Quasiperiodic Potential Research Articles

Critical analysis of the reentrant localization transition in a one-dimensional dimerized quasiperiodic lattice

A re-entrant localization transition has been predicted recently in a one-dimensional quasiperiodic lattice with dimerized hopping between the nearest-neighbour sites (Phys. Rev. Lett. {\bf 126} 106803 (2021)) \cite{PhysRevLett.126.106803}. It has been shown that the interplay between the hopping dimerization and a staggered quasi-periodic disorder manifests two localization transitions through two intermediate phases resulting in four critical points as a function of the quasiperiodic potential. In this paper, we study the phenomenon of the re-entrant localization transition by examining the spectral properties of the states. By performing a systematic finite-size scaling analysis for a fixed value of the hopping dimerization, we obtain accurate critical disorder strengths for different transitions and the associated critical exponents. Moreover, through a multifractal analysis, we study the critical nature of the states across the localization transitions by computing the mass exponents and the corresponding fractal dimensions of the states. Further, we complement the critical nature of the states by computing the Hausdorff dimensions.

Destruction of localization by thermal inclusions: Anomalous transport and Griffiths effects in the Anderson and André-Aubry-Harper models

We discuss and compare two recently proposed toy models for anomalous transport and Griffiths effects in random systems near the Many-Body Localization transitions: the random dephasing model, which adds thermal inclusions in an Anderson Insulator as local Markovian dephasing channels that heat up the system, and the random Gaussian Orthogonal Ensemble (GOE) approach which models them in terms of ensembles of random regular graphs. For these two settings we discuss and compare transport and dissipative properties and their statistics. We show that both types of dissipation lead to similar Griffiths-like phenomenology, with the GOE bath being less effective in thermalising the system due to its finite bandwidth. We then extend these models to the case of a quasi-periodic potential as described by the André-Aubry-Harper model coupled to random thermal inclusions, that we show to display, for large strength of the quasiperiodic potential, a similar phenomenology to the one of the purely random case. In particular, we show the emergence of subdiffusive transport and broad statistics of the local density of states, suggestive of Griffiths like effects arising from the interplay between quasiperiodic localization and random coupling to the baths.

Emergence of multiple localization transitions in a one-dimensional quasiperiodic lattice

Low dimensional quasiperiodic systems exhibit localization transitions by turning all quantum states localized after a critical quasidisorder. While certain systems with modified or constrained quasiperiodic potential undergo multiple localization transitions in one dimension, we predict an emergence of multiple localization transitions without directly imposing any constraints on the quasiperiodic potential. By considering a one-dimensional system described by the Aubry-And\'{r}e (AA) model, we show that an additional staggered onsite potential can drive the system through a series of localization transitions as a function of the staggered potential. Interestingly, we find that the number of localization transitions strongly depends on the strength of the quasiperiodic potential. Moreover, we obtain the signatures of these localization transitions in the expansion dynamics and propose an experimental scheme for their detection in the quantum gas experiment.

Lyapunov exponent, mobility edges, and critical region in the generalized Aubry-André model with an unbounded quasiperiodic potential

In this work, we investigate the Anderson localization problems of the generalized Aubry-Andr\'{e} model (Ganeshan-Pixley-Das Sarma's model) with an unbounded quasi-periodic potential where the parameter $|\alpha|\geq1$. The Lyapunov exponent $\gamma(E)$ and the mobility edges $E_c$ are exactly obtained for the unbounded quasi-periodic potential. With the Lyapunov exponent, we find that there exists a critical region in the parameter $\lambda-E$ plane. The critical region consists of critical states. In comparison with localized and extended states, the fluctuation of spatial extensions of the critical states is much larger. The numerical results show that the scaling exponent of inverse participation ratio (IPR) of critical states $x\simeq0.5$. Furthermore, it is found that the critical indices of localized length $\nu=1$ for bounded ($|\alpha|<1$) case and $\nu=1/2$ for unbounded ($|\alpha|\geq1$) case. The above distinct critical indices can be used to distinguish the localized-extended from localized-critical transitions. At the end, we show that the systems with different $E$ for both cases of $|\alpha|<1$ and $|\alpha|\geq1$ can be classified by the Lyapunov exponent $\gamma(E)$ and Avila's quantized acceleration $\omega(E)$.

