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arXiv:1612.07173 · Physical Review Letters

Observation of Slow Dynamics near the Many-Body Localization Transition in One-Dimensional Quasiperiodic Systems

Henrik P. Lüschen, Pranjal Bordia, Sebastian Scherg, Fabien Alet, Ehud Altman, Ulrich Schneider, Immanuel Bloch

Published Dec 27, 2017• 281 citations
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Abstract

In the presence of sufficiently strong disorder or quasiperiodic fields, an interacting many-body system can fail to thermalize and become many-body localized. The associated transition is of particular interest, since it occurs not only in the ground state but over an extended range of energy densities. So far, theoretical studies of the transition have focused mainly on the case of true-random disorder. In this work, we experimentally and numerically investigate the regime close to the many-body localization transition in quasiperiodic systems. We find slow relaxation of the density imbalance close to the transition, strikingly similar to the behavior near the transition in true-random systems. This dynamics is found to continuously slow down upon approaching the transition and allows for an estimate of the transition point. We discuss possible microscopic origins of these slow dynamics.

DOI
10.1103/physrevlett.119.260401

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In the presence of sufficiently strong disorder or quasiperiodic fields, an interacting many-body system can fail to thermalize and become many-body localized. The associated transition is of particular interest, since it occurs not only in the ground state but over an extended range of energy densities. So far, theoretical studies of the transition have focused mainly on the case of true-random disorder. In this work, we experimentally and numerically investigate the regime close to the many-body localization transition in quasiperiodic systems. We find slow relaxation of the density imbalance close to the transition, strikingly similar to the behavior near the transition in true-random systems. This dynamics is found to continuously slow down upon approaching the transition and allows for an estimate of the transition point. We discuss possible microscopic origins of these slow dynamics.
We study time dynamics of 1D disordered Heisenberg spin-1/2 chain focusing on a regime of large system sizes and a long time evolution. This regime is relevant for observation of many-body localization (MBL), a phenomenon that is expected to freeze the dynamics of the system and prevent it from reaching thermal equilibrium. Performing extensive numerical simulations of the imbalance, a quantity often employed in the experimental studies of MBL, we show that the regime of a slow power-law decay of imbalance persists to disorder strengths exceeding by at least a factor of 2 the current estimates of the critical disorder strength for MBL. Even though we investigate time evolution up to few thousands tunneling times, we observe no signs of the saturation of imbalance that would suggest freezing of system dynamics and provide a smoking gun evidence of MBL. We demonstrate that the situation is qualitatively different when the disorder is replaced by a quasiperiodic potential. In this case, we observe an emergence of a pattern of oscillations of the imbalance that is stable with respect to changes in the system size. This suggests that the dynamics of quasiperiodic systems remain fully local at the longest time scales we reach provided that the quasiperiodic potential is sufficiently strong. Our study identifies challenges in an unequivocal experimental observation of the phenomenon of MBL.
Strong disorder may significantly slow down or even completely hinder the thermalization of quantum many-body systems due to many-body localization (MBL). A sufficiently deep quasiperiodic potential may also inhibit thermalization. In this paper, we numerically demonstrate direct differences in the behavior of standard ergodicity-breaking indicators at the MBL crossover in random and quasiperiodic systems. Our key finding is the exponential increase in the sharpness of the MBL crossover with system size for quasiperiodic systems, a trend that is only linear in disordered systems. The strong tendency towards nonanalytic behavior in quasiperiodic systems is consistent with the existence of dynamical regimes with sharply defined boundaries or an MBL phase transition. It highlights the importance of quasiperiodic systems for our understanding of many-body dynamics.
In a many-body localized (MBL) quantum system, the ergodic hypothesis breaks down, giving rise to a fundamentally new many-body phase. Whether and under which conditions MBL can occur in higher dimensions remains an outstanding challenge both for experiments and theory. Here, we experimentally explore the relaxation dynamics of an interacting gas of fermionic potassium atoms loaded in a two-dimensional optical lattice with different quasiperiodic potentials along the two directions. We observe a dramatic slowing down of the relaxation for intermediate disorder strengths. Furthermore, beyond a critical disorder strength, we see negligible relaxation on experimentally accessible time scales, indicating a possible transition into a two-dimensional MBL phase. Our experiments reveal a distinct interplay of interactions, disorder, and dimensionality and provide insights into regimes where controlled theoretical approaches are scarce.
