Can a closed quantum system generate persistent time-crystal-like dynamics without external driving? Within the Bateman dual oscillator framework, we show that the answer is affirmative. We consider a nonrelativistic (2+1)-dimensional system in which spin-induced spatial deformation generates an effective Bateman oscillator structure. After quantization, the system is governed by a time-independent Hermitian Hamiltonian describing coherent coupling between damped and amplified oscillator sectors while preserving the total energy of the global doubled system. Tracing over the amplified sector, we derive an effective non-Markovian reduced dynamics for the observable subsystem. The resulting memory effects sustain persistent oscillations of subsystem observables and generate emergent time-crystal-like temporal ordering without external periodic driving or equilibrium spontaneous symmetry breaking. Since the oscillatory behavior originates from nonequilibrium reduced subsystem dynamics rather than equilibrium expectation values of the full Hamiltonian, the mechanism lies outside the assumptions of conventional no-go theorems for equilibrium time crystals. The same dynamics further exhibits logarithmic-spiral trajectories and self-similar fractal scaling, revealing a direct connection between coherent dissipative dynamics, non-Markovian memory effects, and emergent temporal ordering in a globally unitary quantum system. In this specific sense, "watching the growth" of these self-similar structures corresponds to observing the gradual formation of time-crystal-like ordering.
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15 pages, 5 figures, 1 table. Discussion of non-Markovian reduced dynamics, dissipative quantum systems, and time-crystal-like temporal ordering
Can a closed quantum system generate persistent time-crystal-like dynamics without external driving? Within the Bateman dual oscillator framework, we show that the answer is affirmative. We consider a nonrelativistic (2+1)-dimensional system in which spin-induced spatial deformation generates an effective Bateman oscillator structure. After quantization, the system is governed by a time-independent Hermitian Hamiltonian describing coherent coupling between damped and amplified oscillator sectors while preserving the total energy of the global doubled system. Tracing over the amplified sector, we derive an effective non-Markovian reduced dynamics for the observable subsystem. The resulting memory effects sustain persistent oscillations of subsystem observables and generate emergent time-crystal-like temporal ordering without external periodic driving or equilibrium spontaneous symmetry breaking. Since the oscillatory behavior originates from nonequilibrium reduced subsystem dynamics rather than equilibrium expectation values of the full Hamiltonian, the mechanism lies outside the assumptions of conventional no-go theorems for equilibrium time crystals. The same dynamics further exhibits logarithmic-spiral trajectories and self-similar fractal scaling, revealing a direct connection between coherent dissipative dynamics, non-Markovian memory effects, and emergent temporal ordering in a globally unitary quantum system. In this specific sense, "watching the growth" of these self-similar structures corresponds to observing the gradual formation of time-crystal-like ordering.
Can a closed quantum system generate time-crystal-like nonequilibrium behavior, self-similar scaling structures, and non-Markovian memory without external driving or coupling to a macroscopic environment? We address this question within the quantum Bateman oscillator formulated in a nonrelativistic (2 + 1)-dimensional phase-space noncommutative framework generated by spin-induced spatial deformation. The resulting doubled quantum dynamics is governed by a time-independent Hermitian Hamiltonian and exhibits an underlying SU(1, 1) structure with amplified and damped collective modes. We show that these modes satisfy an exact discrete scaling covariance, leading to self-similar temporal evolution without external driving. Upon tracing over one oscillator sector, the reduced dynamics becomes intrinsically non-Markovian and is governed by a history-dependent memory kernel. The same scaling structure admits a geometric representation in terms of logarithmic-spiral trajectories associated with the amplified and damped branches of the Bateman system. Because the mechanism relies on nonequilibrium reduced dynamics rather than equilibrium expectation values of local observables, it lies outside the assumptions underlying conventional no-go theorems for equilibrium time crystals. Our results identify spin as the common physical origin of the amplified and damped Bateman dynamics, self-similar scaling periodicity, logarithmic-spiral structures, and non-Markovian memory, also suggesting a natural extension of the mechanism to relativistic anyonic systems.
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.
Boundary Time Crystals (BTCs) represent a distinctive form of nonequilibrium temporal order in which persistent oscillations emerge in open dissipative quantum systems without external periodic driving. Although their dynamical characterization is well established through the spectral properties of the Liouvillian, the thermodynamic meaning of persistent temporal order remains less systematically articulated. This work develops a nonequilibrium-thermodynamic framework for interpreting BTCs as emergent dissipative structures sustained by irreversible energy and entropy exchange. The framework connects Liouvillian temporal dynamics with thermodynamic forces and fluxes, entropy production, dissipative organization, and self-organization, thereby providing a common language for describing temporal persistence and irreversible activity within open quantum systems. In addition, phenomenological measures are introduced to characterize the balance between coherent temporal dynamics and dissipative relaxation, together with a normalized entropy-production measure for comparing dynamical regimes. A concrete numerical demonstration based on the established collective-spin BTC model illustrates the framework by contrasting non-BTC and BTC regimes, finite-size temporal decay, and the associated dissipative activity. The results support a thermodynamic interpretation in which persistent temporal order is maintained as a nonequilibrium organized state rather than as a consequence of equilibrium thermodynamics. The proposed framework is complementary to microscopic quantum-thermodynamic treatments of BTCs and does not replace the established Liouvillian criterion for identifying temporal-crystalline order. Instead, it provides a broader conceptual bridge between temporal symmetry breaking, irreversible thermodynamics, and dissipative self-organization, offering a basis for future quantitative studies of the thermodynamic structure and stability of time-crystalline phases.
The classification of phases of matter has been revolutionised by topological order and nonequilibrium phenomena such as time crystals. However, a unified framework connecting temporal order, topology, and dissipation remains elusive. Here we introduce the concept of "Time-Quasicrystals"-non-equilibrium topological phases that exhibit quasi-periodic temporal order protected by dissipative dynamics. These phases emerge in driven dissipative quantum systems through the interplay of Floquet engineering and non-local dissipation, leading to an effective non-Hermitian Floquet Hamiltonian with topological edge modes in the frequency domain. We demonstrate three hallmarks of Time-Quasicrystals: (1) fractal-like temporal correlations with multiple incommensurate frequencies, (2) a quantised topological invariant robust against disorder, and (3) algebraically decaying memory of phase perturbations. We provide explicit experimental realisations in Rydberg atom arrays and circuit quantum electrodynamics platforms. Our work establishes Time-Quasicrystals as a bridge between topological phases, time crystals, and quantum simulation, opening pathways to dissipativeprotected quantum memory and novel non-equilibrium quantum phenomena.
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
The classification of phases of matter has been revolutionised by topological order and non equilibrium phenomena such as time crystals. However, a unified framework connecting temporal order, topology, and dissipation remains elusive. Here we introduce the concept of "Time-Quasicrystals"—non-equilibrium topological phases that exhibit quasi-periodic temporal order protected by dissipative dynamics. These phases emerge in driven dissipative quantum systems through the interplay of Floquet engineering and non-local dissipation, leading to an effective non-Hermitian Floquet Hamiltonian with topological edge modes in the frequency domain. We demonstrate three hallmarks of Time-Quasicrystals: (1) fractal-like temporal correlations with multiple incommensurate frequencies, (2) a quantised topological invariant robust against disorder, and (3) algebraically decaying memory of phase perturbations. We provide explicit experimental realisations in Rydberg atom arrays and circuit quantum electrodynamics platforms. Our work establishes Time-Quasicrystals as a bridge between topological phases, time crystals, and quantum simulation, opening pathways to dissipative protected quantum memory and novel non-equilibrium quantum phenomena.