Time-crystalline behavior has been predicted and observed in quantum central-spin systems with periodic driving and Ising interactions. Here, we theoretically show that it can also arise in central-spin systems with Heisenberg interactions. We present two methods to achieve this: application of a sufficiently large Zeeman splitting on the central spin compared to the satellite spins, or else by applying additional pulses to the central spin every Floquet period. In both cases, we show that the system exhibits a subharmonic response in spin magnetizations in the presence of disorder for both pure Heisenberg and XXZ interactions. Our results pertain to any XXZ central-spin system, including hyperfine-coupled electron-nuclear systems in quantum dots or color centers.
Time-crystalline behavior has been predicted and observed in quantum central-spin systems with periodic driving and Ising interactions. Here, we theoretically show that it can also arise in central-spin systems with Heisenberg interactions. We present two methods to achieve this: application of a sufficiently large Zeeman splitting on the central spin compared to the satellite spins, or else by applying additional pulses to the central spin every Floquet period. In both cases, we show that the system exhibits a subharmonic response in spin magnetizations in the presence of disorder for both pure Heisenberg and XXZ interactions. Our results pertain to any XXZ central-spin system, including hyperfine-coupled electron-nuclear systems in quantum dots or color centers.
A discrete time crystal is an out-of-equilibrium phase of matter characterized by the spontaneous breaking of discrete time-translation symmetry. Using extensive numerical simulations based on matrix-product-state methods, we provide evidence for discrete time-crystalline behavior in strongly disordered spin chains with Heisenberg interactions, including the isotropic point, subject to periodic driving. Starting from a many-body localized regime, we observe that rotations induced by delta kicks produce a pronounced subharmonic response at half the drive frequency in spin observables. We investigate the stability of this response against rotation-angle errors through entanglement entropy, quantum Fisher information, short-range spin correlations, and restricted-control ergotropy. Increasing the rotation error reveals an intermediate dynamical regime separating the time-crystalline and Floquet-localized responses. In this regime, most observables exhibit signatures of weakly correlated dynamics reminiscent of Anderson localization.
We study the stroboscopic nonequilibrium quantum dynamics of periodically kicked Hamiltonians involving homogeneous central-spin interactions. The system exhibits a strong fragmentation of Hilbert space into four-dimensional Floquet-Krylov subspaces, which oscillate between two disjointed two-dimensional subspaces and thus break the discrete time-translation symmetry of the system. Our analytical and numerical analyses reveal that fully polarized states of the satellite spins exhibit fragmentations that are stable against perturbations and have high overlap with Floquet eigenstates of atypically low bipartite entanglement entropy (scar states). Motivated by the breaking of discrete time translation symmetry by Floquet-Krylov subspaces, we introduce a novel type of time crystal that we call a ``subspace time crystal.'' We present evidence of robust time-crystalline behavior in the form of a period doubling of the total magnetization of fully polarized satellite spin states that persists over long timescales. We compute nonequilibrium phase diagrams with respect to a magnetic field, coupling terms, and pulse error for various interaction types, including Heisenberg, Ising, XXZ, and XX. We also discuss possible experimental realizations of scar time crystals in color center, quantum dot, and rare-earth ion platforms.
A discrete time crystal is a recently discovered nonequilibrium phase of matter that has been shown to exist in disordered, periodically driven Ising spin chains. In this phase, if the system is initially prepared in one of a certain class of pure multispin product states, it periodically returns to this state over very long timescales despite the presence of interactions, disorder, and pulse imperfections. Here, we show that this phase occurs in GaAs quantum dot spin arrays containing as few as three quantum dots, each confining one electron spin, for naturally occurring levels of nuclear spin noise and charge noise. Although the physical interaction in these arrays is a nearest-neighbor Heisenberg exchange interaction, we show that this can be effectively converted into an Ising interaction by applying additional pulses during each drive period. We show that by changing the rotation axis of these pulses, we can select the direction of the Ising interaction and, consequently, the quantization axis of the stabilized multispin states. Moreover, we demonstrate that it is possible to perform coherent rotations of the stabilized states while remaining in the time-crystal phase. These findings open up the intriguing possibility of using time-crystal phases to extend the lifetime of quantum states for information applications.
We show the emergence of Floquet time crystal (FTC) phases in the Floquet dynamics of periodically driven p-spin models, which describe a collection of spin-1/2 particles with all-to-all p-body interactions. Given the mean-field nature of these models, we treat the problem exactly in the thermodynamic limit and show that, for a given p, these systems can host various robust time-crystalline responses with period nT, where T is the period of the drive and n an integer between 2 and p. In particular, the case of four-body interactions (p=4) gives rise to both a usual period-doubling crystal and also a period-quadrupling phase. We develop a comprehensive framework to predict robust subharmonic response in classical area-preserving maps, and use this as a basis to predict the occurrence and characterize the stability of the resulting mean-field FTC phases in the quantum regime. Our analysis reveals that the robustness of the time-crystal behavior is reduced as their period increases, and establishes a connection between the emergence of time crystals, described by eigenstate ordering and robust subharmonic response, and the phenomenology of excited state and dynamical quantum phase transitions. Finally, for the models hosting two or more coexisting time crystal phases, we define protocols where the periodic subharmonic response of the system can be varied in time via the nonperiodic modulation of an external control parameter.
Abstract Discrete time-translational symmetry in a periodically driven many-body system can be spontaneously broken to form a discrete time crystal, an exotic new phase of matter. We present observations characteristic of discrete time crystalline order in a driven system of paramagnetic P-donor impurities in isotopically enriched 28Si cooled below 10 K. The observations exhibit a stable subharmonic peak at half the drive frequency which remains pinned even in the presence of pulse error, a signature of discrete time crystalline order. This signal has a finite lifetime of ∼100 Floquet periods, but this effect is long-lived relative to coherent spin–spin interaction timescales, lasting ∼104 times longer. We present simulations of the system based on the paradigmatic central spin model and show good agreement with experiment. We investigate the role of dissipation and interactions within this model, and show that both are capable of giving rise to discrete time crystal-like behaviour.
Motivated by the search for non-equilibrium phases of matter beyond the Floquet paradigm, we propose a route to realize perfect revivals and time quasicrystals (TQCs) in a Fibonacci-driven central-spin model with Ising interactions between the central and satellite spins in the absence of disorder. We analytically demonstrate that when the interaction strength, λ is tuned to 2mπ, no entanglement is generated between the central and satellite spins at stroboscopic times, independent of the strength of the transverse driving field and the drive sequence. This leads to indefinitely long-lived quasiperiodic oscillations in satellite spin magnetization. Furthermore, for initial states with a definite bath magnetization, no entanglement is generated when the transverse driving field, θ is tuned to π for any value of $ λ$, and the satellite spin magnetization oscillations exhibit period-3 revivals at Fibonacci times. Examining the robustness of these oscillations away from this fine-tuned limit, we identify a parameter regime where the quasiperiodic response does not decay over the timescales probed, giving rise to a long-lived π-pulse-proximate TQC regime. We further show that at special Clifford points, the system exhibits perfect quasiperiodic oscillations of the satellite spin magnetization, with a parity-dependent Fourier response. This response persists over the timescales probed, even away from the Clifford points for an even number of satellite spins, leading to the formation of a Clifford-proximate TQC. Finally, we demonstrate that at this Clifford points, the system is steered along an entangled trajectory, leading to the dynamical generation of Bell-cat and spin-cat states. These states exhibit magnetization of collective spin fluctuates quadratically with the number of satellite spins, thereby enabling Heisenberg-limited metrology.