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arXiv:2405.10975 · Physical review. B./Physical review. B

Describing the critical behavior of the Anderson transition in infinite dimension by random-matrix ensembles: Logarithmic multifractality and critical localization

Weitao Chen, Olivier Giraud, Jiangbin Gong, Gabriel Lemarié

Published Jul 31, 2024• 3 citations
arXivHTMLPDF

Abstract

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 ${D}_{q}$\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.

DOI
10.1103/physrevb.110.014210

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Through a combination of rigorous analytical derivations and extensive numerical simulations, this work reports an exotic multifractal behavior, dubbed “logarithmic multifractality,” in effectively infinite-dimensional systems undergoing the Anderson transition. In contrast to conventional multifractality observed in finite dimensions, logarithmic multifractality at infinite dimension introduces an algebraic behavior with respect to the logarithm of system size or time. We demonstrate this phenomenon across eigenstate statistics, spatial correlations, and wave packet dynamics. Our findings offer crucial insights into strong finite-size effects and slow dynamics in complex systems undergoing the Anderson transition, such as the many-body localization transition. Published by the American Physical Society 2024
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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.
Abstract For short-ranged disordered quantum models in one dimension, the many-body-localization is analyzed via the adaptation to the many-body context (Serbyn et al 2015 Phys. Rev . X 5 041047) of the Thouless point of view on the Anderson transition: the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (many-body-localized phase with an area-law entanglement). The central object is thus the level of hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the strong-multifractality spectrum f ( 0 ⩽ α ⩽ 2 ) = α 2 , well known in the context of Anderson localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. Finally, the possibility of a delocalized non-ergodic phase near criticality is discussed.
For short-ranged disordered quantum models in one dimension, the Many-Body-Localization is analyzed via the adaptation to the Many-Body context [M. Serbyn, Z. Papic and D.A. Abanin, PRX 5, 041047 (2015)] of the Thouless point of view on the Anderson transition : the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (Delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (Many-Body-Localized Phase with an area-law entanglement). The central object is thus the level of Hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the Strong-Multifractality Spectrum f(0≤α≤2)=α2f(0 \leq \alpha \leq 2)=\frac{\alpha}{2}f(0≤α≤2)=2α​, well known in the context of Anderson Localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. As a consequence, the level statistics at criticality is the closest possible to the Poisson statistics of the Localized phase, and the Delocalized phase near the critical point is not ergodic.
Multifractal dimensions allow for characterizing the localization properties of states in complex quantum systems. For ergodic states the finite-size versions of fractal dimensions converge to unity in the limit of large system size. However, the approach to the limiting behavior is remarkably slow. Thus, an understanding of the scaling and finite-size properties of fractal dimensions is essential. We present such a study for random matrix ensembles, and compare with two chaotic quantum systems-the kicked rotor and a spin chain. For random matrix ensembles we analytically obtain the finite-size dependence of the mean behavior of the multifractal dimensions, which provides a lower bound to the typical (logarithmic) averages. We show that finite statistics has remarkably strong effects, so that even random matrix computations deviate from analytic results (and show strong sample-to-sample variation), such that restoring agreement requires exponentially large sample sizes. For the quantized standard map (kicked rotor) the multifractal dimensions are found to follow the random matrix predictions closely, with the same finite statistics effects. For a XXZ spin-chain we find significant deviations from the random matrix prediction-the large-size scaling follows a system-specific path towards unity. This suggests that local many-body Hamiltonians are "weakly ergodic," in the sense that their eigenfunction statistics deviate from random matrix theory.