While the prethermal regime of random many-body localized (MBL) systems is dominated by accidental many-body resonances, another class of resonances, originating from the underlying potential structure, is expected in large-size deterministic systems. It is known that this class of resonances can lead to single-particle critical phases that are neither localized nor extended, but the consequences in interacting systems remain unclear. In this work, we construct an asymptotically solvable model of a one-dimensional nearest-neighbor interacting spin chain, whose spatial structure induces a hierarchy of mirror-like many-body resonances. We derive two phases in the thermodynamic limit, characterized by the satisfaction and violation of a version of the weak eigenstate thermalization hypothesis (ETH). While these two phases are similar to the usual MBL and ETH phases, there exist rare eigenstates that behave like the opposite phase, interpreted as many-body scars and inverted scars. Surprisingly, the two phases can be separated by a finite-temperature phase transition, corresponding to a thermodynamic many-body mobility edge, which was often believed to be impossible. Our results also suggest the existence of delocalized rare regions in an otherwise-localized interacting Aubry-André model, even if there are no low-disorder regions like those in random systems. This challenges the common belief that there is no avalanche instability in quasiperiodic MBL.
While the prethermal regime of random many-body localized (MBL) systems is dominated by accidental many-body resonances, another class of resonances, originating from the underlying potential structure, is expected in large-size deterministic systems. It is known that this class of resonances can lead to single-particle critical phases that are neither localized nor extended, but the consequences in interacting systems remain unclear. In this work, we construct an asymptotically solvable model of a one-dimensional nearest-neighbor interacting spin chain, whose spatial structure induces a hierarchy of mirror-like many-body resonances. We derive two phases in the thermodynamic limit, characterized by the satisfaction and violation of a version of the weak eigenstate thermalization hypothesis (ETH). While these two phases are similar to the usual MBL and ETH phases, there exist rare eigenstates that behave like the opposite phase, interpreted as many-body scars and inverted scars. Surprisingly, the two phases can be separated by a finite-temperature phase transition, corresponding to a thermodynamic many-body mobility edge, which was often believed to be impossible. Our results also suggest the existence of delocalized rare regions in an otherwise-localized interacting Aubry-André model, even if there are no low-disorder regions like those in random systems. This challenges the common belief that there is no avalanche instability in quasiperiodic MBL.
We introduce a multi-scale diagonalization scheme to study the transition between the many-body localized and the ergodic phase in disordered quantum chains. The scheme assumes a sharp dichotomy between subsystems that behave as localized and resonant spots that obey the Eigenstate Thermalization Hypothesis (ETH). We establish a set of microscopic principles defining the diagonalization scheme, and use them to numerically study the transition in very large systems. To a large extent the results are in agreement with an analytically tractable mean-field analysis of the scheme: We find that at the critical point the system is almost surely localized in the thermodynamic limit, hosting a set of thermal inclusions whose sizes are power-law distributed. On the localized side the {\em typical} localization length is bounded from above. The bound saturates upon approach to criticality, entailing that a finite ergodic inclusion thermalizes a region of diverging diameter. The dominant thermal inclusions have a fractal structure, implying that averaged correlators decay as stretched exponentials throughout the localized phase. Slightly on the ergodic side thermalization occurs through an avalanche instability of the nearly localized bulk, whereby rare, supercritically large ergodic spots eventually thermalize the entire sample. Their size diverges at the transition, while their density vanishes. The non-local, avalanche-like nature of this instability entails a breakdown of single parameter scaling and puts the delocalization transition outside the realm of standard critical phenomena.
Whether disordered and quasiperiodic many-body quantum systems host a long-lived localized phase in the thermodynamic limit has been the subject of intense recent debate. While in one dimension, substantial evidence for such a many-body localized (MBL) phase exists, the behavior in higher dimensions remains an open puzzle. In disordered systems, for instance, it has been argued that rare regions may lead to thermalization of the whole system through a mechanism dubbed avalanche instability. In quasiperiodic systems, however, rare regions are altogether absent and the fate of a putative many-body localized phase has hitherto remained largely unexplored. In this work, we investigate the localization properties of two many-body quasiperiodic models, which are two-dimensional generalizations of the Aubry-Andr\'e model. By studying numerically the out-of-equilibrium dynamics of large systems, we find very long-lived localization on experimentally relevant time scales. Surprisingly, we also observe large-scale transport along deterministic lines of weak potential, which appear in the investigated quasiperiodic models. Our results demonstrate that quasiperiodic many-body systems have the remarkable and counter-intuitive capability of exhibiting coexisting localization and transport properties---a phenomenon reminiscent of the behavior of supersolids. Our findings are of direct experimental relevance and can be tested, for instance, using state-of-the-art cold atomic systems.
