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arXiv:2205.07919 · Physical Review Letters

Discrete Time Crystals Enforced by Floquet-Bloch Scars

Biao Huang, Tsz-Him Leung, Dan Stamper-Kurn, W. Vincent Liu

Published Sep 19, 2022• 24 citations
arXivHTMLPDF

Abstract

We analytically identify a new class of quantum scars protected by spatiotemporal translation symmetries, dubbed Floquet-Bloch scars. They are distinguished from previous (quasi-)static scars by a rigid spectral pairing only possible in Floquet systems, where strong interaction and drivings equalize the quasienergy corrections to all scars and maintain their spectral spacings against generic bilinear perturbations. Scars then enforce the spatial localization and rigid discrete time crystal (DTC) oscillations as verified numerically in a trimerized kagome lattice model relevant to recent cold atom experiments. Our analytical solutions offer a potential scheme to understand the mechanisms for more generic translation-invariant DTCs.

DOI
10.1103/physrevlett.129.133001

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We analytically identify a new class of quantum scars protected by spatiotemporal translation symmetries, dubbed Floquet-Bloch scars. They are distinguished from previous (quasi-)static scars by a rigid spectral pairing only possible in Floquet systems, where strong interaction and drivings equalize the quasienergy corrections to all scars and maintain their spectral spacings against generic bilinear perturbations. Scars then enforce the spatial localization and rigid discrete time crystal (DTC) oscillations as verified numerically in a trimerized kagome lattice model relevant to recent cold atom experiments. Our analytical solutions offer a potential scheme to understand the mechanisms for more generic translation-invariant DTCs.
We reconstruct the spectral pairing theories to enable analytical descriptions of eigenstate spatiotemporal orders in translation-invariant system without prethermal conditions. It is shown that the strong Ising interactions and drivings alone stabilize a class of ``cat scar'' eigenstates with tunable patterns, which lead to local discrete time-crystal (DTC) dynamics. They exhibit Fock space localization and long-range correlations robust against generic perturbations in a disorder-free scenario. In particular, we introduce a symmetry indicator method to enumerate cat scars, with which a set of unexpected inhomogeneous scar patterns are identified in addition to the ferromagnetic scars found before. These scars enforce DTC dynamics with rigid inhomogeneous patterns, offering a viable way to verify underlying eigenstate properties experimentally. Further, we prove rigorously that the strong Ising interactions enforces a selection rule for perturbations of different orders, which imposes an exponential suppression of spin fluctuations for Floquet eigenstates. Based on this property, three analytical scaling relations are proved to characterize the amplitudes, Fock space localization, and lifetime for DTC dynamics associated with cat scars. We further provide two practical methods to check whether certain DTC phenomena are dominated by single-spin dynamics or due to genuine interaction effects.
Recent works on the observation of discrete time-crystalline signatures throw up major puzzles on the necessity of localization for stabilizing such out-of-equilibrium phases. Motivated by these studies, we delve into a clean interacting Floquet system whose quasi-spectrum conforms to the ergodic Wigner-Dyson distribution, yet with an unexpectedly robust, long-lived time-crystalline dynamics in the absence of disorder or fine-tuning. We relate such behavior to a measure zero set of nonthermal Floquet eigenstates with long-range spatial correlations, which coexist with otherwise thermal states at near-infinite temperature and develop a high overlap with a family of translationally invariant, symmetry-broken initial conditions. This resembles the notion of ``dynamical scars'' that remain robustly localized throughout a thermalizing Floquet spectrum with fractured structure. We dub such a long-lived discrete time crystal formed in partially nonergodic systems, ``scarred discrete time crystal'' which is distinct by nature from those stabilized by either many-body localization or prethermalization mechanism.
