We present a review of recent theoretical results concerning the many‐body localization (MBL) phenomenon, with the emphasis on dynamical density correlations and transport quantities. They are shown to be closely related, providing a comprehensive description of the ergodic‐to‐nonergodic transition, consistent with experimental findings. While the focus is set mostly on the one‐dimensional model of interacting spinless fermions, we also present evidence for the absence of full MBL in the one‐dimensional Hubbard model and for the density‐wave decay induced by the inter‐chain coupling. image
We present a review of recent theoretical results concerning the many‐body localization (MBL) phenomenon, with the emphasis on dynamical density correlations and transport quantities. They are shown to be closely related, providing a comprehensive description of the ergodic‐to‐nonergodic transition, consistent with experimental findings. While the focus is set mostly on the one‐dimensional model of interacting spinless fermions, we also present evidence for the absence of full MBL in the one‐dimensional Hubbard model and for the density‐wave decay induced by the inter‐chain coupling. image
We present an introductory review of nonergodic dynamics in interacting many-body quantum systems, focusing on the phenomenon of many-body localization (MBL). We describe aspects of MBL and summarize the evidence for a crossover from the ergodic to the MBL regime in finite systems, using the paradigmatic XXZ model as an example. We then broaden the scope to other models to illustrate the generality of the phenomenon. We briefly touch on the largely unexplored relation between MBL and quantum computing.
We present numerical results within the one-dimensional disordered Hubbard model for several characteristic indicators of the many-body localization (MBL). Considering traditionally studied charge disorder (i.e., the same disorder strength for both spin orientations) we find that even at strong disorder all signatures consistently show that while charge degree of freedom is nonergodic, the spin is delocalized and ergodic. This indicates the absence of the full MBL in the model that has been simulated in recent cold-atom experiments. Full localization can be restored if spin-dependent disorder is used instead.
We examine the standard model of many-body localization (MBL), i.e., the disordered chain of interacting spinless fermions, by representing it as the network in the many-body (MB) basis of noninteracting localized Anderson states. By studying eigenstates of the full Hamiltonian, for strong disorders we find that the dynamics is confined up to very long times to disconnected MB clusters in the Fock space. By keeping only resonant contributions and simplifying the quantum problem to rate equations (REs) for MB states, in analogy with percolation problems, the MBL transition is located via the universal cluster distribution and the emergence of the macroscopic cluster. On the ergodic side, our approximate RE approach to the relaxation processes captures well the diffusion transport, as found for the full quantum model. In a broad transient regime, we find an anomalous, i.e., subdiffusivelike, transport, emerging from weak links between MB states.
We study interacting fermions in one dimension subject to random, uncorrelated onsite disorder, a paradigmatic model of many‐body localization (MBL). This model realizes an interaction‐driven quantum phase transition between an ergodic and a many‐body localized phase, with the transition occurring in the many‐body eigenstates. We propose a single‐particle framework to characterize these phases by the eigenstates (the natural orbitals) and the eigenvalues (the occupation spectrum) of the one‐particle density matrix (OPDM) in individual many‐body eigenstates. As a main result, we find that the natural orbitals are localized in the MBL phase, but delocalized in the ergodic phase. This qualitative change in these single‐particle states is a many‐body effect, since without interactions the single‐particle energy eigenstates are all localized. The occupation spectrum in the ergodic phase is thermal in agreement with the eigenstate thermalization hypothesis, while in the MBL phase the occupations preserve a discontinuity at an emergent Fermi edge. This suggests that the MBL eigenstates are weakly dressed Slater determinants, with the eigenstates of the underlying Anderson problem as reference states. We discuss the statistical properties of the natural orbitals and of the occupation spectrum in the two phases and as the transition is approached. Our results are consistent with the existing picture of emergent integrability and localized integrals of motion, or quasiparticles, in the MBL phase. We emphasize the close analogy of the MBL phase to a zero‐temperature Fermi liquid: in the studied model, the MBL phase is adiabatically connected to the Anderson insulator and the occupation‐spectrum discontinuity directly indicates the presence of quasiparticles localized in real space. Finally, we show that the same picture emerges for interacting fermions in the presence of an experimentally‐relevant bichromatic lattice and thereby demonstrate that our findings are not limited to a specific model. image
Many-body localization (MBL), the disorder-induced localization of interacting particles, signals a breakdown of conventional thermodynamics because MBL systems do not thermalize and show nonergodic time evolution. We experimentally observed this nonergodic evolution for interacting fermions in a one-dimensional quasirandom optical lattice and identified the MBL transition through the relaxation dynamics of an initially prepared charge density wave. For sufficiently weak disorder, the time evolution appears ergodic and thermalizing, erasing all initial ordering, whereas above a critical disorder strength, a substantial portion of the initial ordering persists. The critical disorder value shows a distinctive dependence on the interaction strength, which is in agreement with numerical simulations. Our experiment paves the way to further detailed studies of MBL, such as in noncorrelated disorder or higher dimensions.
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.