We construct a solvable spin chain model of many-body localization (MBL) with a tunable mobility edge. This simple model not only demonstrates analytically the existence of mobility edges in interacting one-dimensional (1D) disordered systems, but also allows us to study their physics. By establishing a connection between MBL and a quantum central limit theorem (QCLT), we show that many-body localization-delocalization transitions can be visualized as tuning a mobility edge in the energy spectrum. Since the effective disorder strength for individual eigenstates depends on energy density, we identify "energy-resolved disorder strength" as a physical mechanism for the appearance of mobility edges, and support the universality of this mechanism by arguing its presence in a large class of models including the random-field Heisenberg chain. We also construct models with multiple mobility edges. All our constructions can be made translationally invariant.
Anderson localization, where quenched disorder leads to the localization of single-particle wave functions, has been studied extensively. However, the effect of interaction and non-random potentials on many-body localization (MBL) has gained attention recently. We investigate the dynamics of the interacting Aubry-André model to assess the stability of the many-body localized (MBL) phase. Our motivation to consider the Aubry–André potential as the disorder potential is to eliminate the avalanche mechanism, which often destabilizes MBL in systems with quenched disorder. We analyze the time evolution and the system-size scaling across a range of disorder strengths for the entanglement entropy, number entropy, and the Hartley number entropy. In the non-interacting case, our results confirm an Anderson localization transition at a critical disorder strength Dc = 2, which is consistent with earlier studies. However, when interactions are introduced, the number entropy appears to grow unboundedly as SN ~ ln ln t in the thermodynamic limit, even for large disorder. Meanwhile, the entanglement entropy grows logarithmically, S ~ ln t, and both entropies saturate in a correlated manner at finite times td ~ e^(L·D). These findings suggest that using a quasiperiodic potential to remove the avalanche mechanism does not stabilize true MBL behavior. Instead, the long-time dynamics are qualitatively similar to those found in systems with random disorder. In this thesis, we first introduce the fundamental concepts of thermalization, quantum quench, and symmetry-resolved entropies, then provide a study of localization, specifically in the Aubry-André model. Finally, we place our numerical results in the broader context of the current debate on the stability of many-body localized phases.
We review the physics of many-body localization in models with incommensurate potentials. In particular, we consider one-dimensional quasiperiodic models with single-particle mobility edges. A conventional perspective suggests that delocalized states act as a thermalizing bath for the localized states in the presence of of interactions. However, contrary to this intuition there is evidence that such systems can display non-ergodicity. This is in part due to the fact that the delocalized states do not have any kind of protection due to symmetry or topology and are thus susceptible to localization. A study of such incommensurate models, in the non-interacting limit, shows that they admit extended, partially extended, and fully localized many-body states. Non-interacting incommensurate models cannot thermalize dynamically and remain localized upon the introduction of interactions. In particular, for a certain range of energy, the system can host a non-ergodic extended (i.e. metallic) phase in which the energy eigenstates violate the eigenstate thermalization hypothesis (ETH) but the entanglement entropy obeys volume-law scaling. The level statistics and entanglement growth also indicate the lack of ergodicity in these models. The phenomenon of localization and non-ergodicity in a system with interactions despite the presence of single-particle delocalized states is closely related to the so-called many-body proximity effect and can also be observed in models with disorder coupled to systems with delocalized degrees of freedom. Many-body localization in systems with incommensurate potentials (without single-particle mobility edges) have been realized experimentally, and we show how this can be modified to study the the effects of such mobility edges. Demonstrating the failure of thermalization in the presence of a single-particle mobility edge in the thermodynamic limit would indicate a more robust violation of the ETH.
We propose a multiscale diagonalization scheme to study disordered one-dimensional chains, in particular, the transition between many-body localization (MBL) and the ergodic phase, expected to be governed by resonant spots. Our scheme focuses on the dichotomy of MBL versus validity of the eigenstate thermalization hypothesis. We show that a few natural assumptions imply that the system is localized with probability one at criticality. On the ergodic side, delocalization is induced by a quantum avalanche seeded by large ergodic spots, whose size diverges at the transition. On the MBL side, the typical localization length tends to the inverse of the maximal entropy density at the transition, but there is a divergent length scale related to the response to an inclusion of large ergodic spots. A mean-field approximation analytically illustrates these results and predicts a power-law distribution for thermal inclusions at criticality.