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arXiv:1905.03099 · Physical review. E

Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems

Arnd Bäcker, Masudul Haque, Ivan M. Khaymovich

Published Sep 11, 2019• 91 citations

Abstract

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.

DOI: 10.1103/physreve.100.032117

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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

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\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.

Fractal dimensions have been used as a quantitative measure for structure of eigenstates of quantum many-body systems, useful for comparison to random matrix theory predictions or to distinguish many-body localized systems from chaotic ones. For chaotic systems at midspectrum the states are expected to be ``ergodic'', infinite temperature states with all fractal dimensions approaching 1 in the thermodynamic limit. However, when moving away from midspectrum, the states develop structure, as they are expected to follow the eigenstate thermalization hypothesis, with few-body observables predicted by a finite-temperature ensemble. We discuss how this structure of the observables can be used to bound the fractal dimensions from above, thus explaining their typical arc-shape over the energy spectrum. We then consider how such upper bounds act as a proxy for the fractal dimension over the many-body localization transition, thus formally connecting the single-particle and Fock space pictures discussed in the literature.

It is commonly expected that for quantum chaotic many body systems, the statistical properties approach those of random matrices when increasing the system size. We demonstrate for various kicked spin-1/2 chain models that the average eigenstate entanglement indeed approaches the random matrix result. However, the distribution of the eigenstate entanglement differs significantly. While for autonomous systems such deviations are expected, they are surprising for the more scrambling kicked systems. Similar deviations occur in a tensor-product random matrix model with all-to-all interactions. Therefore, we attribute the origin of the deviations for the kicked spin-1/2 chain models to the tensor-product structure of the Hilbert spaces. As a consequence, this would mean that such many body systems cannot be described by the standard random matrix ensembles.

Finite-size effects have been a major and justifiable source of concern for studies of many-body localization, and several works have been dedicated to the subject.In this Letter, however, we discuss yet another crucial problem that has received much less attention, that of the lack of self-averaging and the consequent danger of reducing the number of random realizations as the system size increases.By taking this into account and considering ensembles with a large number of samples for all system sizes analyzed, we find that the generalized dimensions of the eigenstates of the disordered Heisenberg spin-1/2 chain close to the transition point to localization are described remarkably well by an exact analytical expression derived for the noninteracting Fibonacci lattice, thus providing an additional tool for studies of many-body localization.

The dynamical scaling for statistics of critical multifractal eigenstates proposed by Chalker is analytically verified for the critical random matrix ensemble in the limit of strong multifractality controlled by the small parameter

b⪡1

. The power-law behavior of the quantum return probability ${P}_{N}(\ensuremath{\tau})$ as a function of the matrix size

N

or time $\ensuremath{\tau}$ is confirmed in the limits $\ensuremath{\tau}/N\ensuremath{\rightarrow}\ensuremath{\infty}$ and $N/\ensuremath{\tau}\ensuremath{\rightarrow}\ensuremath{\infty}$, respectively, and it is shown that the exponents characterizing these power laws are equal to each other up to the order

{b}^{2}

. The corresponding analytical expression for the fractal dimension

{d}_{2}

is found.

Fractal dimensions are tools for probing the structure of quantum states and identifying whether they are localized or delocalized in a given basis. These quantities are commonly extracted through finite-size scaling, which limits the analysis to relatively small system sizes. In this work, we demonstrate that the correlation fractal dimension

D_2

can be directly obtained from the long-time dynamics of interacting many-body quantum systems. Specifically, we show that it coincides with the exponent of the power-law decay of the time-averaged survival probability, defined as the fidelity between an initial state and its time-evolved counterpart. This dynamical approach avoids the need for scaling procedures and enables access to larger systems than those typically reachable via exact diagonalization. We test the method on various random matrix ensembles, including full random matrices, the Rosenzweig-Porter model, and power-law banded random matrices, and extend the analysis to interacting many-body systems described by the one-dimensional Aubry-André model and the disordered spin-1/2 Heisenberg chain. In the case of full random matrices, we also derive an analytical expression for the entire evolution of the time-averaged survival probability.