Discrete time crystals (DTCs) are non-equilibrium phases of matter that break the discrete time-translation symmetry and is characterized by a robust subharmonic response in periodically driven quantum systems. Here, we explore the DTC in a disorder-free, periodically kicked XXZ spin chain, which is stabilized by the Floquet strong Hilbert space fragmentation. We numerically show the period-doubling response of the conventional DTC order, and uncover a multiple-period response with beating dynamics due to the coherent interplay of multiple πππ-pairs in the Floquet spectrum of small-size systems. The lifetime of the DTC order exhibits independence of the driving frequency and a power-law dependence on the ZZ interaction strength. It also grows exponentially with the system size, as a hallmark of the strong fragmentation inherent to the Floquet model. We analytically reveal the approximate conservation of the magnetization and domain-wall number in the Floquet operator for the emergent strong fragmentation, which is consistent with numerical results of the dimensionality ratio of symmetry subspaces. The rigidity and phase regime of the DTC order are identified through finite-size scaling of the Floquet-spectrum-averaged mutual information, as well as via dynamical probes. Our work establishes the Floquet Hilbert space fragmentation as a disorder-free mechanism for sustaining nontrivial temporal orders in out-of-equilibrium quantum many-body systems.
We investigate the emergence of a scarred discrete time crystal (SDTC) phase in a periodically driven dimerized spin chain. While generic interacting Floquet systems are expected to thermalize according to the eigenstate thermalization hypothesis (ETH), we demonstrate that this system hosts quantum many-body scars (QMBS) that induce a regime of weak ergodicity breaking. Through an analysis of Floquet level statistics, entanglement entropy, and eigenstate fidelity, we identify a manifold of low-entanglement states characterized by semi-Poisson statistics embedded within an otherwise thermal spectrum. These scarred states support robust subharmonic oscillations with period doubling, signaling the spontaneous breaking of discrete time-translation symmetry. We show that the SDTC response is robust against a variety of initial state configurations, demonstrating its stability beyond fine-tuned conditions. A finite-size scaling analysis reveals that the time-crystalline lifetime grows with system size within the range accessible to our exact-diagonalization calculations. However, drawing on the general phenomenology of approximate many-body scars, we expect that hybridization between Floquet scars and the thermal continuum will eventually curtail this growth, causing the lifetime to saturate at system sizes beyond our current numerical reach. This characterizes the SDTC as a long-lived metastable dynamical regime rather than a strictly stable thermodynamic phase, providing a comprehensive framework for understanding the interplay between periodic driving and constrained many-body dynamics in disorder-free systems.
The Floquet time crystal, which breaks discrete time-translation symmetry, is an intriguing phenomenon in nonequilibrium systems. It is crucial to understand the rigidity and robustness of discrete time crystal (DTC) phases in a many-body system, and finding a precisely solvable model can pave a way for understanding of the DTC phase. Here, we propose and study a solvable spin chain model by mapping it to a Floquet superconductor through the Jordan-Wigner transformation. The phase diagrams of Floquet topological systems are characterized by topological invariants and tell the existence of anomalous edge states. The subharmonic oscillation, which is the typical signal of the DTC, can be generated from such edge states and protected by topology and solvability. We also examine the robustness of the DTC by adding symmetry-preserving and symmetry-breaking perturbations. Our results on the topologically protected DTC can provide a deep understanding of the DTC when generalized to other interacting or dissipative systems.
The control of many-body quantum dynamics in complex systems is a key challenge in the quest to reliably produce and manipulate large-scale quantum entangled states. Recently, quench experiments in Rydberg atom arrays [Bluvstein et al. Science 371, 1355 (2021)SCIEAS0036-807510.1126/science.abg2530] demonstrated that coherent revivals associated with quantum many-body scars can be stabilized by periodic driving, generating stable subharmonic responses over a wide parameter regime. We analyze a simple, related model where these phenomena originate from spatiotemporal ordering in an effective Floquet unitary, corresponding to discrete time-crystalline behavior in a prethermal regime. Unlike conventional discrete time crystals, the subharmonic response exists only for Néel-like initial states, associated with quantum scars. We predict robustness to perturbations and identify emergent timescales that could be observed in future experiments. Our results suggest a route to controlling entanglement in interacting quantum systems by combining periodic driving with many-body scars.