Prethermal quantum many-body Kapitza phases of periodically driven spin systems
Alessio Lerose, Jamir Marino, Andrea Gambassi, Alessandro Silva
备注：
Significantly improved and enlarged version (12 pages, 8 figures). Detailed discussion of the effects of algebraically decaying interactions added. Discussion of quantum simulations with trapped ions added. Selected as "Editors' suggestion" in Physical Review B
As realized by Kapitza long ago, a rigid pendulum can be stabilized upside down by periodically driving its suspension point with tuned amplitude and frequency. While this dynamical stabilization is feasible in a variety of instances in systems with few degrees of freedom, it is natural to search for generalizations to multi-particle systems. In particular, a fundamental question is whether, by periodically driving a single parameter in a many-body system, one can stabilize an otherwise unstable phase of matter against all possible fluctuations of its microscopic degrees of freedom. In this work we show that such stabilization occurs in experimentally realizable quantum many-body systems: a periodic modulation of a transverse magnetic field can make ferromagnetic spin systems with long-range interactions stably trapped around unstable paramagnetic configurations as well as in other unconventional dynamical phases with no equilibrium counterparts. We demonstrate that these quantum Kapitza phases have a long lifetime and can be observed in current experiments with trapped ions.
Periodically driven (Floquet) systems are said to prethermalize when their energy absorption is very slow for 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, first, numerically study the dependence of the lifetime on the initial con…
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 th…
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…
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…
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 quasi-conserved quantity in the prethermal state and the ensemble is characterized by the temperature of the state. An explicit exact expression …
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 relat…
We develop a flow renormalization approach for periodically-driven quantum systems, which reveals prethermal dynamical regimes and associated timescales via direct correspondence between real time and flow time behavior. In this formalism, the dynamical problem is recast in terms of coupling constants of the theory flowing towards an attractive fixed point that represents the thermal Floquet Hamiltonian at long times, while narrowly avoiding a series of unstable fixed points which determine distinct prethermal regimes at intermediate times. We study a class of relevant perturbations that trigg…
We discuss the effect of dissipation on heating which occurs in periodically driven quantum many body systems. We especially focus on a periodically driven Bose-Hubbard model coupled to an energy and particle reservoir. Without dissipation, this model is known to undergo parametric instabilities which can be considered as an initial stage of heating. By taking the weak on-site interaction limit as well as the weak system-reservoir coupling limit, we find that parametric instabilities are suppressed if the dissipation is stronger than the on-site interaction strength and stable steady states ap…
We establish some general dynamical properties of lattice many-body systems that are subject to a high-frequency periodic driving. We prove that such systems have a quasi-conserved extensive quantity $H_*$, 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 $H_*$ up to time $τ_*$, which is exponentially long in the driving frequency. We further show that the energy absorption rate is exponentially small in the driving frequency. In cases where $H_*$ is ergodic, …
Prethermalization refers to the transient phenomenon where a system thermalizes according to a Hamiltonian that is not the generator of its evolution. We provide here a rigorous framework for quantum spin systems where prethermalization is exhibited for very long times. First, we consider quantum spin systems under periodic driving at high frequency $ν$. We prove that up to a quasi-exponential time $τ_* \sim e^{c \fracν{\log^3 ν}}$, the system barely absorbs energy. Instead, there is an effective local Hamiltonian $\hat D$ that governs the time evolution up to $τ_*$, and hence this effective H…
Having established the fact that interacting classical kicked rotor systems exhibit long-lived prethermal phase with quasi-conserved average Hamiltonian before entering into chaotic heating regime, we use spatio-temporal fluctuation correlation of kinetic energy to probe the above dynamic phases. We remarkably find the diffusive transport of fluctuation in the prethermal regime reminding us the underlying hydrodynamic picture in a generalized Gibbs ensemble with a definite temperature that depends on the driving parameter and initial conditions. On the other hand, the heating regime, character…
We prove the existence of non-equilibrium phases of matter in the prethermal regime of periodically-driven, long-range interacting systems, with power-law exponent $α> d$, where $d$ is the dimensionality of the system. In this context, we predict the existence of a disorder-free, prethermal discrete time crystal in one dimension -- a phase strictly forbidden in the absence of long-range interactions. Finally, using a combination of analytic and numerical methods, we highlight key experimentally observable differences between such a prethermal time crystal and its many-body localized counterpar…
