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arXiv:1905.02017 · Physical Review Letters

Einstein Relation for a Driven Disordered Quantum Chain in the Subdiffusive Regime

Marcin Mierzejewski, Peter Prelovšek, J. Bonča

Published May 24, 2019• 8 citations
arXivHTMLPDF

Abstract

A quantum particle propagates subdiffusively on a strongly disordered chain when it is coupled to itinerant hard-core bosons. We establish a generalized Einstein relation (GER) that relates such subdiffusive spread to an unusual time-dependent drift velocity, which appears as a consequence of a constant electric field. We show that GER remains valid much beyond the regime of the linear response. Qualitatively, it holds true up to strongest drivings when the nonlinear field effects lead to the Stark-like localization. Numerical calculations based on full quantum evolution are substantiated by much simpler rate equations for the boson-assisted transitions between localized Anderson states.

DOI
10.1103/physrevlett.122.206601

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A quantum particle propagates subdiffusively on a strongly disordered chain when it is coupled to itinerant hard-core bosons. We establish a generalized Einstein relation (GER) that relates such subdiffusive spread to an unusual time-dependent drift velocity, which appears as a consequence of a constant electric field. We show that GER remains valid much beyond the regime of the linear response. Qualitatively, it holds true up to strongest drivings when the nonlinear field effects lead to the Stark-like localization. Numerical calculations based on full quantum evolution are substantiated by much simpler rate equations for the boson-assisted transitions between localized Anderson states.
We consider the propagation of a single particle in a random chain, assisted by the coupling to dispersive bosons. Time evolution treated with rate equations for hopping between localized states reveals a qualitative difference between dynamics due to noninteracting bosons and hard-core bosons. In the first case the transient dynamics is subdiffusive, but multiboson processes allow for long-time normal diffusion, while hard-core effects suppress multiboson processes leading to persistent subdiffusive transport, consistent with numerical results for a full many-body evolution. In contrast, analogous study for a quasiperiodic potential reveals a stable long-time diffusion.
We study a quantum particle coupled to hard-core bosons and propagating on disordered ladders with RRR legs, ranging from R=1R=1R=1 (chains) to $R\ensuremath{\gg}1$ (planes). The particle dynamics is studied within the framework of rate equations for the boson-assisted transitions between the Anderson states. We demonstrate that for finite $R<\ensuremath{\infty}$ and sufficiently strong disorder the dynamics is nondiffusive, while two-dimensional planar systems with $R\ensuremath{\rightarrow}\ensuremath{\infty}$ remain diffusive for arbitrarily strong disorder. The transition from diffusive to subdiffusive regimes may be identified via statistical fluctuations of resistivity. Close to the transition, the corresponding distribution function in the diffusive regime has fat tails which decrease much slower than 1/L1/\sqrt{L}1/L​, where LLL is the system size. Finally, we present evidence that similar non-Gaussian fluctuations arise also in standard models of many-body localization.
We investigate chains of interacting spinless fermions subject to a finite external field FFF (also called Stark chains) and focus on the regime where the charge thermalization follows the subdiffusive hydrodynamics. First, we study reduced models conserving the dipole moment and derive an explicit Einstein relation which links the subdiffusive transport coefficient with the correlations of the dipolar current. This relation explains why the decay rate ${\mathrm{\ensuremath{\Gamma}}}_{q}$ of the density modulation with wave vector qqq shows q4{q}^{4}q4 dependence. In the case of the Stark model, a similar Einstein relation is also derived and tested using various numerical methods. They confirm an exponential reduction of the transport coefficient with increasing FFF. On the other hand, our study of the Stark model indicates that upon increasing qqq there is a crossover from subdiffusive behavior, ${\mathrm{\ensuremath{\Gamma}}}_{q}\ensuremath{\propto}{q}^{4}$, to the normal diffusive relaxation, ${\mathrm{\ensuremath{\Gamma}}}_{q}\ensuremath{\propto}{q}^{2}$, at the wave vector q∗{q}^{*}q∗ which vanishes for $F\ensuremath{\rightarrow}0$.
We study a particle which propagates in a one-dimensional strong random potential and is coupled to a bosonic bath. We independently test various properties of bosons (hopping term, hard-core effects, and generic boson-boson interaction) and show that bosonic itineracy is the essential ingredient governing the dynamics of the particle. Coupling of the particle to itinerant phonons or hard-core bosons alike leads to delocalization of the particle by virtue of a subdiffusive (or diffusive) spread from the initially localized state. Delocalization remains in effect even when the boson frequency and the bandwidth of itinerant bosons remain an order of magnitude smaller than the magnitude of the random potential. When the particle is coupled to localized bosons, its spread remains logarithmic or even sublogarithmic. The latter result together with the survival probability shows that the particle remains localized despite being coupled to bosons.
We study transport in a one-dimensional boundary-driven Anderson insulator (the XX spin chain with onsite disorder) with randomly positioned onsite dephasing, observing a transition from diffusive to subdiffusive spin transport below a critical density of sites with dephasing. This model is intended to mimic the passage of an excitation through (many-body) insulating regions or ergodic bubbles, therefore providing a toy model for the diffusion-subdiffusion transition observed in the disordered Heisenberg model by \ifmmode \check{Z}\else \v{Z}\fi{}nidari\ifmmode \check{c}\else \v{c}\fi{} et al. [Phys. Rev. Lett. 117, 040601 (2016)]. We also present the exact solution of a semiclassical model of conductors and insulators introduced by Agarwal et al. [Phys. Rev. Lett. 114, 160401 (2015)], which exhibits both diffusive and subdiffusive phases, and qualitatively reproduces the results of the quantum system. The critical properties of both models, when passing from diffusion to subdiffusion, are interpreted in terms of ``Griffiths effects.'' We show that the finite-size scaling comes from the interplay of three characteristic lengths: one associated with disorder (the localization length), one with dephasing, and the third with the percolation problem defining large, rare, insulating regions. We conjecture that the latter, which grows logarithmically with system size, may potentially be responsible for the fact that heavy-tailed resistance distributions typical of Griffiths effects have not been observed in subdiffusive interacting systems.
We investigate the influence of the on-site Hubbard interaction UUU on the eigenstates and dynamics of two electrons restricted to move in a linear chain with long-range correlated disorder. We solve the time-dependent Schr\"odinger equation to follow the time evolution of an initially localized Gaussian two-electron wave packet. In the regime of strongly correlated disorder, for which one-electron extended eigenstates emerge near the band center, the electron-electron coupling promotes the trapping of a finite portion of the wave packet. In the presence of a uniform electric field, the wave packet develops complex Bloch oscillations. The power spectrum of the centroid's velocity trace shows a splitting near the typical semiclassical Bloch frequency, as well as a frequency doubling phenomenon for intermediate couplings which is related to the bounded states components that are present in the wave packet. Finally, we show that localized and extended two-electron eigenstates coexist near the band center with the level spacing distribution showing a universal Poissonian form irrespective to the Hubbard coupling.