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Physical review. E

Reservoir-induced stabilization of a periodically driven many-body system

Thomas Veness, Kay Brandner

Published Oct 27, 2023• 3 citations
Publisher

Abstract

Exploiting the rich phenomenology of periodically driven many-body systems is notoriously hindered by persistent heating in both the classical and the quantum realm. Here, we investigate to what extent coupling to a large thermal reservoir makes stabilization of a nontrivial steady state possible. To this end, we model both the system and the reservoir as classical spin chains where driving is applied through a rotating magnetic field, and we simulate the Hamiltonian dynamics of this setup. We find that the intuitive limits of infinite frequency and vanishing frequency, where the system dynamics is governed by the average and the instantaneous Hamiltonian, respectively, can be smoothly extended into entire regimes separated only by a small crossover region. At high frequencies, the driven system stroboscopically attains a Floquet-type Gibbs state at the reservoir temperature. At low frequencies, a global synchronized Gibbs state emerges, whose temperature may depart significantly from the initial temperature of the reservoir. Although our analysis in some parts relies on the specific properties of our setup, we argue that much of its phenomenology could be generic.

DOI
10.1103/physreve.108.l042102

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Exploiting the rich phenomenology of periodically driven many-body systems is notoriously hindered by persistent heating in both the classical and the quantum realm. Here, we investigate to what extent coupling to a large thermal reservoir makes stabilization of a nontrivial steady state possible. To this end, we model both the system and the reservoir as classical spin chains where driving is applied through a rotating magnetic field, and we simulate the Hamiltonian dynamics of this setup. We find that the intuitive limits of infinite frequency and vanishing frequency, where the system dynamics is governed by the average and the instantaneous Hamiltonian, respectively, can be smoothly extended into entire regimes separated only by a small crossover region. At high frequencies, the driven system stroboscopically attains a Floquet-type Gibbs state at the reservoir temperature. At low frequencies, a global synchronized Gibbs state emerges, whose temperature may depart significantly from the initial temperature of the reservoir. Although our analysis in some parts relies on the specific properties of our setup, we argue that much of its phenomenology could be generic.
Exploiting the rich phenomenology of periodically-driven many-body systems is notoriously hindered by persistent heating in both the classical and quantum realm. Here, we investigate to what extent coupling to a large thermal reservoir makes stabilisation of a non-trivial steady state possible. To this end, we model both the system and the reservoir as classical spin chains where driving is applied through a rotating magnetic field, and simulate the Hamiltonian dynamics of this setup. We find that the intuitive limits of infinite and vanishing frequency, where the system dynamics is governed by the average and the instantaneous Hamiltonian, respectively, can be smoothly extended into entire regimes separated only by a small crossover region. At high frequencies, the driven system stroboscopically attains a Floquet-type Gibbs state at the reservoir temperature. At low frequencies, a synchronised Gibbs state emerges, whose temperature may depart significantly from that of the reservoir. Although our analysis in some parts relies on the specific properties our setup, we argue that much of its phenomenology should be generic for a large class of systems.
Floquet theory is an indispensable tool for analyzing periodically driven quantum many-body systems. Although it does not universally extend to classical systems, some of its methodologies can be adopted in the presence of well-separated timescales. Here we use these tools to investigate the stroboscopic behaviors of a classical spin chain that is driven by a periodic magnetic field and coupled to a thermal reservoir. We detail and expand our previous work: we investigate the significance of higher-order corrections to the classical Floquet-Magnus expansion in both the high- and low-frequency regimes; explicitly probe the evolution dynamics of the reservoir; and further explore how the driven system and the reservoir synchronize with the applied field at low frequencies. In line with our earlier results, we find that the high-frequency regime is characterized by a local Floquet-Gibbs ensemble with the reservoir acting as a nearly-reversible heat sink. At low frequencies, the driven system rapidly enters a synchronized state, which can only be fully described in a global picture accounting for the concurrent relaxation of the reservoir in a fictitious magnetic field arising from the drive. We highlight how the evolving nature of the reservoir may still be incorporated in a local picture by introducing an effective temperature. Finally, we show that generic local-dissipation models that account for the influence of the reservoir on the driven system phenomenologically through Markovian dissipative equations of motion can generally not reproduce the rich behavior that our microscopic simulations reveal. In particular, such models prove insufficient to account for the suppression of overall energy absorption that is induced by the here observed synchronization between driven system and reservoir.
