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  • Open Access

Aperiodically Driven Integrable Systems and Their Emergent Steady States

Sourav Nandy1, Arnab Sen1, and Diptiman Sen2

  • 1Department of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
  • 2Centre for High Energy Physics, Indian Institute of Science, Bengaluru 560012, India

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Phys. Rev. X 7, 031034 – Published 22 August, 2017

DOI: https://doi.org/10.1103/PhysRevX.7.031034

Abstract

Does a closed quantum many-body system that is continually driven with a time-dependent Hamiltonian finally reach a steady state? This question has only recently been answered for driving protocols that are periodic in time, where the long-time behavior of the local properties synchronizes with the drive and can be described by an appropriate periodic ensemble. Here, we explore the consequences of breaking the time-periodic structure of the drive with additional aperiodic noise in a class of integrable systems. We show that the resulting unitary dynamics leads to new emergent steady states in at least two cases. While any typical realization of random noise causes eventual heating to an infinite-temperature ensemble for all local properties in spite of the system being integrable, noise that is self-similar in time leads to an entirely different steady state (which we dub the “geometric generalized Gibbs ensemble”) that emerges only after an astronomically large time scale. To understand the approach to the steady state, we study the temporal behavior of certain coarse-grained quantities in momentum space that fully determine the reduced density matrix for a subsystem with size much smaller than the total system. Such quantities provide a concise description for any drive protocol in integrable systems that are reducible to a free-fermion representation.

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Exposing an ensemble of atoms or molecules to a changing field—such as pulses from a laser or a varying magnetic field—can lead to some intriguing behavior that is not seen when the system is left alone. In particular, if the changes are periodic in time, that can be a useful experimental tool for generating novel nonequilibrium phases. Much work has been devoted to understanding the steady-state behavior of periodically driven systems, but the consequences of breaking that periodicity are largely unexplored. Aperiodic perturbations are also inevitable in experimental setups because of imperfect control over the time variation of the driving field. Our mathematical analysis reveals the emergence of steady states that arise only in the presence of aperiodic driving.

We consider a class of minimal models of interacting quantum spins that allow for a tractable analysis when driven aperiodically in time. We find that new steady states occur in at least two cases. Along the way, we also introduce new coarse-grained quantities in momentum space that completely determine the local properties of the system and are useful for studying driven systems in general.

Our work shows that aperiodically driven systems can have much richer behaviors over long time scales than their periodically driven counterparts and that classifying the full range of possible steady states is a worthwhile endeavor. One interesting research direction would be to study if quasiperiodic (i.e., neither fully random nor periodic) driving can avoid heating that is seen in generic driven systems.

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Phys. Rev. X 7, 031034– Published 22 August, 2017
  • Received 23 February 2017
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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