Distinguishing an Anderson Insulator from a Many-Body Localized phase through space-time snapshots with Neural Networks

Florian Kotthoff, Frank Pollmann, Giuseppe De Tomasi
Phys. Rev. 104, 224307 & arXiv:2108.04244

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-André 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.

Giuseppe De Tomasi, Ivan M. Khaymovich
Phys. Rev. Lett. 124, 200602 (2020)

In this work, we built up a bridge between ergodic properties extracted form entanglement measurements and the ones from multifractal analysis. We generalised the work of Don. N. Page  [Phys. Rev. Lett. 71, 1291] for the entanglement entropy, to the case of non-ergodic but extended (NEE) states. In particular, by implementing the NEE states with a new and simple class of random states, which live in a fractal of the Fock space, we compute, both analytically and numerically, its von Neumann/Renyi entropy. Remarkably, we show that the entanglement, both Renyi and von Neumann, entropy  can still show a fully ergodic behaviour, even tough the wave function lives in a vanishing ratio of the full Hilbert space in the thermodynamic limit.

Giuseppe De Tomasi, Daniel Hetterich, Pablo Sala, and Frank Pollmann
Phys. Rev. B 100, 214313 (2019)

We study the t−V disordered spinless fermionic chain in the strong coupling regime, t/V→0. Strong interactions highly hinder the dynamics of the model, fragmenting its Hilbert space into exponentially many blocks in system size. Macroscopically, these blocks can be characterized by the number of new degrees of freedom, which we refer to as movers. We focus on two limiting cases: Blocks with only one mover and the ones with a finite density of movers. The former many-particle block can be exactly mapped to a single-particle Anderson model with correlated disorder in one dimension. As a result, these eigenstates are always localized for any finite amount of disorder. The blocks with a finite density of movers, on the other side, show an MBL transition that is tuned by the disorder strength. Moreover, we provide numerical evidence that its ergodic phase is diffusive at weak disorder. Approaching the MBL transition, we observe sub-diffusive dynamics at finite time scales and find indications that this might be only a transient behavior before crossing over to diffusion.

Giuseppe De Tomasi, Soumya Bera, Antonello Scardicchio, Ivan M. Khaymovich.
Phys. Rev. B 101, 100201 (R) (2020)

We study the finite-time dynamics of an initially localized wave-packet in the Anderson model on the random regular graph (RRG). Considering the full probability distribution  of a particle to be at some distance x from the initial state at time t, we give evidence that dynamics is sub-diffusively over a range of disorder strengths, wider than a putative non-ergodic phase. We provide a detailed analysis of the propagation in space-time (x,t) domain, identifying four different regimes. These regimes in (x,t) are determined by the position of a wave-front, which moves sub-diffusively to the most distant sites. We support our numerical results by a self-consistent semi-classical picture of wavepacket. Importantly, the Anderson model on the RRG can be considered as proxy of the many-body localization transition (MBL) on the Fock space of a generic interacting system. In the final discussion, we outline possible implications of our findings for MBL.

Giuseppe De Tomasi, Frank Pollmann, and Markus Heyl.
Phys. Rev. B 99, 241114 (R) (2019)

We introduce a method to efficiently study the dynamical properties of many-body localized systems in the regime of strong disorder and weak interactions. Our method reproduces qualitatively and quantitatively the real-time evolution with a polynomial effort in system size and independent of the desired time scales. We use our method to study quantum information propagation, correlation functions, and temporal fluctuations in one- and two-dimensional MBL systems. Moreover, we outline strategies for a further systematic improvement of the accuracy and we point out relations of our method to recent attempts to simulate the real-time dynamics of quantum many-body systems in classical or artificial neural networks.

Giuseppe De Tomasi, Soumya Bera, Jens H. Bardarson, Frank Pollmann.
Phys. Rev. Lett. 118, 016804 (2017)

We demonstrate that the quantum mutual information (QMI) is a useful probe to study many-body localization (MBL). First, we focus on the detection of a metal--insulator transition for two different models, the noninteracting Aubry-André-Harper model and the spinless fermionic disordered Hubbard chain. We find that the QMI in the localized phase decays exponentially with the distance between the regions traced out, allowing us to define a correlation length, which converges to the localization length in the case of one particle. Second, we show how the QMI can be used as a dynamical indicator to distinguish an Anderson insulator phase from an MBL phase. By studying the spread of the QMI after a global quench from a random product state, we show that the QMI does not spread in the Anderson insulator phase but grows logarithmically in time in the MBL phase.