We explore the limitations of using imbalance dynamics as a diagnostic tool for many-body localization (MBL) and show that spatial averaging can mask important microscopic features. Focusing on the strongly disordered regime of the random-field XXZ chain, we use state-of-the-art numerical techniques (Krylov time evolution and full diagonalization) to demonstrate that site-resolved spin autocorrelators reveal a rich and complex dynamical behavior that is obscured by the imbalance observable. By analyzing the time evolution and infinite-time limits of these local probes, we reveal resonant structures and rare local instabilities within the MBL phase. These numerical findings are supported by an analytical, few-site toy model that captures the emergence of a multiple-peak structure in local magnetization histograms, which is a hallmark of local resonances. These few-body local effects provide a more detailed understanding of ergodicity-breaking dynamics, and also allow us to explain the finite-size effects of long-time imbalance, and its sensitivity to the initial conditions in quench protocols. Overall, our experimentally testable predictions highlight the necessity of a refined, site-resolved approach to fully understand the complexities of MBL and its connection to rare-region effects.
We explore the limitations of using imbalance dynamics as a diagnostic tool for many-body localization (MBL) and show that spatial averaging can mask important microscopic features. Focusing on the strongly disordered regime of the random-field XXZ chain, we use state-of-the-art numerical techniques (Krylov time evolution and full diagonalization) to demonstrate that site-resolved spin autocorrelators reveal a rich and complex dynamical behavior that is obscured by the imbalance observable. By analyzing the time evolution and infinite-time limits of these local probes, we reveal resonant structures and rare local instabilities within the MBL phase. These numerical findings are supported by an analytical, few-site toy model that captures the emergence of a multiple-peak structure in local magnetization histograms, which is a hallmark of local resonances. These few-body local effects provide a more detailed understanding of ergodicity-breaking dynamics, and also allow us to explain the finite-size effects of long-time imbalance, and its sensitivity to the initial conditions in quench protocols. Overall, our experimentally testable predictions highlight the necessity of a refined, site-resolved approach to fully understand the complexities of MBL and its connection to rare-region effects.
Motivated by recent debates around the many-body localization (MBL) problem, and in particular its stability against systemwide resonances, we investigate long-distance spin-spin correlations across the phase diagram of the random-field XXZ model, with a particular focus on the strong disorder regime. Building on state-of-the-art shift-invert diagonalization techniques, we study the high-energy behavior of transverse and longitudinal correlation functions, computed at the largest possible distance, for a broad range of disorder and interaction strengths. Our results show that while transverse correlations display a fairly stable exponential decay over the entire XXZ phase diagram, longitudinal correlations exhibit markedly different behavior, revealing distinct physical regimes. More precisely, we identify an intermediate disorder region where standard observables show well-converged MBL behavior [Colbois et al., Phys. Rev. Lett. 133, 116502 (2024)] while the distributions of longitudinal correlations reveal unexpected fat tails toward large values. These rare events strongly influence the average decay of longitudinal correlations, which we find to be algebraic in a broad region inside the supposed MBL phase, whereas the typical decay remains mostly exponential. At stronger disorder and weaker interactions, this intermediate regime is replaced by a more conventional exponential decay with short correlation lengths for both typical and average correlators, as expected for standard localization. Our findings shed light on the systemwide instabilities and raise important questions about the impact of such rare but large long-range correlations on the stability of the MBL phase. Finally, we discuss the possible fate of the intermediate region in the context of recent perspectives in the field.