Periodically driven Floquet quantum systems provide a promising platform to investigate novel physics out of equilibrium. Unfortunately, the drive generically heats up the system to a featureless infinite temperature state. For large driving frequency, the heat absorption rate is predicted to be exponentially small, giving rise to a long-lived prethermal regime which exhibits all the intriguing properties of Floquet systems. Here we experimentally observe Floquet prethermalization using nuclear magnetic resonance techniques. We first show the relaxation of a far-from-equilibrium initial state …
Prethermalization, where quasi-steady states are realized in the intermediate long time regime (prethermal regime), in periodically driven (Floquet) systems is an important phenomenon since it provides a platform of nontrivial Floquet many-body physics. In this Letter, we consider Floquet systems with dual energy scales: the Hamiltonian consists of two different terms whose amplitude is either comparable or much smaller than the frequency. As a result, when the larger-amplitude drive induces a $\mathbb{Z}_N$ symmetry operation, we obtain the effective static Hamiltonian respecting a new emerge…
We demonstrate that the prethermal regime of periodically driven (Floquet), classical many-body systems can host nonequilibrium phases of matter. In particular, we show that there exists an effective Hamiltonian that captures the dynamics of ensembles of classical trajectories despite the breakdown of this description at the single trajectory level. In addition, we prove that the effective Hamiltonian can host emergent symmetries protected by the discrete time-translation symmetry of the drive. The spontaneous breaking of such an emergent symmetry leads to a subharmonic response, characteristi…
I demonstrate the potential of reinforcement learning (RL) to prepare quantum states of strongly periodically driven non-linear single-particle models. The ability of Q-Learning to control systems far away from equilibrium is exhibited by steering the quantum Kapitza oscillator to the Floquet-engineered stable inverted position in the presence of a strong periodic drive within several shaking cycles. The study reveals the potential of the intra-period (micromotion) dynamics, often neglected in Floquet engineering, to take advantage over pure stroboscopic control at moderate drive frequencies. …
In periodically-driven quantum systems, resonances can induce exotic nonequilibrium behavior and new phases of matter without static analog. We report on the emergence of fractional and integer resonances in a broad class of many-body Hamiltonians with a modulated hopping with a frequency that is either a fraction or an integer of the on-site interaction. We contend that there is a fundamental difference between these resonances when interactions bring the system to a Floquet prethermal state. Second-order processes dominate the dynamics in the fractional resonance case, leading to less entang…
We analyze a version of the sine-Gordon model in which the strength of the cosine potential has a periodic dependence on time. This model can be considered as the continuum limit of the many body generalization of the Kapitza pendulum. Based on the perturbed CFT point of view, we develop a truncated conformal space approach (TCSA) to investigate the Floquet quasienergy spectrum. We focus on the effective behaviour for large driving frequencies, which we also derive exactly. Depending on the driving protocol, we can recover the original sine-Gordon model or its two-frequency version. The rich s…
The conventional framework for defining and understanding phases of matter requires thermodynamic equilibrium. Extensions to non-equilibrium systems have led to surprising insights into the nature of many-body thermalization and the discovery of novel phases of matter, often catalyzed by driving the system periodically. The inherent heating from such Floquet drives can be tempered by including strong disorder in the system, but this can also mask the generality of non-equilibrium phases. In this work, we utilize a trapped-ion quantum simulator to observe signatures of a non-equilibrium driven …
Long-range interacting systems such as nitrogen vacancy centers in diamond and trapped ions serve as useful experimental setups to probe a range of nonequilibrium many-body phenomena. In particular, via driving, various effective Hamiltonians with physics potentially quite distinct from short-range systems can be realized. In this Letter, we derive general bounds on the linear response energy absorption rates of periodically driven systems of spins or fermions with long-range interactions that are sign changing and fall off as $1/r^α$ with $α> d/2$. We show that the disordered averaged energy …
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 c…
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 quasi-conserved quantity for long time scale. These two general properties lead to intriguing classification on the initial stage of relaxation, one of which is similar to the prethermalization phenomenon i…
