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arXiv:2402.04078 · Physical review. B./Physical review. B

Metronome spin stabilizes time-crystalline dynamics

Niklas Euler, Adrian Braemer, Luca Benn, Martin Gärttner

Published Jun 3, 2024• 4 citations
arXivHTMLPDF

Abstract

We investigate a disorder-free quantum Ising chain subject to a time-periodic drive that rotates each spin by an angle $\ensuremath{\pi}(1\ensuremath{-}{\ensuremath{\epsilon}}_{i})$. In case all spins experience the same deviation $\ensuremath{\epsilon}$ and the system is initialized in a fully polarized state, the dynamics is known to be time crystalline: the magnetization of the system exhibits period-doubled oscillations for timescales that grow exponentially with the length of the chain. In this work, we study the effect of a deviation $\ensuremath{\epsilon}$ that differs between spins. We find that reducing $\ensuremath{\epsilon}$ for a single spin drastically enhances the lifetime of spatiotemporal order, suggesting the name metronome spin. Employing perturbative arguments in an average Hamiltonian picture, we explain this observation for initial states with macroscopic bulk magnetization. Furthermore, in the case of random bit-string initial states, we report the enhancement of the lifetime of a topological edge mode, which can also be understood in the same picture. Finally, we discuss an altered geometry in which the metronome spin is not directly part of the chain, affecting the dynamics in different ways in the two scenarios considered. Our findings unveil the intricate dynamics that emerge in Floquet systems under the influence of a spatially varying drive, thereby uncovering new avenues for Floquet engineering.

DOI
10.1103/physrevb.109.224301

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Periodically driven (Floquet) systems are described by time-dependent Hamiltonians that possess discrete time translation symmetry. The spontaneous breaking of this symmetry leads to the emergence of a novel non-equilibrium phase of matter—the Discrete Time Crystal (DTC). In this paper, we propose a scheme to extend the lifetime of a DTC in a paradigmatic model—a translation-invariant Ising spin chain with nearest-neighbor interaction J, subjected to a periodic kick by a transverse magnetic field with frequency 2πT. This system exhibits the hallmark signature of a DTC—persistent sub-harmonic oscillations with frequency πT—for a wide parameter regime. Employing both analytical arguments as well as exact diagonalization calculations, we demonstrate that the lifetime of the DTC is maximized, when the interaction strength is tuned to an optimal value, JT=π. Our proposal essentially relies on an interaction-induced quantum interference mechanism that suppresses the creation of excitations, and thereby enhances the DTC lifetime. Intriguingly, we find that the period doubling oscillations can last eternally in even size systems. This anomalously long lifetime can be attributed to a time reflection symmetry that emerges at JT=π. Our work provides a promising avenue for realizing a robust DTC in various quantum emulator platforms.