We study time dynamics of 1D disordered Heisenberg spin-1/2 chain focusing on a regime of large system sizes and a long time evolution. This regime is relevant for observation of many-body localization (MBL), a phenomenon that is expected to freeze the dynamics of the system and prevent it from reaching thermal equilibrium. Performing extensive numerical simulations of the imbalance, a quantity often employed in the experimental studies of MBL, we show that the regime of a slow power-law decay of imbalance persists to disorder strengths exceeding by at least a factor of 2 the current estimates of the critical disorder strength for MBL. Even though we investigate time evolution up to few thousands tunneling times, we observe no signs of the saturation of imbalance that would suggest freezing of system dynamics and provide a smoking gun evidence of MBL. We demonstrate that the situation is qualitatively different when the disorder is replaced by a quasiperiodic potential. In this case, we observe an emergence of a pattern of oscillations of the imbalance that is stable with respect to changes in the system size. This suggests that the dynamics of quasiperiodic systems remain fully local at the longest time scales we reach provided that the quasiperiodic potential is sufficiently strong. Our study identifies challenges in an unequivocal experimental observation of the phenomenon of MBL.
Abstract Statistical mechanics provides a framework for describing the physics of large, complex many-body systems using only a few macroscopic parameters to determine the state of the system. For isolated quantum many-body systems, such a description is achieved via the eigenstate thermalization hypothesis (ETH), which links thermalization, ergodicity and quantum chaotic behavior. However, tendency towards thermalization is not observed at finite system sizes and evolution times in a robust many-body localization (MBL) regime found numerically and experimentally in the dynamics of interacting many-body systems at strong disorder. Although the phenomenology of the MBL regime is well-established, the central question remains unanswered: under what conditions does the MBL regime give rise to an MBL phase , in which the thermalization does not occur even in the asymptotic limit of infinite system size and evolution time? This review focuses on recent numerical investigations aiming to clarify the status of the MBL phase, and it establishes the critical open questions about the dynamics of disordered many-body systems. The last decades of research have brought an unprecedented new variety of tools and indicators to study the breakdown of ergodicity, ranging from spectral and wave function measures, matrix elements of observables, through quantities probing unitary quantum dynamics, to transport and quantum information measures. We give a comprehensive overview of these approaches and attempt to provide a unified understanding of their main features. We emphasize general trends towards ergodicity with increasing length and time scales, which exclude naive single-parameter scaling hypothesis, necessitate the use of more refined scaling procedures, and prevent unambiguous extrapolations of numerical results to the asymptotic limit. Providing a concise description of numerical methods for studying ETH and MBL, we explore various approaches to tackle the question of the MBL phase. Persistent finite size drifts towards ergodicity consistently emerge in quantities derived from eigenvalues and eigenvectors of disordered many-body systems. The drifts are related to continuous inching towards ergodicity and non-vanishing transport observed in the dynamics of many-body systems, even at strong disorder. These phenomena impede the understanding of microscopic processes at the ETH-MBL crossover. Nevertheless, the abrupt slowdown of dynamics with increasing disorder strength provides premises suggesting the proximity of the MBL phase. This review concludes that the questions about thermalization and its failure in disordered many-body systems remain a captivating area open for further explorations.
Despite the exponentially large amount of information required in the quantum description of many-body systems, finite size numerical simulations have paved the way for recent progress in the understanding of quantum phases of matter both in and out-of-equilibrium. \n \nIn this spirit, using exact diagonalization for a small number of spins ~ 20, first, we investigate the fate of both ergodic quantum many-body systems that thermalize at long times and systems with quenched disorder that exhibit many-body localization (MBL) and fail to thermalize (due to an extensive set of local conserved degrees of freedom) under periodic driving. Ergodic systems always delocalize in energy space and heat up to infinite temperature, for both local driving, in which the time dependence of the Hamiltonian is restricted to a few contiguous degrees of freedom, and global driving. On the other hand, MBL systems remain localized at finite energy density for local driving, while in the case of global driving these systems can either heat up to infinite temperature or remain localized depending on the parameters of the drive. Underlying the latter is the emergence of an effective time independent MBL Hamiltonian describing