Lieb–Robinson bound in one-dimensional inhomogeneous quantum systems

Lieb–Robinson bound (LRB) in one-dimensional noninteracting many-electron systems with the disordered and quasiperiodic on-site potentials is studied numerically. For the short-range hopping system, a logarithmic light cone (i.e. |x|=βlogt+x0) is found in the system with the disordered on-site potential for small time. The coefficient β decreases with the increasing strength of disordered. When time is large, the bound does not change with time (i.e. |x|=C). For the generalized Fibonacci quasiperiodic (GFQ) system, the on-site potential is taken as V or −V according to two kinds of GFQ sequences. It is found that the system has a power-law light cone (i.e. |x|∝tγ, with 0<γ<1). The exponent γ decreases with the increasing V. We also find that γ for the first class of GFQ system is larger than that for the second class of GFQ system with the same V. Finally, the effects of the long-range hopping which decays like r−α with the distance r on LRB are discussed.

Emergence of quasiperiodic Bloch wave functions in quasicrystals

We study the emergence of quasiperiodic Bloch wave functions in quasicrystals, employing the one-dimensional Fibonacci model as a test case. We find that despite the fact that Bloch functions are not eigenfunctions themselves, superpositions of relatively small numbers of nearly degenerate eigenfunctions give rise to extended quasiperiodic Bloch functions. These functions possess the structure of earlier ancestors of the underlying Fibonacci potential, and it is often possible to obtain different ancestors as different superpositions around the same energy. There exists an effective crystal momentum that characterizes these ancestors, which is determined by the mean energy of the superimposed eigenfunctions, giving rise to an effective dispersion curve. We also find that quasiperiodic Bloch functions do emerge as eigenfunctions when weak disorder is introduced into the otherwise perfect quasiperiodic potential. These theoretical results may explain a number of experimental observations, and may have practical consequences on emerging theories of band topology and correlated electrons in quasicrystals.

Topological gap labeling with third Chern numbers in three-dimensional quasicrystals

We study the topological gap labeling of general 3D quasicrystals and we find that every gap in the spectrum is characterized by a set of the third Chern numbers. We show that a quasi-periodic structure has multiple Brillouin zones defined by redundant wavevectors, and the number of states below a gap is quantized as an integer linear combination of volumes of these Brillouin zones. The associated quantum numbers to characterize energy gaps can be expressed as third Chern numbers by considering a formal relationship between an adiabatic charge pumping under cyclic deformation of the quasi-periodic potential and a topological nonlinear electromagnetic response in 6D insulators.

Zonal Estimators for Quasiperiodic Bosonic Many-Body Phases.

In this work, we explore the relevant methodology for the investigation of interacting systems with contact interactions, and we introduce a class of zonal estimators for path-integral Monte Carlo methods, designed to provide physical information about limited regions of inhomogeneous systems. We demonstrate the usefulness of zonal estimators by their application to a system of trapped bosons in a quasiperiodic potential in two dimensions, focusing on finite temperature properties across a wide range of values of the potential. Finally, we comment on the generalization of such estimators to local fluctuations of the particle numbers and to magnetic ordering in multi-component systems, spin systems, and systems with nonlocal interactions.

Real-complex transition driven by quasiperiodicity: A class of non- PT symmetric models

We study a one-dimensional lattice model subject to non-Hermitian quasiperiodic potentials. First, we strictly demonstrate that there exists an interesting dual mapping relation between $|a|&lt;1$ and $|a|&gt;1$ with regard to the potential tuning parameter $a$. The localization property of $|a|&lt;1$ can be directly mapped to that of $|a|&gt;1$, the analytical expression of the mobility edge of $|a|&gt;1$ is therefore obtained through the spectral properties of $|a|&lt;1$. More impressive, we prove rigorously that, even if the phase $\ensuremath{\theta}\ensuremath{\ne}0$ in quasiperiodic potentials, the model becomes non-$\mathcal{PT}$ symmetric, however, there still exists a type of real-complex transition driven by non-Hermitian disorder, which is a class beyond the $\mathcal{PT}$-symmetric class.

