu}=1.590(1.579,1.602)$. We find that the multifractal exponents ${\ensuremath{\Delta}}_{q}$ exhibit a previously predicted symmetry relation and we confirm the nonparabolic nature of their spectrum. We explain in detail the MFSS procedure first introduced in our Letter [Phys. Rev. Lett. 105, 046403 (2010)] and, in addition, we show how to take account of correlations in the simulation data. The MFSS procedure is applicable to any continuous phase transition exhibiting multifractal fluctuations in the vicinity of the critical point.
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.
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.
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 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.
Power-law random banded matrix ensembles have proven effective in predicting critical properties at finite-dimensional Anderson transitions. Here, the authors extend the use of random matrices to infinite dimensions by introducing ensembles specifically designed to capture infinite-dimensional Anderson transitions. Using both analytical arguments and advanced numerical simulations, they uncover two unique critical behaviors: logarithmic multifractality and critical localization. As shown in the image, logarithmic multifractality displays an algebraic dependence of the large moments of wavefunction amplitudes on the logarithm of system size, characterized by a spectrum of log-multifractal dimensions $d\phantom{\rule{0}{0ex}}q>0$, while the conventional multifractal dimensions Dq{D}_{q}Dq​\hskip-0.22em, which describe scaling with system size, are zero. These insights provide a structured framework for understanding the pronounced finite-size effects and slow dynamics in these Anderson transitions, similar to the many-body localization transition.