the long time dynamics of the system. These numerical predictions have recently been verified experimentally in a one dimensional quasi-disordered optical lattice of interacting fermions wherein unitary time evolution can be experimentally probed. \n \nSubsequently, we study the stability of MBL systems with respect to another perturbation, in this case by thermal inclusions, which remains poorly understood. We consider a simple model to address this question: a two level system interacting with 𝑁 localized bits subject to random fields. In the large 𝑁 limit, the localized bits are thermalized for any nonzero ratio of interactions to disorder strength. Moreover, at finite 𝑁, in the transition region, the single-site eigenstate entanglement entropies exhibit bimodal distributions, so that localized bits are either "on" (strongly entangled) or "off" (weakly entangled) in eigenstates. The clusters of "on" bits vary significantly between eigenstates of the same sample, which provides evidence for a heterogenous discontinuous transition out of the localized phase in single site observables. These results also imply the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions 𝑑 that are high but finite. \n \nThe second part of this thesis is motivated by the recent breakthroughs of artificial intelligence in domains such as computer vision involving highly dimensional data of complexity not unlike the one found in the description of many-body systems. Using Monte Carlo simulations to generate synthetic data, we find that kernel methods, a class of machine learning algorithms, can learn the mathematical \nform of physical discriminators associated with phase transitions, such as order parameters and Hamiltonian constraints, for a set of two dimensional spin models: the ferromagnetic Ising model, a conserved-order-parameter Ising model, and the Ising gauge theory. \n \nOur work shows that the combination of numerical simulations with the ability of machine learning algorithms for automatic feature detection hold promise to become standard tools in the exploration and characterization of different phases of matter in condensed matter and other many-body systems.
The effect of rare system-wide resonances in the many-body localization (MBL) transition has recently attracted significant attention. They are expected to play a prominent role in the stability of the MBL phase, prompting the development of new theoretical frameworks to properly account for their statistical weight. We employ a method based on an analogy with mean-field disordered glassy systems to characterize the statistics of transmission amplitudes between distant many-body configurations in Hilbert space, and apply it to the random-field XXZ spin chain. By introducing a Lagrange multiplier, which formally plays the role of an effective temperature controlling the influence of extreme outliers in the heavy-tailed distribution of propagators, we identify three distinct regimes: (i) an ergodic phase with uniform spreading in Hilbert space, (ii) an intermediate regime where delocalization is driven by rare, disorder-dependent long-range resonances, and (iii) a robust MBL phase where such resonances cannot destabilize localization. We derive a finite-size phase diagram in the disorder--interaction plane both in the spin and in the Anderson basis that quantitatively agrees with recent numerical results based on real-space spin-spin correlation functions. We further demonstrate that even infinitesimal interactions can destroy the Anderson insulator at finite disorder, with the critical disorder remaining finite down to small interaction strengths. By visualizing resonant transmission pathways on the Hilbert space graph, we provide a complementary perspective to real-space and spectral probes, revealing how the destabilization of the MBL phase at finite sizes stems from the emergence of resonant paths that become progressively rarer and shorter-ranged deep in the localized phase.
Disordered quantum systems undergoing a many-body localization (MBL) transition fail to reach thermal equilibrium under their own dynamics. Distinguishing between asymptotically localized or delocalized dynamics based on numerical results is, however, nontrivial due to finite-size effects. Numerical linked cluster expansions (NLCE) provide a means to tackle quantum systems directly in the thermodynamic limit but are challenging for models without translational invariance. Here we demonstrate that NLCE provides a powerful tool to explore MBL by simulating quench dynamics in disordered spin-1/2 two-leg ladders and Fermi-Hubbard chains. Combining NLCE with an efficient real-time evolution of pure states, we obtain converged results for the decay of the imbalance on long time scales and show that, especially for intermediate disorder below the putative MBL transition, NLCE outperforms direct simulations of finite systems with open or periodic boundaries. Furthermore, while spin is delocalized even in strongly disordered Hubbard chains with frozen charge, we unveil that an additional tilted potential leads to a drastic slowdown of the spin imbalance and nonergodic behavior on accessible times. Our work sheds light on MBL in systems beyond the well-studied disordered Heisenberg chain and emphasizes the usefulness of NLCE for this purpose.