Topological superconductors and exact mobility edges in non-Hermitian quasicrystals

We study a class of non-Hermitian topological superconductors described by one-dimensional Aubry-Andr\'e Harper and mosaic quasiperiodic models with $p$-wave superconducting pairing, where the non-Hermiticity is introduced by on-site complex quasiperiodic potentials. We generalize two topological invariants, one is based on the transfer matrix method and the other is the generalized Majorana polarization, to characterize the topological superconducting phases and verify the existence of Majorana zero modes in non-Hermitian quasiperiodic superconductors. By combing the Lyapunov exponent, the fractional dimension of wave functions, and topological invariants, we investigate the localization phenomena, topological superconductivity, and topological phase transitions. In the non-Hermitian Aubry-Andr\'e Harper model with $p$-wave pairing, the system undergoes an extended-critical-localized phase transition with increasing the complex phase. The localization transition is consistent with the topological phase transition and unconventional real-complex transition of eigenenergy. In the non-Hermitian mosaic model, we provide an analytical expression of mobility edges and prove the intrinsic relation between the mobility edges and unconventional real-complex transitions. Our discoveries unveil the richness of topological and localization phenomena in non-Hermitian quasicrystals with $p$-wave pairing.

Finite-temperature phases of trapped bosons in a two-dimensional quasiperiodic potential

We study a system of 2D trapped bosons in a quasiperiodic potential via ab initio Path Integral Monte Carlo simulations, focusing on its finite temperature properties, which have not yet been explored. Alongside the superfluid, normal fluid and insulating phases, we demonstrate the existence of a Bose glass phase, which is found to be robust to thermal fluctuations, up to about half of the critical temperature of the non-interacting system. Local quantities in the trap are characterized by employing zonal estimators, allowing us to trace a phase diagram; we do so for a set of parameters within reach of current experiments with quasi-2D optical confinement.

Mobility edges and reentrant localization induced by superradiance

We study a Bose-Einstein condensate trapped by a ladder lattice in a high-fitness cavity. The ladder lattice is loaded in the &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$x\text-y$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M1.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M1.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; plane and the cavity is along the &lt;i&gt;x&lt;/i&gt; direction. A pump laser shines on atoms from the &lt;i&gt;z&lt;/i&gt; direction. Under the mean-field approximation, we consider the emergence of the quasi-periodic potentials induced by superradiance in the ladder lattice, which is described by &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$\hat{H}_{\text{MF}}=\hat{H}_{\text{Lad}}+\hat{V}_{\text{eff}}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M2.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M2.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; with the effective potential &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$\hat{V}_{\text{eff}}(\alpha)={\displaystyle \sum\nolimits_{i = 1}^{N}}\displaystyle \sum\nolimits_{\sigma = 1,2}\left[\lambda_{\rm{D}}\cos({2\pi\beta i})+U_{\rm{D}}\cos^{2}({2\pi\beta i})\right]\hat{c}^{†}_{i,\sigma}\hat{c}_{i,\sigma}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M3.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M3.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. We find that the quasi-periodic potential can induce the reentrant localization transition and the regime with mobility edges. In the smaller &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$U_{\rm{D}}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M4.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M4.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; case, the system exhibits a localization transition. The transition is associated with an intermediate regime with mobility edges. When &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$U_{\rm{D}}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M5.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M5.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; goes beyond a critical value &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$U_{\rm{D}}^{(\rm c)}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M6.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M6.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, with the increase of &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$\lambda_{\rm{D}}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M7.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M7.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the system undergoes a reentrant localization transition. This indicates that after the first transition, some of the localized eigenstates change back to the extended ones for a range of &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$\lambda_{\rm{D}}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M8.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M8.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. For a larger &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$\lambda_{\rm{D}}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M9.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20212246_M9.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the system experiences the second localization transition, then all states become localized again. Finally, the local phase diagram of the system is also discussed. This work builds a bridge between the reentrant localization and the superradiance, and it provides a new perspective for the reentrant localization.

