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arXiv:2505.23910

Dynamical detection of extended nonergodic states in many-body quantum systems

David A. Zarate-Herrada, Isaías Vallejo-Fabila, Léa Ferreira dos Santos, Eduardo Jonathan Torres-Herrera

Published May 29, 2025• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2505.23910

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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 D2D_2D2​ 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.
Models with correlated disorder are rather common in physics. In some of them, like the Aubry-André (AA) model, the localization phase diagram can be found from the (self)duality with respect to the Fourier transform. In others, like the all-to-all translation-invariant Rosenzweig-Porter (TI RP) ensemble or the many-body localization (MBL), one needs to develop more sophisticated and usually phenomenological methods to obtain the localization transition. In addition, such models contain not only localization, but also the ergodicity-breaking transition in the Hilbert-space eigenstate structure, giving way to the nonergodic (i.e., MBL) phase of states with nontrivial fractal dimensions D q > 0 . In this work, we suggest a method to calculate both the above transitions and a lower bound to the fractal dimensions D 2 and D ∞ , relevant for the physical observables. To verify this method, we apply it to the class of long-range (self-)dual models, interpolating between a tight-binding ballistic model and TI RP via both power-law dependencies of the hopping terms and the on-site disorder correlations, and, thus, being out of the validity range of the previously developed methods. We show that the interplay of the correlated disorder and the power-law decaying hopping terms leads to the emergence of the two types of nonergodic phases in the entire range of parameters, even without having any quasiperiodicity of the AA potential. The analytical results of the above method are in full agreement with the extensive numerical calculations.
We study numerically and analytically the quench dynamics of isolated many-body quantum systems. Using full random matrices from the Gaussian orthogonal ensemble, we obtain analytical expressions for the evolution of the survival probability, density imbalance, and out-of-time-ordered correlator. They are compared with numerical results for a one-dimensional-disordered model with two-body interactions and shown to bound the decay rate of this realistic system. Power-law decays are seen at intermediate times, and dips below the infinite time averages (correlation holes) occur at long times for all three quantities when the system exhibits level repulsion. The fact that these features are shared by both the random matrix and the realistic disordered model indicates that they are generic to nonintegrable interacting quantum systems out of equilibrium. Assisted by the random matrix analytical results, we propose expressions that describe extremely well the dynamics of the realistic chaotic system at different time scales.
We investigate how the dynamical fluctuations of many-body quantum systems out of equilibrium can be mitigated when they are opened to a dephasing environment. We consider the survival probability (spectral form factor with a filter) evolving under different kinds of random matrices and under a spin-1/2 model with weak and strong disorder. In isolated many-body quantum systems, the survival probability is non-self-averaging at any timescale, that is, the relative variance of its fluctuations does not decrease with system size. By opening the system, we find that the fluctuations are always reduced, but self-averaging can only be ensured away from critical points. Self-averaging is achieved for the long-time dynamics of full random matrices, power-law banded random matrices deep in the delocalized phase, and the Rosenzweig-Porter ensemble in all phases except at the localization transition point. For the spin model, the survival probability becomes self-averaging only in the chaotic regime provided the initial states are in the middle of the spectrum. Overall, a strongly non-self-averaging survival probability in open systems is an indicator of criticality.
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.
This work supports the existence of extended nonergodic states in the intermediate region between the chaotic (thermal) and the many‐body localized phases. These states are identified through an extensive analysis of static and dynamical properties of a finite one‐dimensional system with onsite random disorder. The long‐time dynamics is particularly sensitive to changes in the spectrum and in the structures of the eigenstates. The study of the evolution of the survival probability, Shannon information entropy, and von Neumann entanglement entropy enables the distinction between the chaotic and the intermediate region. image
The isolated one-dimensional Heisenberg model with static random magnetic fields has become paradigmatic for the analysis of many-body localization. Here, we study the dynamics of this system initially prepared in a highly-excited nonstationary state. Our focus is on the probability for finding the initial state later in time, the so-called survival probability. Two distinct behaviors are identified before equilibration. At short times, the decay is very fast and equivalent to that of clean systems. It subsequently slows down and develops a power-law behavior with an exponent that coincides with the multifractal dimension of the eigenstates.