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arXiv:2404.06667

Slow relaxation of quasi-periodically driven integrable quantum many-body systems

Souradeep Ghosh, Sourav Bhattacharjee, Souvik Kumar Bandyopadhyay

Published Apr 10, 2024• 0 citations
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Abstract

We study the emergence and stability of a prethermal phase in an integrable many-body system subjected to a Fibonacci drive. Despite not being periodic, Fibonacci drives have been shown to introduce dynamical constraints due to their self-similar structure, unlike random driving protocols. From perturbative analysis, this has been argued to result in an exponentially long prethermal phase in the high frequency limit of driving. Examining higher order terms in the perturbative expansion, we show that the perturbative description breaks down eventually in such systems at a finite universal order, which depends solely on features of the Fibonacci sequence. This leads to an onset of energy absorption at long time scales for intermediate and low driving frequencies. Interestingly, in spite of the breakdown of an effective Hamiltonian in the perturbative analysis, we still observe slow logarithmic heating time-scales, unlike purely random drives.

DOI
10.48550/arxiv.2404.06667

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We study the emergence and stability of a prethermal phase in an integrable many-body system subjected to a Fibonacci drive. Despite not being periodic, Fibonacci drives have been shown to introduce dynamical constraints due to their self-similar structure, unlike random driving protocols. From perturbative analysis, this has been argued to result in an exponentially long prethermal phase in the high frequency limit of driving. Examining higher order terms in the perturbative expansion, we show that the perturbative description breaks down eventually in such systems at a finite universal order, which depends solely on features of the Fibonacci sequence. This leads to an onset of energy absorption at long time scales for intermediate and low driving frequencies. Interestingly, in spite of the breakdown of an effective Hamiltonian in the perturbative analysis, we still observe slow logarithmic heating time-scales, unlike purely random drives.
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.
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.
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 w.r.t.~random perturbations in the drive period. Considering exclusively nonintegrable Floquet Hamiltonians, we find that heating rates are non-universal: 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 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.
We simulate the dynamics of a disordered interacting spin chain subject to a quasiperiodic time-dependent drive, corresponding to a stroboscopic Fibonacci sequence of two distinct Hamiltonians. Exploiting the recursive drive structure, we can efficiently simulate exponentially long times. After an initial transient, the system exhibits a long-lived glassy regime characterized by a logarithmically slow growth of entanglement and decay of correlations analogous to the dynamics at the many-body delocalization transition. Ultimately, at long time scales, which diverge exponentially for weak or rapid drives, the system thermalizes to infinite temperature. The slow relaxation enables metastable dynamical phases, exemplified by a "time quasicrystal" in which spins exhibit persistent oscillations with a distinct quasiperiodic pattern from that of the drive. We show that in contrast with Floquet systems, a high-frequency expansion strictly breaks down above fourth order, and fails to produce an effective static Hamiltonian that would capture the prethermal glassy relaxation.
We discuss the universal nature of relaxation in isolated many-body quantum systems subjected to global and strong periodic driving. Our rigorous Floquet analysis shows that the energy of the system remains almost constant up to an exponentially long time in frequency for arbitrary initial states and that an effective Hamiltonian obtained by a truncation of the Floquet-Magnus expansion is a quasiconserved quantity in a long time scale. These two general properties lead to an intriguing classification on the initial stage of relaxation, one of which is similar to the prethermalization phenomenon in nearly integrable systems.
It is well understood that many-body systems driven at high frequency heat up only exponentially slowly and exhibit a long prethermalization regime. We prove rigorously that a certain relevant class of systems heat very slowly under weak periodic driving at intermediate frequency as well. This class of systems are those whose time-dependent, possibly translation-invariant, Hamiltonian is a weak perturbation of a sum of mutually commuting terms. This condition covers several periodically kicked systems that have been considered in the literature recently, in particular kicked Ising models. In contrast to the high-frequency regime, the prethermalization dynamics of our systems is in general not related to any time-independent effective Floquet Hamiltonian. Our results also have non-trivial implications for closed (time-independent) systems. We use the example of an Ising model with transversal and longitudinal field to show how they imply confinement of excitations. More generally, they show how "glassy" kinetically constrained models emerge naturally from simple many-body Hamiltonians, thus connecting to the topic of 'translation-invariant localization'.