Boundedness of Solutions for a Bouncing Ball Model with Quasiperiodic Potential

Boundedness of Solutions for a Bouncing Ball Model with Quasiperiodic Potential

Electrical circuit simulation of nonreciprocal Aubry-André models

&lt;sec&gt;The introduction of non-Hermiticity into traditional Hermitian quantum systems generalizes their basic notions and brings about many novel phenomena, e.g., the non-Hermitian skin effect that is exclusive to non-Hermitian systems, attracting enormous attention from almost all branches of physics. Contrary to the quantum platforms, classical systems have the advantages of low cost and mature techniques under room temperature. Among them, the classical electrical circuits are more flexible on simulating quantum tight-binding models in principle with any range of hopping under any boundary conditions in any dimension, and have become a powerful platform for the simulation of quantum matters. In this paper, by constructing an electrical circuit, we simulate by SPICE the static properties of a prototypical non-Hermitian model—the nonreciprocal Aubry-André (AA) model that has the nonreciprocal hopping and on-site quasiperiodic potentials. &lt;/sec&gt;&lt;sec&gt;The paper is organized as follows: Following the introduction, in Sec. II we review in detail the Laplacian formalism of electrical circuits and the mapping to the quantum tight-binding model. Then, in Sec. III, an electrical circuit is proposed with resistors, capacitors, inductors, and the negative impedance converters with current inversion (INICs), establishing a mapping between the circuit's Laplacian and the non-reciprocal AA model's Hamiltonian under periodic boundary conditions (PBCs) or open boundary conditions (OBCs). Especially, the nonreciprocity, the key of this model, is realized by INICs. In Sec IV, based on the mapping, for the proposed circuit under PBCs, we reconstruct the circuit's Laplacian via SPICE by measuring voltage responses of an AC current input at each node. The complex spectrum and its winding number &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$\nu$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220219_M1.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220219_M1.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; can be calculated by the measured Laplacian, which are consistent with the theoretical prediction, showing &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$\nu=\pm 1$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220219_M2.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220219_M2.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; for non-Hermitian topological regimes with complex eigenenergies and extended eigenstates, and &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$\nu=0$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220219_M3.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220219_M3.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; for topologically trivial regimes with real eigenenergies and localized eigenstates. In Sec V, for the circuit under OBCs, a similar method is used for measuring the node distribution of voltage response, which simulates the competition of non-Hermitian skin effects and the Anderson localization, depending on the strength of quasiperiodic potentials; the phase transition points also appear in the inverse participation ratios of voltage responses. &lt;/sec&gt;&lt;sec&gt;During the design process, the parameters of auxiliary resistors and capacitors are evaluated for obtaining stable responses, because the complex eigenfrequecies of the circuits are inevitable under PBCs. Our detailed scheme can directly instruct further potential experiments, and the designing method of the electrical circuit is universal and can in principle be applied to the simulation for other quantum tight-binding models. &lt;/sec&gt;

Minimal families of limit operators

We study two abstract scenarios, where an operator family has a certain minimality property. In both scenarios, it is shown that norm, spectrum and resolvent are the same for all family members. Both abstract settings are illustrated by practically relevant examples, including discrete Schr\odinger operators with periodic, quasiperiodic, almost-periodic, Sturmian and pseudo-ergodic potential. The main tool is the method of limit operators, known from studies of Fredholm operators and convergence of projection methods. We close by connecting this tool to the study of subwords of the operator potential.

Distinguishing an Anderson insulator from a many-body localized phase through space-time snapshots with neural networks

Distinguishing the dynamics of an Anderson insulator from a Many-Body Localized (MBL) phase is an experimentally challenging task. In this work, we propose a method based on machine learning techniques to analyze experimental snapshot data to separate the two phases. We show how to train $3D$ convolutional neural networks (CNNs) using space-time Fock-state snapshots, allowing us to obtain dynamic information about the system. We benchmark our method on a paradigmatic model showing MBL ($t-V$ model with quenched disorder), where we obtain a classification accuracy of $\approx 80 \%$ between an Anderson insulator and an MBL phase. We underline the importance of providing temporal information to the CNNs and we show that CNNs learn the crucial difference between an Anderson localized and an MBL phase, namely the difference in the propagation of quantum correlations. Particularly, we show that the misclassified MBL samples are characterized by an unusually slow propagation of quantum correlations, and thus the CNNs label them wrongly as Anderson localized. Finally, we apply our method to the case with quasi-periodic potential, known as the Aubry-Andr\'e model (AA model). We find that the CNNs have more difficulties in separating the two phases. We show that these difficulties are due to the fact that the MBL phase of the AA model is characterized by a slower information propagation for numerically accessible system sizes.

