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arXiv:1609.06640 · Physical Review A

Slow relaxation and sensitivity to disorder in trapped lattice fermions after a quench

Maximilian Schulz, C. Hooley, Roderich Moessner

Published Dec 27, 2016• 3 citations
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Abstract

We consider a system of noninteracting fermions in one dimension subject to a single-particle potential consisting of (a) a strong optical lattice, (b) a harmonic trap, and (c) uncorrelated on-site disorder. After a quench, in which the center of the harmonic trap is displaced, we study the occupation function of the fermions and the time evolution of experimental observables. Specifically, we present numerical and analytical results for the postquench occupation function of the fermions, and analyze the time evolution of the real-space density profile. Unsurprisingly for a noninteracting (and therefore integrable) system, the infinite-time limit of the density profile is nonthermal. However, due to Bragg localization of the higher-energy single-particle states, the approach to even this nonthermal state is extremely slow. We quantify this statement, and show that it implies a sensitivity to disorder parametrically stronger than that expected from Anderson localization.

DOI
10.1103/physreva.94.063643

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We consider a system of noninteracting fermions in one dimension subject to a single-particle potential consisting of (a) a strong optical lattice, (b) a harmonic trap, and (c) uncorrelated on-site disorder. After a quench, in which the center of the harmonic trap is displaced, we study the occupation function of the fermions and the time evolution of experimental observables. Specifically, we present numerical and analytical results for the postquench occupation function of the fermions, and analyze the time evolution of the real-space density profile. Unsurprisingly for a noninteracting (and therefore integrable) system, the infinite-time limit of the density profile is nonthermal. However, due to Bragg localization of the higher-energy single-particle states, the approach to even this nonthermal state is extremely slow. We quantify this statement, and show that it implies a sensitivity to disorder parametrically stronger than that expected from Anderson localization.
In the context of an isolated three-dimensional noninteracting fermionic lattice system, we study the effects of a sudden quantum quench between a disorder-free situation and one in which disorder results in a mobility edge and associated Anderson localization. Salient post-quench features hinge upon the overlap between momentum states and post-quench eigenstates and whether these latter states are extended or localized. We find that the post-quench momentum distribution directly reflects these overlaps. For the local density, we show that disorder generically prevents the equilibration of quantum expectation values to a steady state and that the persistent fluctuations have a nonmonotonic dependence on the strength of disorder. We identify two distinct types of fluctuations, namely, temporal fluctuations describing the time-dependent fluctuations of the local density around its time average and sample-to-sample fluctuations characterizing the variations of these time averages from one realization of disorder to another. We demonstrate that both of these fluctuations vanish for extremely extended as well as extremely localized states, peaking at some intermediate value.
In the context of an isolated three-dimensional noninteracting fermionic lattice system, we study the effects of a sudden quench between a disorder-free situation and one in which disorder results in a mobility edge and associated Anderson localization. Salient post-quench features hinge upon the overlap between momentum states and post-quench eigenstates and whether these latter states are extended or localized. We find that the post-quench momentum distribution directly reflects these overlaps. For the local density, we show that disorder generically prevents the equilibration of quantum expectation values to a steady state and that the persistent fluctuations have a \textit{nonmonotonic} dependence on the strength of disorder. We identify two distinct types of fluctuations, namely, temporal fluctuations describing the time-dependent fluctuations of the local density around its time average and sample-to-sample fluctuations characterizing the variations of these time averages from one realization of disorder to another. We demonstrate that both of these fluctuations vanish for extremely extended as well as extremely localized states, peaking at some intermediate value. The post-quench behavior captures crucial features of the Anderson localization transition through these fluctuations.
