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.
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.
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.
A cornerstone assumption that most literature on discrete time crystals has relied on is that homogeneous Floquet systems generally heat to a featureless infinite temperature state, an expectation that motivated researchers in the field to mostly focus on many-body localized systems. Some works have, however, shown that the standard diagnostics for time crystallinity apply equally well to clean settings without disorder. This fact raises the question whether a homogeneous discrete time crystal is possible in which the originally expected heating is evaded. Studying both a localized and an homogeneous model with short-range interactions, we clarify this issue showing explicitly the key differences between the two cases. On the one hand, our careful scaling analysis confirms that, in the thermodynamic limit and in contrast to localized discrete time crystals, homogeneous systems indeed heat. On the other hand, we show that, thanks to a mechanism reminiscent of quantum scars, finite-size homogeneous systems can still exhibit very crisp signatures of time crystallinity. A subharmonic response can in fact persist over timescales that are much larger than those set by the integrability-breaking terms, with thermalization possibly occurring only at very large system sizes (e.g., of hundreds of spins). Beyond clarifying the emergence of heating in disorder-free systems, our work casts a spotlight on finite-size homogeneous systems as prime candidates for the experimental implementation of nontrivial out-of-equilibrium physics.
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.
The presence of quantum scars, athermal eigenstates of a many-body Hamiltonian with finite-energy density, leads to an absence of ergodicity and long-time coherent dynamics starting from initial states that have a high overlap with scars as experimentally observed in a chain of ultracold Rydberg atoms. We show, via study of a periodically driven Rydberg chain, that the drive frequency acts as a tuning parameter for several reentrant transitions between ergodic and weak ergodicity breaking regimes. The former regime shows rapid thermalization of correlation functions and absence of scars in the Floquet spectrum of the system. The latter regime, in contrast, has scars; they lead to long-time coherent dynamics of correlation functions. We provide an analytical explanation of the existence of these regimes by going beyond a high-frequency Magnus expansion and using a novel perturbative approach valid at large drive amplitudes to derive a Floquet Hamiltonian which qualitatively explains the behavior of the driven system at arbitrary frequencies. We also discuss experiments involving finite Rydberg chains which can validate our theory. Our results demonstrate the possibility of drive-frequency-induced tuning between ergodic and weak ergodicity breaking dynamics in a disorder-free quantum many-body system.
Time crystals, a phase showing spontaneous breaking of time-translation symmetry, has been an intriguing subject for systems far away from equilibrium. Recent experiments found such a phase in both the presence and the absence of localization, while in theories localization by disorder is usually assumed a priori. In this work, we point out that time crystals can generally exist in systems without disorder. A series of clean quasi-one-dimensional models under Floquet driving are proposed to demonstrate this unexpected result in principle. Robust time crystalline orders are found in the strongly interacting regime along with the emergent integrals of motion in the dynamical system, which can be characterized by level statistics and the out-of-time-ordered correlators. We propose two cold atom experimental schemes to realize the clean Floquet time crystals, one by making use of dipolar gases and another by synthetic dimensions.
Motivated by experimental observations of time-symmetry breaking behavior in a periodically driven (Floquet) system, we study a one-dimensional spin model to explore the stability of such Floquet discrete time crystals (DTCs) under the interplay between interaction and the microwave driving. For intermediate interactions and high drivings, from the time evolution of both stroboscopic spin polarization and mutual information between two ends, we show that Floquet DTCs can exist in a prethermal time regime without the tuning of strong disorder. For much weak interactions the system is a symmetry-unbroken phase, while for strong interactions it gives its way to a thermal phase. Through analyzing the entanglement dynamics, we show that large driving fields protect the prethermal DTCs from many-body localization and thermalization. Our results suggest that by increasing the spin interaction, one can drive the experimental system into optimal regime for observing a robust prethermal DTC phase.