arXiv:2104.05201•Atoms•Apr 12, 2021•7 citations
We investigate a disorder-free quantum Ising chain subject to a time-periodic drive that rotates each spin by an angle \ensuremathπ(1\ensuremath−\ensuremathϵi)\ensuremath{\pi}(1\ensuremath{-}{\ensuremath{\epsilon}}_{i})\ensuremathπ(1\ensuremath−\ensuremathϵi​). In case all spins experience the same deviation \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ and the system is initialized in a fully polarized state, the dynamics is known to be time crystalline: the magnetization of the system exhibits period-doubled oscillations for timescales that grow exponentially with the length of the chain. In this work, we study the effect of a deviation \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ that differs between spins. We find that reducing \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ for a single spin drastically enhances the lifetime of spatiotemporal order, suggesting the name metronome spin. Employing perturbative arguments in an average Hamiltonian picture, we explain this observation for initial states with macroscopic bulk magnetization. Furthermore, in the case of random bit-string initial states, we report the enhancement of the lifetime of a topological edge mode, which can also be understood in the same picture. Finally, we discuss an altered geometry in which the metronome spin is not directly part of the chain, affecting the dynamics in different ways in the two scenarios considered. Our findings unveil the intricate dynamics that emerge in Floquet systems under the influence of a spatially varying drive, thereby uncovering new avenues for Floquet engineering.
Time crystals break the discrete time translational invariance of an external periodic drive by oscillating at an integer multiple of the driving period. In addition to this fundamental property, other aspects are often considered to be essential characteristics of a time crystal, such as the presence of disorder or interactions, robustness against small variations of system parameters, and the free choice of the initial quantum state. We study a finite-length polarized XX spin chain engineered to display a spectrum of equidistant energy levels without drive and show that it keeps a spectrum of equidistant Floquet quasienergies when subjected to a large variety of periodic driving schemes. Arbitrary multiples of the driving period can then be reached by adjusting parameters of the drive, for arbitrary initial states. This behavior is understood by mapping the XX spin chain with N+1N+1N+1 sites to a single large spin with S=N/2S=N/2S=N/2 invoking the closure of the group SU(2). Our simple model is neither intrinsically disordered nor is it an interacting many-body system (after suitable mapping), and it does not have a thermodynamic limit in the conventional sense. It does, however, show controllable discrete time translational symmetry breaking for arbitrary initial states and a degree of robustness against perturbations, thereby carrying some characteristic traits of a discrete time crystal.
Abstract Floquet spin chains have been a venue for understanding topological states of matter that are qualitatively different from their static counterparts by, for example, hosting π edge modes that show stable period-doubled dynamics. However the stability of these edge modes to interactions has traditionally required the system to be many-body localized in order to suppress heating. In contrast, here we show that even in the absence of disorder, and in the presence of bulk heating, π edge modes are long lived. Their lifetime is extracted from exact diagonalization and is found to be non-perturbative in the interaction strength. A tunneling estimate for the lifetime is obtained by mapping the stroboscopic time-evolution to dynamics of a single particle in Krylov subspace. In this subspace, the π edge mode manifests as the quasi-stable edge mode of an inhomogeneous Su-Schrieffer-Heeger model whose dimerization vanishes in the bulk of the Krylov chain.
Long-lived π\piπ edge modes of interacting and disorder-free Floquet spin chains
Floquet spin chains have been a venue for understanding topological states of matter that are qualitatively different from their static counterparts by, for example, hosting π\piπ edge modes that show stable period-doubled dynamics. However the stability of these edge modes to interactions has traditionally required the system to be many-body localized in order to suppress heating. In contrast, here we show that even in the absence of disorder, and in the presence of bulk heating, π\piπ edge modes are long lived. Their lifetime is extracted from exact diagonalization and is found to be non-perturbative in the interaction strength. A tunneling estimate for the lifetime is obtained by mapping the stroboscopic time-evolution to dynamics of a single particle in Krylov subspace. In this subspace, the π\piπ edge mode manifests as the quasi-stable edge mode of an inhomogeneous Su-Schrieffer-Heeger model whose dimerization vanishes in the bulk of the Krylov chain.
Discrete time crystals are non-equilibrium phases of matter with spatiotemporal order and a response that breaks the time-translation symmetry of a periodic external drive. Interactions, spatial disorder and correlated random driving can in theory stabilize this order. In experiments, however, coupling to the environment limits the time crystal lifetime. Here we use stochastic local bit-flip error correction to prolong the time-crystalline response of a driven, spatially disordered two-dimensional Ising model on a superconducting quantum processor, using a repetition code with distances three and five on a honeycomb lattice with up to 68 physical qubits. Even though ancilla qubits, midcircuit syndrome measurements and decoding increase the required quantum and classical resources, the stochastic drive increases the memory decay time by a factor of up to two. At late times we observe that recovery stabilizes spatial order across the lattice. The spin-configuration distributions explain this separation, as recovery suppresses disagreeing bonds while broadening occupies completely reversed states. Syndrome measurements without correction accelerate the memory decay, demonstrating the importance of conditional correction. Our results show that local bit-flip error correction can preserve time-crystalline memory over more drive cycles than the bare dynamics.
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.
Periodically driven (Floquet) systems are described by time-dependent Hamiltonians that possess discrete time translation symmetry. The spontaneous breaking of this symmetry leads to the emergence of a novel non-equilibrium phase of matter—the Discrete Time Crystal (DTC). In this paper, we propose a scheme to extend the lifetime of a DTC in a paradigmatic model—a translation-invariant Ising spin chain with nearest-neighbor interaction J, subjected to a periodic kick by a transverse magnetic field with frequency 2πT. This system exhibits the hallmark signature of a DTC—persistent sub-harmonic oscillations with frequency πT—for a wide parameter regime. Employing both analytical arguments as well as exact diagonalization calculations, we demonstrate that the lifetime of the DTC is maximized, when the interaction strength is tuned to an optimal value, JT=π. Our proposal essentially relies on an interaction-induced quantum interference mechanism that suppresses the creation of excitations, and thereby enhances the DTC lifetime. Intriguingly, we find that the period doubling oscillations can last eternally in even size systems. This anomalously long lifetime can be attributed to a time reflection symmetry that emerges at JT=π. Our work provides a promising avenue for realizing a robust DTC in various quantum emulator platforms.