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arXiv:1408.6235 · Physical Review B

Many-body localization and delocalization in the two-dimensional continuum

Rahul Nandkishore

Published Nov 25, 2014• 35 citations
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Abstract

We discuss whether localization in the two-dimensional continuum can be stable in the presence of short-range interactions. We conclude that, for an impurity model of disorder, if the system is prepared below a critical temperature $T<{T}_{c}$, then perturbation theory about the localized phase converges almost everywhere. As a result, the system is at least asymptotically localized and perhaps even truly many-body localized, depending on how certain rare regions behave. Meanwhile, for $T>{T}_{c}$, perturbation theory fails to converge, which we interpret as interaction-mediated delocalization. We calculate the boundary of the region of perturbative stability of localization in the interaction-strength-temperature plane. We also discuss the behavior in a speckle disorder (relevant for cold-atom experiments) and conclude that perturbation theory about the noninteracting phase diverges for arbitrarily weak interactions with speckle disorder, suggesting that many-body localization in the two-dimensional continuum cannot survive away from the impurity limit.

DOI
10.1103/physrevb.90.184204

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We discuss whether localization in the two-dimensional continuum can be stable in the presence of short-range interactions. We conclude that, for an impurity model of disorder, if the system is prepared below a critical temperature T<TcT<{T}_{c}T<Tc​, then perturbation theory about the localized phase converges almost everywhere. As a result, the system is at least asymptotically localized and perhaps even truly many-body localized, depending on how certain rare regions behave. Meanwhile, for T>TcT>{T}_{c}T>Tc​, perturbation theory fails to converge, which we interpret as interaction-mediated delocalization. We calculate the boundary of the region of perturbative stability of localization in the interaction-strength-temperature plane. We also discuss the behavior in a speckle disorder (relevant for cold-atom experiments) and conclude that perturbation theory about the noninteracting phase diverges for arbitrarily weak interactions with speckle disorder, suggesting that many-body localization in the two-dimensional continuum cannot survive away from the impurity limit.
I discuss the quantum dynamics of strongly disordered quantum systems with critically long range interactions, decaying as 1/r2d1/{r}^{2d}1/r2d in ddd spatial dimensions. I argue that, contrary to expectations, localization in such systems is stable at low orders in perturbation theory, giving rise to an unusual ``critically many-body localized (MBL) regime.'' I discuss the phenomenology of this critical MBL regime, which includes distinctive signatures in entanglement, charge statistics, noise, and transport. Experimentally, such a critically localized regime can be realized in three-dimensional systems with van der Waals interactions, such as Rydberg atoms, and in one-dimensional systems with 1/r21/{r}^{2}1/r2 interactions, such as trapped ions. I estimate timescales on which high-order perturbative and nonperturbative (avalanche) phenomena may destabilize this critically MBL regime and conclude that the avalanche sets the limiting timescale, in the limit of strong disorder or weak interactions.
We consider an extended electronic system with localized single‐particle states coupled by short‐range interactions, in the absence of coupling to an external bath. We show that many‐body localization, which exists in tight‐binding models, is unstable in a continuum. Irrespective of the dimensionality of the system, many‐body localization does not survive the unbounded growth of the single‐particle localization length with increasing energy that is characteristic of the continuum limit. The system remains delocalized down to arbitrarily small temperature T , although its dynamics slows down as T decreases. Remarkably, the conductivity vanishes with decreasing T faster than in the Arrhenius law. The system can be characterized by an effective T ‐dependent single‐particle mobility edge which diverges in the limit of . Delocalization is driven by interactions between hot electrons above the mobility edge and the “bath” of thermal electrons in the vicinity of the Fermi level. image
Closed generic quantum many-body systems may fail to thermalize under certain conditions even after long times, a phenomenon called many-body localization (MBL). Numerous studies support the stability of the MBL phase in strongly disordered one-dimensional systems. However, the situation is much less clear when a small part of the system is ergodic, a scenario which also has important implications for the existence of many-body localization in higher dimensions. Here we address this question experimentally using a large-scale quantum simulator of ultracold bosons in a two-dimensional optical lattice. We prepare two-component mixtures of varying relative population and implement a disorder potential which is only experienced by one of the components. The second non-disordered ''clean'' component plays the role of a bath of adjustable size that is collisionally coupled to the ''dirty'' component. Our experiments show how the dynamics of the dirty component, which, when on its own, show strong evidence of localization, become affected by the coupling to the clean component. For a high clean population, the clean component appears to behave as an effective bath for the system which leads to its delocalization, while for a smaller clean population, the ability of the bath to destabilize the system becomes strongly reduced. Our results reveal how a finite-sized quantum system can bring another one towards thermalization, in a regime of complex interplay between disorder, tunneling and intercomponent interactions. They provide a new benchmark for effective theories aiming to capture the complex physics of MBL in the weakly localized regime.
In the presence of strong disorder and weak interactions, closed quantum systems can enter a many-body localized phase wherein the system does not conduct heat or charge, does not equilibrate even for arbitrarily long times, and robustly violates quantum statistical mechanics. While this is well established in closed quantum systems, the interplay of such systems with delocalized degrees of freedom is much less understood. Here, the authors consider a model for which, in the noninteracting limit, some degrees of freedom are localized while others remain delocalized. Such a system can be viewed as a model for a many-body localized system brought into contact with a small bath of a comparable number of degrees of freedom. The authors numerically and analytically study the effect of interactions on this system and find that in certain parameter regimes, results are consistent with interaction-induced localization of the entire system.
Closed generic quantum many-body systems may fail to thermalize under certain conditions even after long times, a phenomenon called many-body localization (MBL).Numerous studies support the stability of the MBL phase in strongly disordered one-dimensional systems.However, the situation is much less clear when a small part of the system is ergodic, a scenario which also has important implications for the existence of many-body localization in higher dimensions.Here we address this question experimentally using a largescale quantum simulator of ultracold bosons in a two-dimensional optical lattice.We prepare two-component mixtures of varying relative population and implement a disorder potential which is experienced only by one of the components.The second nondisordered "clean" component plays the role of a bath of adjustable size that is collisionally coupled to the "dirty" component.Our experiments show how the dynamics of the dirty component, which, when on its own, show strong evidence of localization, become affected by the coupling to the clean component.For a high clean population, the clean component appears to behave as an effective bath for the system which leads to its delocalization, while for a smaller clean population, the ability of the bath to destabilize the system becomes strongly reduced.Our results reveal how a finite-sized quantum system can bring another one towards thermalization, in a regime of complex interplay between disorder, tunneling, and intercomponent interactions.They provide a new benchmark for effective theories aiming to capture the complex physics of MBL in the weakly localized regime.