The study of collective behaviour in many-body systems often explores fundamentally new ideas absent from the mere constituents of such a system. A paradigmatic model for these studies is the spin-1/2 XXZ chain and its fermionic equivalent. This thesis can be broadly divided into the study of two fundamental aspects of this model. Firstly, we discuss localisation phenomena in one dimensional lattices as often experimentally realised in cold atom systems. Secondly, we investigate how disorder and symmetry influence heat transport in spin chains. More specifically, in the first part we consider a system of non-interacting fermions in one dimension subject to a single-particle potential consisting of a strong optical lattice, a harmonic trap, and uncorrelated on-site disorder. We investigate a global inhomogeneous quantum quench and present numerical and analytical results for static and dynamical properties. We show that the approach to the non-thermal equilibrium state is extremely slow and that it implies a sensitivity to disorder parametrically stronger than that expected from Anderson localisation. We also consider the above system in a strong non-uniform electric field. In the non-interacting case, due to Wannier-Stark localisation, the single-particle wave functions are exponentially localised without quenched disorder. We show that this system remains localised in the presence of nearest-neighbour interactions and exhibits physics analogous to models of conventional many-body localisation. The second part explores the hydrodynamics of the disordered XYZ spin chain. Using time-evolving block decimation on open chains of up to 400 spins attached to thermal baths, we probe the energy transport of this system. Our principal findings are as follows. For weak disorder there is a stable diffusive region that persists up to a critical disorder strength that depends on the XY anisotropy. Then, for disorder strengths above this critical value energy transport becomes increasingly subdiffusive.
We analyze the effect of a linear time variation of the interaction strength on a trapped one-dimensional Bose gas confined to an optical lattice. The evolution of different observables such as the experimentally accessible on site particle distribution are studied as a function of the ramp time by using time-dependent numerical techniques. We find that the dynamics of a trapped system typically displays two regimes: For long ramp times, the dynamics is governed by density redistribution, while at short ramp times, local dynamics dominates as the evolution is identical to that of an homogeneous system. In the homogeneous limit, we also discuss the nontrivial scaling of the energy absorbed with the ramp time.
We study the real-time dynamics of local occupation numbers in a one-dimensional model of spinless fermions with a random on-site potential for a certain class of initial states. The latter are thermal (mixed or pure) states of the model in the presence of an additional static force but become nonequilibrium states after a sudden removal of this static force. For this class and high temperatures, we show that the induced dynamics is given by a single correlation function at equilibrium, independent of the initial expectation values being prepared close to equilibrium (by a weak static force) or far away from equilibrium (by a strong static force). Remarkably, this type of universality holds true in both the ergodic phase and the many-body localized regime. Moreover, it does not depend on the specific choice of a unit cell for the local density. We particularly discuss two important consequences. First, the long-time expectation value of the local density is uniquely determined by the fluctuations of its diagonal matrix elements in the energy eigenbasis. Thus, the validity of the eigenstate thermalization hypothesis is not only a sufficient but also a necessary condition for thermalization. Second, the real-time broadening of density profiles is always given by the current autocorrelation function at equilibrium via a generalized Einstein relation. In the context of transport, we discuss the influence of disorder for large particle-particle interactions, where normal diffusion is known to occur in the disorder-free case. Our results suggest that normal diffusion is stable against weak disorder, while they are consistent with anomalous diffusion for stronger disorder below the localization transition. Particularly, for weak disorder, Gaussian density profiles can be observed for single disorder realizations, which we demonstrate for finite lattices up to 31 sites.
Finite systems of bosons and/or fermions described by the Hubbard model can be realized using ultracold atoms confined in optical lattices. The ground states of these systems often exhibit a coexistence of compressible superfluid and incompressible Mott insulating domains. We analyze such systems by studying the out-of-equilibrium dynamics following a weak sudden quench of the trapping potential. In particular, we show how the temporal variance of the site occupations reveals the location of spatial boundaries between compressible and incompressible regions. The feasibility of this approach is demonstrated for several models using numerical simulations. We first consider integrable systems, hard-core bosons (spinless fermions) confined by a harmonic potential, where space-separated Mott and superfluid phases coexist. Then, we analyze a nonintegrable system, a $J\text{\ensuremath{-}}V\text{\ensuremath{-}}{V}^{\ensuremath{'}}$ model, with coexisting charge density wave and superfluid phases. We find that the temporal variance of the site occupations is a more effective measure than other standard indicators of phase boundaries such as a local compressibility. Finally, to make contact with experiments, we propose a consistent estimator for such temporal variance. Our numerical experiments show that the phase boundary is correctly spotted using as little as 30 measurements. Based on these results, we argue that analyzing temporal fluctuations is a valuable experimental tool for exploring phase boundaries in trapped atom systems.