Fate of Many-Body Localization Under Periodic Driving

Achilleas Lazarides, Arnab Das, and Roderich Moessner

Phys. Rev. Lett. 115, 030402 – Published 13 July 2015

Abstract

We study many-body localized quantum systems subject to periodic driving. We find that the presence of a mobility edge anywhere in the spectrum is enough to lead to delocalization for any driving strength and frequency. By contrast, for a fully localized many-body system, a delocalization transition occurs at a finite driving frequency. We present numerical studies on a system of interacting one-dimensional bosons and the quantum random energy model, as well as simple physical pictures accounting for those results.

Received 13 October 2014

DOI:https://doi.org/10.1103/PhysRevLett.115.030402

Authors & Affiliations

Achilleas Lazarides¹, Arnab Das², and Roderich Moessner¹

¹Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany
²Theoretical Physics Department, Indian Association for the Cultivation of Science, Kolkata 700032, India

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Vol. 115, Iss. 3 — 17 July 2015

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Images

Figure 1
Plot of driving frequency $ω_{c}$ below which the system delocalizes as a function of disorder amplitude $w$ . The shaded areas correspond to delocalization. The red dots are obtained from finite-size studies of the level statistics of the system. The disorder amplitude $w_{c}$ is the value below which the undriven system is delocalized in the absence of driving. The blue line is a guide to the eye.
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Figure 2
Main plot: Probability distribution of the EEVs of the density at site $j = 3$ . Left (right): Driving with a low (high) frequency (see Fig. 1) results in the probability distributions which does (does not) develop a central peak upon increasing system size, signaling delocalization [8]. Data are disorder averaged over $10^{4}$ (100) realizations for $L = 14$ ( $L = 16$ ). Inset: Level statistics parameter versus inverse system size in the localized (bottom, red) delocalized (top, blue) phases. The parameter $η = \int d s s P (s)$ with $P (s)$ the probability distribution of the level statistics [3, 7], taking the value $η_{P / C U E}$ in the localized or delocalized regime. Data averaged over 1000 disorder realizations for $L = 8, 10, 12$ , 100 realizations for $L = 14$ .
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Figure 3
Schematic illustration of subsystems and their energy levels in a MBL system. The horizontal axis indexes the conserved quantity (increasing $n$ corresponds to increasing spatial index $i$ ); the vertical indexes energy. $ξ_{c}$ is some localization length, setting the typical spatial size of the subsystems. A periodic coupling of the subsystems with frequency $ω$ couples the red state in the middle block to both the purple and the green levels in the two neighboring blocks, while a time-independent coupling would only couple it to the green ones. The width of the purple and green strips is set by the amplitude of the driving. Critically, the limit of $ω$ greater than the typical subsystem bandwidth is indistinguishable from a time-independent driving. By contrast, the limit of $ω \to 0$ collapses the local spectra, wiping out the effect of disorder.
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Figure 4
Driving the QREM. The top left-hand figure shows the participation ratio $ϕ$ for the eigenstates of the undriven model, showing a mobile region (blue) surrounded by a localized region (red). Driving with frequency $ω / J = 0.1$ and amplitude $δ / J = 0.2$ (top right) causes all states at a given $Γ_{0}$ to become as delocalized as the least localized state at that $Γ_{0}$ in the undriven model. This is also shown in the bottom panel, which shows $ϕ$ for $Γ_{0} = 0.01, 0.1, 0.5$ (red, blue, and green lines, from top to bottom) in the absence (presence) of driving with darker (lighter) color. The driven points always lie below the undriven points for the corresponding $Γ_{0}$ . This is due to the strong mixing of all undriven eigenstates by the driving. All data in this figure are for eight spins and averaged over 1000 disorder realizations.
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