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arXiv:2403.08476 · Communications Physics

Emergent continuous time crystal in dissipative quantum spin system without driving

Shu Yang, Zeqing Wang, Libin Fu, Jianwen Jie

Published Mar 24, 2025• 12 citations
arXivHTMLPDF

Abstract

Time crystals are a nonequilibrium phase of matter that extend fundamental spontaneous symmetry breaking into the temporal dimension, typically requiring external driving for their realization. Here, we explore the nonequilibrium phase diagram of a two-dimensional dissipative Heisenberg spin system without external coherent or incoherent driving. Through numerical analysis of spin dynamics, we identify nonstationary steady states, some of which are limit cycles with persistent periodic oscillations, while others exhibit chaotic, aperiodic behavior. The emergence of limit cycle steady states breaks the continuous time-translation symmetry of this time-independent many-body system, classifying them as continuous time crystals. We further validate these oscillatory behaviors by testing their stability against local perturbations and assess the robustness of the emergent continuous time crystals by introducing isotropic Gaussian white noise. This work provides insights into the intricate interplay between the dissipation and spin interaction, and opens possibilities for realizing dissipation-induced, heating-immune time crystals. Time crystals represent a nonequilibrium phase of matter without an equilibrium counterpart. The authors investigate the nonequilibrium steadystate phase diagram of a two-dimensional dissipative Heisenberg model in the absence of driving, and identify a continuous time crystalline phase exhibiting persistent oscillations in time.

