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arXiv:2108.11421 · Physical review. B./Physical review. B

From prethermalization to chaos in periodically driven coupled rotors

Yonathan Sadia, Emanuele G. Dalla Torre, Atanu Rajak

Published May 3, 2022• 6 citations
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Abstract

Periodically driven (Floquet) systems are said to prethermalize when their energy absorption is very slow for a long time. This effect was first discovered in quantum spin models, where the heating rate is exponentially small in the ratio between the driving frequency and the spin bandwidth. Recently, it was shown that prethermalization occurs also in classical systems with an infinite bandwidth. Here, we address the open question of which small parameter controls the lifetime of the prethermal state in these systems. We show that the lifetime is controlled by the temperature of the prethermal state, which is quasiconserved when the heating is slow. We substantiate this finding in systems of periodically driven coupled rotors, by studying the dependence of the prethermal lifetime on both the initial conditions and the connectivity. This result allows us to develop a simple analytical model that describes the crossover from prethermalization to chaos in many-body classical systems.

DOI
10.1103/physrevb.105.184302

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Periodically driven (Floquet) systems are said to prethermalize when their energy absorption is very slow for a long time. This effect was first discovered in quantum spin models, where the heating rate is exponentially small in the ratio between the driving frequency and the spin bandwidth. Recently, it was shown that prethermalization occurs also in classical systems with an infinite bandwidth. Here, we address the open question of which small parameter controls the lifetime of the prethermal state in these systems. We show that the lifetime is controlled by the temperature of the prethermal state, which is quasiconserved when the heating is slow. We substantiate this finding in systems of periodically driven coupled rotors, by studying the dependence of the prethermal lifetime on both the initial conditions and the connectivity. This result allows us to develop a simple analytical model that describes the crossover from prethermalization to chaos in many-body classical systems.
It is proved that the energy absorption in a periodically driven classical spin system is exponentially slow in frequency, which results in a two-step relaxation called the Floquet prethermalization. This result is shown by establishing the classical limit of the quantum spin dynamics. The Floquet prethermal state is well described by the Gibbs ensemble with respect to the static effective Hamiltonian obtained by a truncation of the Floquet-Magnus expansion. On the other hand, the same effective Hamiltonian does not reproduce the local dynamics for an exponentially long time. This is due to the chaoticity of classical dynamics and in stark contrast to quantum spin systems, in which the effective Hamiltonian well reproduces the exact quantum dynamics up to an exponentially long time.
Prethermalization, by introducing emergent quasiconserved observables, plays a crucial role in protecting periodically driven (Floquet) many-body phases over an exponentially long time, while the ultimate fate of such quasiconserved operators can signal thermalization to infinite temperature. To elucidate the properties of prethermal quasiconservation in many-body Floquet systems, here we systematically analyze infinite-temperature correlations between observables. We numerically show that the late-time behavior of the autocorrelations unambiguously distinguishes quasiconserved observables from nonconserved ones, allowing one to single out a set of linearly independent quasiconserved observables. By investigating two Floquet spin models, we identify two different mechanisms underlying the quasiconservation law. First, we numerically verify energy quasiconservation when the driving frequency is large, so that the system dynamics is approximately described by a static prethermal Hamiltonian. More interestingly, under moderate driving frequency, another quasiconserved observable can still persist if the Floquet driving contains a large global rotation. We show theoretically how to calculate this conserved observable and provide numerical verification. Having systematically identified all quasiconserved observables, we can finally investigate their behavior in the infinite-time limit and thermodynamic limit, using autocorrelations obtained from both numerical simulation and experiments in solid-state nuclear magnetic resonance systems.
We reveal a continuous dynamical heating transition between a prethermal and an infinite-temperature stage in a clean, chaotic periodically driven classical spin chain. The transition time is a steep exponential function of the drive frequency, showing that the exponentially long-lived prethermal plateau, originally observed in quantum Floquet systems, survives the classical limit. Even though there is no straightforward generalization of Floquet's theorem to nonlinear systems, we present strong evidence that the prethermal physics is well described by the inverse-frequency expansion. We relate the stability and robustness of the prethermal plateau to drive-induced synchronization not captured by the expansion. Our results set the pathway to transfer the ideas of Floquet engineering to classical many-body systems, and are directly relevant for photonic crystals and cold atom experiments in the superfluid regime.
We introduce well-defined characterizations of prethermal states in realistic periodically driven many-body systems with unbounded chaotic diffusion of the kinetic energy. These systems, interacting arrays of periodically kicked rotors, are paradigmatic models of many-body chaos theory. We show that the prethermal states in these systems are well described by a generalized Gibbs ensemble based essentially on the average Hamiltonian. The latter is the quasiconserved quantity in the prethermal state and the ensemble is characterized by the temperature of the state. An explicit exact expression for this temperature is derived. Using also arguments based on chaos theory, we demonstrate that the lifetime of the prethermal state is exponentially long in the inverse of the temperature, in units of the driving frequency squared. Our analytical results, in particular those for the temperature and the lifetime of the prethermal state, agree well with numerical observations.
We establish some general dynamical properties of quantum many-body systems that are subject to a high-frequency periodic driving. We prove that such systems have a quasiconserved extensive quantity H∗{H}_{*}H∗​, which plays the role of an effective static Hamiltonian. The dynamics of the system (e.g., evolution of any local observable) is well approximated by the evolution with the Hamiltonian H∗{H}_{*}H∗​ up to time ${\ensuremath{\tau}}_{*}$, which is exponentially large in the driving frequency. We further show that the energy absorption rate is exponentially small in the driving frequency. In cases where H∗{H}_{*}H∗​ is ergodic, the driven system prethermalizes to a thermal state described by H∗{H}_{*}H∗​ at intermediate times $t\ensuremath{\lesssim}{\ensuremath{\tau}}_{*}$, eventually heating up to an infinite-temperature state after times $t\ensuremath{\sim}{\ensuremath{\tau}}_{*}$. Our results indicate that rapidly driven many-body systems generically exhibit prethermalization and very slow heating. We briefly discuss implications for experiments which realize topological states by periodic driving.
A many-body interacting system of classical kicked rotor serves as a prototypical model for studying Floquet heating dynamics. Having established the fact that this system exhibits a long-lived prethermal phase with a quasiconserved average Hamiltonian before entering into the chaotic heating regime, we use spatiotemporal fluctuation correlation of kinetic energy as a two-point observable to probe the above dynamic phases. We remarkably find the diffusive transport of fluctuation in the prethermal regime suggesting an underlying hydrodynamic picture in a generalized Gibbs ensemble with a definite temperature that depends on the driving parameter and the initial conditions. On the other hand, the heating regime is characterized by a diffusive growth of kinetic energy where the correlation is sharply localized around the fluctuation center for all time. Consequently, we attribute nondiffusive and nonlocalized structure of correlation to the crossover regime, connecting the prethermal phase to the heating phase, where the kinetic energy displays a complicated growth structure. We understand these numerical findings using the notion of relative phase matching where the prethermal phase (heating regime) refers to an effectively coupled (isolated) nature of the rotors. We exploit the statistical uncorrelated nature of the angles of the rotors in the heating regime to find the analytical form of the correlator that mimics our numerical results in a convincing way.