Floquet theory is an indispensable tool for analysing periodically-driven quantum many-body systems. Although it does not universally extend to classical systems, some of its methodologies can be adopted in the presence of well-separated timescales. Here we use these tools to investigate the stroboscopic behaviours of a classical spin chain that is driven by a periodic magnetic field and coupled to a thermal reservoir. We detail and expand our previous work: we investigate the significance of higher-order corrections to the classical Floquet-Magnus expansion in both the high- and low-frequency regimes; explicitly probe the evolution the dynamics of the reservoir; and further explore how the driven system synchronises with the applied field at low frequencies. In line with our earlier results, we find that the high-frequency regime is characterised by a local Floquet-Gibbs ensemble with the reservoir acting as a nearly-reversible heatsink. At low frequencies, the driven system rapidly enters a synchronised state, which can only be fully described in a global picture accounting for the concurrent relaxation of the reservoir in a fictitious magnetic field arising from the drive. We highlight how the evolving nature of the reservoir may still be incorporated in a local picture by introducing an effective temperature. Finally, we argue that dissipative equations of motion for periodically-driven many-body systems, at least at intermediate frequencies, must generically be non-Markovian.
We study the dynamics of periodically kicked many-body systems away from the high-frequency regime, and discuss a family of Floquet systems where the notion of prethermalization can be naturally extended to intermediate and low driving frequencies. We investigate numerically the dynamics of both integrable and nonintegrable systems, and report on the formation of a long-lived prethermal plateau, akin to the high-frequency limit, where the system thermalizes with respect to an effective Hamiltonian captured by the inverse-frequency expansion (IFE). Unlike the high-frequency regime, we find that the relevant heating times are model dependent: we analyze the stability of the prethermal plateau to small perturbations in the drive period and show that, in a spin chain whose IFE is intractable, the plateau duration is insensitive to the perturbation strength, in contrast to a chain where the IFE admits the resummation of an entire subseries. Infinitesimal perturbations are enough to restore the ergodic properties of the system, and decrease residual finite-size effects. Although the regime where the Floquet system leaves the prethermal plateau and starts heating up to infinite temperature is not captured by the IFE, we provide evidence that the evolved subsystem is described well by a thermal state with respect to the IFE Hamiltonian, with a gradually changing temperature, in accord with the eigenstate thermalization hypothesis.
We give a general overview of the high-frequency regime in periodically driven systems and identify three distinct classes of driving protocols in which the infinite-frequency Floquet Hamiltonian is not equal to the time-averaged Hamiltonian. These classes cover systems, such as the Kapitza pendulum, the Harper–Hofstadter model of neutral atoms in a magnetic field, the Haldane Floquet Chern insulator and others. In all setups considered, we discuss both the infinite-frequency limit and the leading finite-frequency corrections to the Floquet Hamiltonian. We provide a short overview of Floquet theory focusing on the gauge structure associated with the choice of stroboscopic frame and the differences between stroboscopic and non-stroboscopic dynamics. In the latter case, one has to work with dressed operators representing observables and a dressed density matrix. We also comment on the application of Floquet Theory to systems described by static Hamiltonians with well-separated energy scales and, in particular, discuss parallels between the inverse-frequency expansion and the Schrieffer–Wolff transformation extending the latter to driven systems.
We investigate a class of periodically driven many-body systems that allows us to extend the phenomenon of prethermalization to the vicinity of isolated intermediate-to-low drive frequencies away from the high-frequency limit. We provide numerical evidence for the formation of a parametrically long-lived prethermal plateau, captured by an effective Floquet Hamiltonian computed using the replica inverse-frequency expansion, and demonstrate its stability with respect to random perturbations in the drive period. Considering exclusively nonintegrable Floquet Hamiltonians, we find that heating rates are nonuniversal: we observe Fermi's golden rule scaling, power-law scaling inconsistent with the golden rule, and non-power-law scaling, depending on the drive. Despite the asymptotic character of the inverse-frequency expansion, we show that it describes the thermostatic properties of the state all along the evolution up to infinite temperature, with higher-order terms improving the accuracy. Our results suggest a dynamical mechanism to gradually increase the temperature in isolated quantum simulators, such as ultracold atoms, and open up an alternative possibility to investigate thermal phase transitions and the interplay between thermal and quantum criticality using Floquet drives.