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arXiv:1405.3279 · Physical Review B

Scenario for delocalization in translation-invariant systems

Wojciech De Roeck, François Huveneers

Published Oct 28, 2014• 84 citations
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Abstract

We investigate the possibility of many-body localization in translation-invariant Hamiltonian systems, which was recently brought up by several authors. A key feature of many-body localized disordered systems is recovered, namely the fact that resonant spots are rare and far-between. However, we point out that resonant spots are mobile, unlike in models with strong quenched disorder, and that these mobile spots constitute a possible mechanism for delocalization, albeit possibly only on very long timescales. In some models, this argument for delocalization can be made very explicit in first order of perturbation theory in the hopping. For models where this does not work, we present instead a nonperturbative argument that relies solely on ergodicity inside the resonant spots.

DOI
10.1103/physrevb.90.165137

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We investigate the possibility of many-body localization in translation-invariant Hamiltonian systems, which was recently brought up by several authors. A key feature of many-body localized disordered systems is recovered, namely the fact that resonant spots are rare and far-between. However, we point out that resonant spots are mobile, unlike in models with strong quenched disorder, and that these mobile spots constitute a possible mechanism for delocalization, albeit possibly only on very long timescales. In some models, this argument for delocalization can be made very explicit in first order of perturbation theory in the hopping. For models where this does not work, we present instead a nonperturbative argument that relies solely on ergodicity inside the resonant spots.
We investigate the possibility of many-body localization in translation invariant Hamiltonian systems. A key feature of many-body localized disordered systems is recovered, namely the fact that resonant spots are rare and far-between. However, we point out that resonant spots are mobile, unlike in models with quenched disorder, and that they could therefore in principle delocalize the system. The motion of these resonant spots is reminiscent of motion in kinetically constrained models, the frustration effect being caused by a mismatch of energies rather than by geometric constraints. We provide examples where, in first order in the hopping, the resonant spots are effectively trapped (analogous to a jammed phase) and therefore localization is not ruled out, but we also have examples where, despite the rareness of resonant spots, the resonant spots are not trapped and hence this obvious scenario for many-body localization is not realized. We comment on the analysis of higher orders, but we did not reach a final conclusion yet.
Recently it has been suggested that many-body localization (MBL) can occur in translationally-invariant systems, and candidate 1D models have been proposed. We find that such models typically exhibit strong finite-size effects due to the presence of two or more vastly different energy scales, including an artificial separation of density of states into bands separated by large energy gaps. By carefully choosing the parameters of the models such that finite-size effects are minimized, we perform extensive numerical studies of small systems using exact diagonalization. Based on the variety of diagnostics, including entanglement properties and the behavior of local observables, we find the systems exhibit thermal (ergodic), rather than MBL-like behavior. Our results suggest that MBL in translationally-invariant systems with two very different energy scales is less robust than perturbative arguments suggest, possibly pointing to the importance of non-perturbative effects which induce delocalization in the thermodynamic limit.
The phenomenon of localization is usually accompanied with the presence of quenched disorder. To what extent disorder is necessary for localization is a well-known open problem. In this paper, we prove the instability of localization in translation-invariant systems. For any translation-invariant local Hamiltonian exhibiting either Anderson or many-body localization, an arbitrarily small translation-invariant random local perturbation almost surely leads to the following manifestations of delocalization: (i) Transport: For any (inhomogeneous) initial state, the spatial distribution of energy or any other local conserved quantity becomes uniform at late times. (ii) Scrambling: The out-of-time-ordered correlator of any traceless local operators decays to zero at late times. (iii) Thermalization: Random product states locally thermalize to the infinite temperature state with overwhelming probability.
Many-body localization provides a mechanism to avoid thermalization in isolated interacting quantum systems. The breakdown of thermalization may be complete, when all eigenstates in the many-body spectrum become localized, or partial, when the so-called many-body mobility edge separates localized and delocalized parts of the spectrum. Previously, De Roeck et al. [Phys. Rev. B 93, 014203 (2016)] suggested a possible instability of the many-body mobility edge in energy density. The local ergodic regions---so-called ``bubbles''---resonantly spread throughout the system, leading to delocalization. In order to study such instability mechanism, in this work we design a model featuring many-body mobility edge in particle density: the states at small particle density are localized, while increasing the density of particles leads to delocalization. Using numerical simulations with matrix product states, we demonstrate the stability of many-body localization with respect to small bubbles in large dilute systems for experimentally relevant timescales. In addition, we demonstrate that processes where the bubble spreads are favored over processes that lead to resonant tunneling, suggesting a possible mechanism behind the observed stability of many-body mobility edge. We conclude by proposing experiments to probe particle density mobility edge in the Bose-Hubbard model.
Many-body quantum systems typically display fast dynamics and ballistic spreading of information. Here we address the open problem of how slow the dynamics can be after a generic breaking of integrability by local interactions. We develop a method based on degenerate perturbation theory that reveals slow dynamical regimes and delocalization processes in general translation invariant models, along with accurate estimates of their delocalization time scales. Our results shed light on the fundamental questions of the robustness of quantum integrable systems and the possibility of many-body localization without disorder. As an example, we construct a large class of one-dimensional lattice models where, despite the absence of asymptotic localization, the transient dynamics is exceptionally slow, i.e., the dynamics is indistinguishable from that of many-body localized systems for the system sizes and time scales accessible in experiments and numerical simulations.
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.