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arXiv:1906.06989 · Physical review. B./Physical review. B

Floquet time spirals and stable discrete-time quasicrystals in quasiperiodically driven quantum many-body systems

Hongzheng Zhao, Florian Mintert, Johannes Knolle

Published Oct 2, 2019• 48 citations
arXivHTMLPDF

Abstract

We analyze quasiperiodically driven quantum systems that can be mapped exactly to periodically driven ones and find Floquet time spirals in analogy with spatially incommensurate spiral magnetic states. Generalizing the mechanism to many-body systems, we discover that a form of discrete-time translation symmetry breaking can also occur in quasiperiodically driven systems. We construct a discrete-time quasicrystal stabilized by many-body localization. Crucially, it persists also under perturbations that break the equivalence with periodic systems. As such, it provides evidence of a stable quasiperiodically driven many-body quantum system which does not heat up to the featureless infinite-temperature state.

DOI
10.1103/physrevb.100.134302

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We analyze quasiperiodically driven quantum systems that can be mapped exactly to periodically driven ones and find Floquet time spirals in analogy with spatially incommensurate spiral magnetic states. Generalizing the mechanism to many-body systems, we discover that a form of discrete-time translation symmetry breaking can also occur in quasiperiodically driven systems. We construct a discrete-time quasicrystal stabilized by many-body localization. Crucially, it persists also under perturbations that break the equivalence with periodic systems. As such, it provides evidence of a stable quasiperiodically driven many-body quantum system which does not heat up to the featureless infinite-temperature state.
Floquet (periodically driven) systems can give rise to unique nonequilibrium phases of matter without equilibrium analogs. The most prominent example is the realization of discrete time crystals. An intriguing question emerges: What other novel phases can manifest when the constraint of time periodicity is relaxed? In this study, we explore quantum systems subjected to a quasiperiodic drive. Leveraging a strongly interacting spin ensemble in diamond, we identify the emergence of long-lived discrete time quasicrystals. Unlike conventional time crystals, time quasicrystals exhibit robust subharmonic responses at multiple incommensurate frequencies. Furthermore, we show that the multifrequency nature of the quasiperiodic drive allows for the formation of diverse patterns associated with different discrete time quasicrystalline phases. Our findings demonstrate the existence of nonequilibrium phases in quasi-Floquet settings, significantly broadening the catalog of novel phenomena in driven many-body quantum systems.
We study prethermal time quasicrystalline (TQC) order in a quasiperiodically driven chain of non-interacting spin-1/2 particles. The drive consists of two parts, switched on and off periodically with frequency ωdω_dωd​: (i) disordered Ising interactions, with exchange couplings chosen from a symmetric interval [−J/2,J/2][-J/2, J/2][−J/2,J/2], allowing random antiferromagnetic or ferromagnetic nearest-neighbor couplings, together with a random transverse field; and (ii) a rotating transverse magnetic field with frequency ΩΩΩ. The ratio ωd/Ωω_d/Ωωd​/Ω is chosen to be irrational, producing multiple incommensurate frequencies and yielding quasiperiodic dynamics beyond Floquet theory. Using exact diagonalization, we analyze the time autocorrelation function, dynamical structure factor, and entanglement entropy (EE). In the high-frequency regime, robust spectral peaks at incommensurate frequencies (not integer multiples of the fundamental drives) signal quasiperiodic time-translation symmetry breaking (QTTSB). The EE exhibits sublinear power-law growth followed by a prethermal plateau, indicating suppressed resonant heating due to an energy scale mismatch. The nonequilibrium lifetime increases rapidly with driving frequency. Unlike symmetric disorder sampling, an asymmetric distribution of the Ising exchange couplings induces collective spin rigidity, enhancing the system's resistance to heating. The TQC phase remains stable against next-nearest-neighbor (NNN) exchange perturbations and rotational imperfections, with robustness comparable to discrete time crystals (TCs) under periodic driving. Our results establish this quasiperiodically driven system as a platform for long-lived nonequilibrium temporal order, revealing the interplay of disorder, collective rigidity, and quasiperiodic driving.