Finite-size scaling analysis of the many-body localization transition in quasiperiodic spin chains

We analyze the finite-size scaling of the average gap-ratio and the entanglement entropy across the many-body localization (MBL) transition in one dimensional Heisenberg spin-chain with quasi-periodic (QP) potential. By using the recently introduced cost-function approach, we compare different scenarios for the transition using exact diagonalization of systems up to 22 lattice sites. Our findings suggest that the MBL transition in the QP Heisenberg chain belongs to the class of Berezinskii-Kosterlitz-Thouless (BKT) transition, the same as in the case of uniformly disordered systems as advocated in recent studies. Moreover, we observe that the critical disorder strength shows a clear sub-linear drift with the system-size as compared to the linear drift seen in random disordered models, suggesting that the finite-size effects in the MBL transition for the QP systems are less severe than that in the random disordered scenario. Moreover, deep in the ergodic regime, we find an unexpected double-peak structure of distribution of on-site magnetizations that can be traced back to the strong correlations present in the QP potential.

Features of the Motion of Ultracold Atoms in Quasiperiodic Potentials

We consider here quasiperiodic potentials on the plane, which can serve as a “transitional link” between ordered (periodic) and chaotic (random) potentials. As can be shown, in almost any family of quasiperiodic potentials depending on a certain set of parameters, it is possible to distinguish a set (in the parameter space) where, according to a certain criterion, potentials with features of ordered potentials arise, and a set where we have potentials with features of random potentials. These sets complement each other in the complete parameter space, and each of them has its own specific structure. The difference between “ordered” and “chaotic” potentials will manifest itself, in particular, in the transport properties at different energies, which we consider here in relation to systems of ultracold atoms. It should be noted here also that the transport properties of particles in the considered potentials can be accompanied by the phenomena of “partial integrability” inherent in two-dimensional Hamiltonian systems.

Unconventional real-complex spectral transition and Majorana zero modes in nonreciprocal quasicrystals

Non-Hermiticity from nonreciprocal hoppings has recently been shown to induce exotic non-Hermitian phenomena. Here, we study a class of nonreciprocal quasicrystal superconductors described by one-dimensional Kitaev and dimerized Kitaev chains with complex quasiperiodic potentials. We unveil that the quasiperiodic potentials lead to a localization transition and all states become localized, but the nonreciprocal hopping is favorable to the extension of these states. We also uncover the existence of correspondence between the localization transition and the unconventional real-complex transition of the energy spectrum. Moreover, the localization transition takes place accompanied by a topological phase transition, which can be determined by calculating the energy gap closing point and topological invariants. Especially in the dimerized Kitaev chain, three distinguishable regions of the wave function are observed, i.e., an extended region, a localized region, and an intermediate region with mobility edges, which originates from the interplay between the dimerized hopping and the non-Hermitian quasiperiodic potential. In this case, the topological phase transition point is consistent with the boundary between the intermediate and localized regions. Furthermore, the dimerized hopping can induce a second localization transition, where the extended bulk states become localized and then extended again. These unusual properties provide a distinctive paradigm compared with the diagonal quasiperiodic systems.

Detecting ergodic bubbles at the crossover to many-body localization using neural networks

The transition between ergodic and many-body localized phases is expected to occur via an avalanche mechanism, in which \emph{ergodic bubbles} that arise due to local fluctuations in system properties thermalize their surroundings leading to delocalization of the system, unless the disorder is sufficiently strong to stop this process. We propose an algorithm based on neural networks that allows to detect the ergodic bubbles using experimentally measurable two-site correlation functions. Investigating time evolution of the system, we observe a logarithmic in time growth of the ergodic bubbles in the MBL regime. The distribution of the size of ergodic bubbles converges during time evolution to an exponentially decaying distribution in the MBL regime, and a power-law distribution with a thermal peak in the critical regime, supporting thus the scenario of delocalization through the avalanche mechanism. Our algorithm permits to pin-point quantitative differences in time evolution of systems with random and quasiperiodic potentials, as well as to identify rare (Griffiths) events. Our results open new pathways in studies of the mechanisms of thermalization of disordered many-body systems and beyond.