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arXiv:1903.07352 · Physical Review A

Many-body localization in continuum systems: Two-dimensional bosons

Giulio Bertoli, B. L. Altshuler, Gora V. Shlyapnikov

Published Jul 24, 2019• 9 citations
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Abstract

We demonstrate that many-body localization of two-dimensional weakly interacting bosons in disorder remains stable in the thermodynamic limit at sufficiently low temperatures. Highly energetic particles destroy the localized state only above a critical temperature, which increases with the strength of the disorder. If the particle distribution is truncated at high energies, as it is for cold-atom systems, the localization can be stable at any temperature.

DOI
10.1103/physreva.100.013628

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We demonstrate that many-body localization of two-dimensional weakly interacting bosons in disorder remains stable in the thermodynamic limit at sufficiently low temperatures. Highly energetic particles destroy the localized state only above a critical temperature, which increases with the strength of the disorder. If the particle distribution is truncated at high energies, as it is for cold-atom systems, the localization can be stable at any temperature.
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.
Recent theoretical and numerical evidence suggests that localization can survive in disordered many-body systems with very high energy density, provided that interactions are sufficiently weak. Stronger interactions can destroy localization, leading to a so-called many-body localization transition. This dynamical phase transition is relevant to questions of thermalization in extended quantum systems far from the zero-temperature limit. It separates a many-body localized phase, in which localization prevents transport and thermalization, from a conducting (``ergodic'') phase in which the usual assumptions of quantum statistical mechanics hold. Here, we present numerical evidence that many-body localization also occurs in models without disorder but rather a quasiperiodic potential. In one dimension, these systems already have a single-particle localization transition, and we show that this transition becomes a many-body localization transition upon the introduction of interactions. We also comment on possible relevance of our results to experimental studies of many-body dynamics of cold atoms and nonlinear light in quasiperiodic potentials.
We study phase transitions in a two dimensional weakly interacting Bose gas in a random potential at finite temperatures. We identify superfluid, normal fluid, and insulator phases and construct the phase diagram. At T=0 one has a tricritical point where the three phases coexist. The truncation of the energy distribution at the trap barrier, which is a generic phenomenon in cold atom systems, limits the growth of the localization length and in contrast to the thermodynamic limit the insulator phase is present at any temperature.
Physics Disordered interacting quantum many-body systems can become hopelessly localized. This so-called many-body localization has been studied in one-dimensional systems. In two dimensions, experiments indicate that the localization persists, but because it is difficult to tell the difference between no thermalization and slow thermalization, more theoretical work is needed. Relying on an approximate numerical method, Wahl et al. undertook large-scale simulations of a two-dimensional lattice of bosonic atoms in the presence of interactions and disorder. At intermediate disorder strengths, the on-site entanglement entropies exhibited a bimodal distribution, indicating a coexistence of localized and thermalized states; as disorder was increased, localized states took over. The authors were able to extract a critical disorder strength and set a benchmark for future experiments. Nat. Phys. 10.1038/s41567-018-0339-x (2018).
We use thermalization indicators and numerical linked cluster expansions to probe the onset of many-body localization in a disordered one-dimensional hard-core boson model in the thermodynamic limit. We show that after equilibration following a quench from a delocalized state, the momentum distribution indicates a freezing of one-particle correlations at higher values than in thermal equilibrium. The position of the delocalization to localization transition, identified by the breakdown of thermalization with increasing disorder strength, is found to be consistent with the value from the level statistics obtained via full exact diagonalization of finite chains. Our results strongly support the existence of a many-body localized phase in the thermodynamic limit.
When pushed out of equilibrium, generic interacting quantum systems equilibrate locally and are expected to evolve towards a locally thermal description despite their unitary time evolution. Systems in which disorder competes with interactions and transport can violate this expectation by exhibiting many-body localization. The strength of the disorder with respect to the other two parameters drives a transition from a thermalizing system towards a nonthermalizing one. The existence of this transition is well established both in experimental and numerical studies for finite systems. However, the stability of many-body localization in the thermodynamic limit is largely unclear. With increasing system size, a generic disordered system will contain with high probability areas of low disorder variation. If large and frequent enough, those areas constitute ergodic grains which can hybridize and thus compete with localization. While the details of this process are not yet settled, it is conceivable that if such regions appear sufficiently often, they might be powerful enough to restore thermalization. We set out to shed light on this problem by constructing potential landscapes with low disorder regions and numerically investigating their localization behavior in the Heisenberg model. Our findings suggest that many-body localization may be more stable than anticipated in other recent theoretical works.