Boundary time crystals exhibit spontaneous breaking of continuous time-translation symmetry through persistent periodic oscillations in driven-dissipative many-body systems. Here, we show that multilevel interference provides a natural route beyond periodic order, enabling dissipative time quasicrystals without externally imposed quasiperiodic driving. We consider a collectively driven-dissipative four-level ensemble with two degenerate excited states and two degenerate ground states. In the thermodynamic limit, the exact mean-field dynamics reduces to an irrational flow on a two-dimensional torus, yielding quasiperiodic order parameters with discrete spectra generated by two incommensurate fundamental frequencies. Vanishing maximal Lyapunov exponents demonstrate that the nonlinear self-consistent dynamics remains nonchaotic. Our results establish a minimal interference-induced mechanism for time-quasiperiodic order and open a route toward higher-dimensional quasiperiodic dynamics in multilevel systems.
Quasi-periodic driving protocols provide a powerful route to realize novel non-equilibrium phases of matter beyond the Floquet paradigm. However, these protocols inevitably lead to infinite-temperature heat death in isolated systems, which poses a major challenge to their experimental realization. We demonstrate that dissipation can be harnessed to stabilize quasi-periodically driven systems, enabling the realization of robust non-equilibrium phases. Using the paradigmatic open Dicke model, we provide a blueprint for realizing a stable time quasicrystal (TQC) by Fibonacci driving. This TQC is characterized by a robust sub-harmonic quasi-periodic response that is dictated by, but qualitatively distinct from the external Fibonacci drive. By directly analyzing the time evolution in the thermodynamic limit, we establish the existence of TQC order in this system for a wide parameter regime. We trace the origin of the stability of the TQC to the attractor structure induced by dissipation. Strikingly, the TQC order persists in the deep quantum regime with as few as two qubits. We systematically study the dependence of the TQC lifetime, τ∗τ^{\ast}τ∗, on the number of qubits and demonstrate that τ∗τ^{\ast}τ∗ increases monotonically with the system size. Crucially, the TQC is not observed in the absence of dissipation. Our work establishes dissipation as a mechanism for stabilizing non-equilibrium phases of matter under quasi-periodic drive.
We show that locally interacting, periodically driven (Floquet) Hamiltonian dynamics coupled to a Langevin bath support finite-temperature discrete time crystals (DTCs) with an infinite autocorrelation time. By contrast to both prethermal and many-body localized DTCs, the time crystalline order we uncover is stable to arbitrary perturbations, including those that break the time translation symmetry of the underlying drive. Our approach utilizes a general mapping from probabilistic cellular automata to open classical Floquet systems undergoing continuous-time Langevin dynamics. Applying this mapping to a variant of the Toom cellular automaton, which we dub the "π-Toom time crystal," leads to a 2D Floquet Hamiltonian with a finite-temperature DTC phase transition. We provide numerical evidence for the existence of this transition, and analyze the statistics of the finite temperature fluctuations. Finally, we discuss how general results from the field of probabilistic cellular automata imply the existence of discrete time crystals (with an infinite autocorrelation time) in all dimensions, d≥1.
Despite being forbidden in equilibrium, spontaneous breaking of time translation symmetry can occur in periodically driven, Floquet systems with discrete time-translation symmetry.The period of the resulting discrete time crystal is quantized to an integer multiple of the drive period, arising from a combination of collective synchronization and many body localization.Here, we consider a simple model for a onedimensional discrete time crystal which explicitly reveals the rigidity of the emergent oscillations as the drive is varied.We numerically map out its phase diagram and compute the properties of the dynamical phase transition where the time crystal melts into a trivial Floquet insulator.Moreover, we demonstrate that the model can be realized with current experimental technologies and propose a blueprint based upon a one dimensional chain of trapped ions.Using experimental parameters (featuring long-range interactions), we identify the phase boundaries of the ion-time-crystal and propose a measurable signature of the symmetry breaking phase transition.