We theoretically study transport properties in one-dimensional interacting quasiperiodic systems at infinite temperature. We compare and contrast the dynamical transport properties across the many-body localization (MBL) transition in quasiperiodic and random models. Using exact diagonalization we compute the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ and the return probability $R(\ensuremath{\tau})$ and study their average low-frequency and long-time power-law behavior, respectively. We show that the low-energy transport dynamics is markedly distinct in both the thermal and MBL phases in quasiperiodic and random models and find that the diffusive and MBL regimes of the quasiperiodic model are more robust than those in the random system. Using the distribution of the dc conductivity, we quantify the contribution of sample-to-sample and state-to-state fluctuations of $\ensuremath{\sigma}(\ensuremath{\omega})$ across the MBL transition. We find that the activated dynamical scaling ansatz works poorly in the quasiperiodic model but holds in the random model with an estimated activation exponent $\ensuremath{\psi}\ensuremath{\approx}0.9$. We argue that near the MBL transition in quasiperiodic systems, critical eigenstates give rise to a subdiffusive crossover regime on finite-size systems.
We study localization and many-body localization transition in one-dimensional systems in the presence of deterministic quasiperiodic potential. We use single-particle excitations obtained through single-particle Green's function in real space to characterize the localization to delocalization transition. A single parameter scaling analysis of the ratio of the typical to average value of the local density of states (LDOS) of single-particle excitations shows that the critical exponent with which the correlation length $\ensuremath{\xi}$ diverges at the transition point $\ensuremath{\xi}\ensuremath{\sim}|h\ensuremath{-}{h}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, coming from the localized side, satisfies the inequality $\ensuremath{\nu}\ensuremath{\ge}1$ for the noninteracting Aubry-Andre (AA) model. For the interacting system with AA potential, we study single-particle excitations produced in highly excited many-body eigenstates across the MBL transition and found that the critical exponent obtained from finite-size scaling of the ratio of the typical to average value of the LDOS satisfies $\ensuremath{\nu}\ensuremath{\ge}1$ here as well. This analysis of the local density of states shows that the localization and MBL transition in systems with quasiperiodic potential belong to a different universality class than the localization and MBL transition in systems with random disorder where $\ensuremath{\nu}\ensuremath{\ge}2$. In complete contrast to this, finite-size scaling of the level spacing ratio is known to support the same universality class for MBL transitions in systems with quasiperiodic as well as random disorder potentials. For the interacting systems with quasiperiodic potentials, though finite-size scaling of the level spacing ratio shows a transition at hclsr{h}_{c}^{\mathrm{lsr}}hclsr​ which is close to the transition point obtained from LDOS within numerical precision, the critical exponent obtained from finite-size scaling of level spacing ratio is $\ensuremath{\nu}\ensuremath{\sim}0.54$ in close similarity to the MBL systems with random disorder.
We investigate charge relaxation in disordered and quasiperiodic quantum wires of spinless fermions $(t\ensuremath{-}V$ model) at different inhomogeneity strength WWW in the localized and nearly localized regime. Our observable is the time-dependent density correlation function, $\mathrm{\ensuremath{\Phi}}(x,t)$, at infinite temperature. We find that disordered and quasiperiodic models behave qualitatively similar: Although even at longest observation times the width $\mathrm{\ensuremath{\Delta}}x(t)$ of $\mathrm{\ensuremath{\Phi}}(x,t)$ does not exceed significantly the noninteracting localization length, ${\ensuremath{\xi}}_{0}$, strong finite-size effects are encountered. Our findings appear difficult to reconcile with the rare-region physics (Griffiths effects) that often is invoked as an explanation for the slow dynamics observed by us and earlier computational studies. Motivated by our numerical data we discuss a scenario in which the MBL-phase splits into two subphases: in ${\mathrm{MBL}}_{\text{A}}\phantom{\rule{4pt}{0ex}}\mathrm{\ensuremath{\Delta}}x(t)$ diverges slower than any power, while it converges towards a finite value in MBLB{\mathrm{MBL}}_{\text{B}}MBLB​. Within the scenario the transition between MBLA{\mathrm{MBL}}_{\text{A}}MBLA​ and the ergodic phase is characterized by a length scale, $\ensuremath{\xi}$, that exhibits an essential singularity $ln\ensuremath{\xi}\ensuremath{\sim}1/|W\ensuremath{-}{W}_{{\text{c}}_{1}}|$. Relations to earlier numerics and proposals of two-phase scenarios will be discussed.