DOI
10.1038/s42005-025-02040-1

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Time crystals are a nonequilibrium phase of matter that extend fundamental spontaneous symmetry breaking into the temporal dimension, typically requiring external driving for their realization. Here, we explore the nonequilibrium phase diagram of a two-dimensional dissipative Heisenberg spin system without external coherent or incoherent driving. Through numerical analysis of spin dynamics, we identify nonstationary steady states, some of which are limit cycles with persistent periodic oscillations, while others exhibit chaotic, aperiodic behavior. The emergence of limit cycle steady states breaks the continuous time-translation symmetry of this time-independent many-body system, classifying them as continuous time crystals. We further validate these oscillatory behaviors by testing their stability against local perturbations and assess the robustness of the emergent continuous time crystals by introducing isotropic Gaussian white noise. This work provides insights into the intricate interplay between the dissipation and spin interaction, and opens possibilities for realizing dissipation-induced, heating-immune time crystals. Time crystals represent a nonequilibrium phase of matter without an equilibrium counterpart. The authors investigate the nonequilibrium steadystate phase diagram of a two-dimensional dissipative Heisenberg model in the absence of driving, and identify a continuous time crystalline phase exhibiting persistent oscillations in time.
Time crystals are classified as discrete or continuous depending on whether they spontaneously break discrete or continuous time translation symmetry. Although discrete time crystals have been extensively studied in periodically driven systems, the experimental realization of a continuous time crystal is still pending. We report the observation of a limit cycle phase in a continuously pumped dissipative atom-cavity system that is characterized by emergent oscillations in the intracavity photon number. The phase of the oscillation was found to be random for different realizations, and hence, this dynamical many-body state breaks continuous time translation symmetry spontaneously. Furthermore, the observed limit cycles are robust against temporal perturbations and therefore demonstrate the realization of a continuous time crystal.
Continuous time crystals (CTCs) are characterized by sustained oscillations that break the time-translation symmetry. Since the ruling out of equilibrium CTCs by no-go theorems, the emergence of such dynamical phases has been observed in various driven-dissipative quantum platforms. The current understanding of CTCs is mainly based on mean-field theories, which fail to address the problem of whether the continuous time-translation symmetry can be broken in noisy, spatially extended systems absent in all-to-all couplings. Here, we propose a CTC realized in a quantum contact model through self-organized bistability. The CTCs stem from the interplay between collective dissipation induced by the first-order absorbing phase transitions and slow constant driving provided by an incoherent pump. The stability of such oscillatory phases in finite dimensions under the action of intrinsic quantum fluctuations is scrutinized by the functional renormalization group method and numerical simulations. Occurring at the edge of many-body synchronization, the CTC phase exhibits an inherent period and amplitude with a coherence time linearly diverging with system size, thus also constituting a boundary time crystal. Our results serve as a solid route towards self-protected CTCs in strongly interacting open systems. Published by the American Physical Society 2024
Time crystals, as temporal analogs of space crystals, manifest as stable and periodic behavior that breaks time translation symmetry. In an open quantum system, many-body interaction subjected to dissipation allows one to develop the time crystalline order in an unprecedented way, as refer to dissipative time crystals. Here we report the observation of multiple time crystals in the continuously driven-dissipative and strongly interacting Rydberg thermal gases, in which continuous time crystals, sub-harmonic time crystals, and high-harmonic time crystals are observed in the same system by manipulating the Rydberg excitation. Our work provides new ways to explore the nonequilibrium phases of matter in open systems. Such time crystals with persistent oscillation rooted in emergent quantum correlations, may emerge as a ubiquitous tool in quantum metrology, for instance, continuous sensing and parameter estimation surpassing the standard quantum limit.
While a generic open quantum system decays to its steady state, continuous time crystals (CTCs) develop spontaneous oscillation and never converge to a stationary state. Just as crystals develop correlations in space, CTCs do so in time. Here, we introduce a continuous quasi time crystals (CQTC). Despite being characterized by the presence of nondecaying oscillations, this phase does not retain its long-range order, making it time analogous to quasicrystal structures. We investigate the emergence of this phase in a minimal model, made of two coupled collective spin subsystems, each developing a CTC phase upon the action of a strong enough drive. The addition of a coupling enables the emergence of different synchronized phases, where both subsystems oscillate at the same frequency. In the transition between different CTC orders, the system develops chaotic dynamics with aperiodic oscillations. These chaotic features differ from those of closed quantum systems, as the dynamics is not characterized by a unitary evolution. We investigate the connection between chaos and this quasicrystalline phase using mean-field techniques, and we confirm these results by including quantum fluctuations at the lowest order.
The breaking of the continuous time-translation symmetry manifests, in Markovian open quantum systems, through the emergence of nonstationary dynamical phases. Systems that display nonequilibrium transitions into these phases are referred to as time crystals, and they can be realized, for example, in many-body systems governed by collective dissipation and long-ranged interactions. Here, we provide a complete analytical characterization of a boundary time-crystal phase transition. This involves exact expressions for the order parameter and for the dynamics of quantum fluctuations, which, in the time-crystalline phase, remains asymptotically non-Markovian as a consequence of the time-translation symmetry breaking. We demonstrate that boundary time crystals are intrinsically critical phases, where fluctuations exhibit a power-law divergence with time. Our results show that a dissipative time-crystal phase is far more than merely a classical nonlinear and nonstationary (limit cycle) dynamics of a macroscopic order parameter. It is rather a genuine many-body phase where the properties of correlations distinctly differ from those of stationary ones.
Spontaneous symmetry breaking plays a pivotal role in physics ranging from the emergence of elementary particles to the phase transitions of matter. The spontaneous breaking of continuous time translation symmetry leads to a novel state of matter named continuous time crystal (CTC). It exhibits periodic oscillation without the need for periodic driving, and the relative phases for repetitively realized oscillations are random. However, the mechanism behind the spontaneous symmetry breaking in CTCs, particularly the random phases, remains elusive. Here we propose and experimentally realize two types of CTCs based on distinct mechanisms: manifold topology and near-chaotic motion. We observe both types of CTCs in thermal atomic ensembles by artificially synthesizing spin-spin nonlinear interactions through a measurement-feedback scheme. Our work provides general recipes for the realization of CTCs, and paves the way for exploring CTCs in various systems.