Periodically driven quantum matter can realize exotic dynamical phases. In order to understand how ubiquitous and robust these phases are, it is pertinent to investigate the heating dynamics of generic interacting quantum systems. Here we study the thermalization in a periodically-driven generalized Sachdev-Ye-Kitaev (SYK)-model, which realizes a crossover from a heavy Fermi liquid (FL) to a non-Fermi liquid (NFL) at a tunable energy scale. Developing an exact field theoretic approach, we determine two distinct regimes in the heating dynamics. While the NFL heats exponentially and thermalizes …
From stable periodic orbits to many-body chaos: doubly tunable prethermalization via engineering of an emergent band structure
State Key Laboratory of Artificial Microstructure and Mesoscopic Physics, School of Physics, Peking University（北京大学物理学院人工微结构和介观物理国家重点实验室）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Cavendish Laboratory, University of Cambridge（剑桥大学卡文迪许实验室）；Technical University of Munich, TUM School of Natural Sciences, Physics Department（慕尼黑工业大学自然科学院物理系）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）
The critical properties characterizing the formation of the Floquet time crystal in the prethermal phase are investigated analytically in the periodically driven $O(N)$ model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and absence of long-range spatial order from the non-trivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for $N\to\infty$, we determine the exponent $ν$ that characterizes the di…
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 t…
It is by now well established that periodically driven quantum many-body systems can realize topological nonequilibrium phases without any equilibrium counterpart. Here we show that, even in the absence of time translation symmetry, nonequilibrium topological phases of matter can exist in aperiodically driven systems for tunably parametrically long prethermal lifetimes. As a prerequisite, we first demonstrate the existence of longlived prethermal Anderson localization in two dimensions under random multipolar driving. We then show that the localization may be topologically nontrivial with a qu…
This work explores a fundamental dynamical structure for a wide range of many-body quantum systems under periodic driving. Generically, in the thermodynamic limit, such systems are known to heat up to infinite temperature states after infinite-time evolution, irrespective of dynamical details. In the present study, instead of considering infinitely long-time scale, we aim to provide a framework to understand the long but finite time behavior, namely the transient dynamics. In the analysis, we focus on the Floquet-Magnus (FM) expansion that gives a formal expression of the effective Hamiltonian…
Localization may survive in periodically driven (Floquet) quantum systems, but is generally unstable for aperiodic drives. In this work, we identify a hidden conservation law originating from a chiral symmetry in a disordered spin-1/2 XX chain. This protects indefinitely long-lived localization for general--even aperiodic--drives. Therefore, rather counter-intuitively, adding further potential disorder which spoils the conservation law delocalizes the system, via a controllable parametrically long-lived prethermal regime. This provides a first example of persistent single-particle `localizatio…
The nonequilibrium quantum dynamics of closed many-body systems is a rich yet challenging field. While recent progress for periodically driven (Floquet) systems has yielded a number of rigorous results, our understanding on quantum many-body systems driven by rapidly varying but a- and quasi-periodic driving is still limited. Here, we derive rigorous, non-perturbative, bounds on the heating rate in quantum many-body systems under Thue-Morse quasi-periodic driving and under random multipolar driving, the latter being a tunably randomized variant of the former. In the process, we derive a static…
A tremendous amount of recent attention has focused on characterizing the dynamical properties of periodically driven many-body systems. Here, we use a novel numerical tool termed `density matrix truncation' (DMT) to investigate the late-time dynamics of large-scale Floquet systems. We find that DMT accurately captures two essential pieces of Floquet physics, namely, prethermalization and late-time heating to infinite temperature. Moreover, by implementing a spatially inhomogeneous drive, we demonstrate that an interplay between Floquet heating and diffusive transport is crucial to understandi…
Demonstrating the ability of existing quantum platforms to perform certain computational tasks intractable to classical computers represents a cornerstone in quantum computing. Despite the growing number of such proposed "quantum supreme" tasks, it remains an important challenge to identify their direct applications. In this work, we describe how the approach proposed in Ref. [arXiv:2002.11946] for demonstrating quantum supremacy in generic driven analog many-body systems, such as those found in cold atom and ion setups, can be extended to explore dynamical quantum phase transitions. We show h…
The idea of breaking time-translation symmetry has fascinated humanity at least since ancient proposals of the perpetuum mobile. Unlike the breaking of other symmetries, such as spatial translation in a crystal or spin rotation in a magnet, time translation symmetry breaking (TTSB) has been tantalisingly elusive. We review this history up to recent developments which have shown that discrete TTSB does takes place in periodically driven (Floquet) systems in the presence of many-body localization. Such Floquet time-crystals represent a new paradigm in quantum statistical mechanics --- that of an…
