Abstract Open many-body quantum systems can exhibit intriguing nonequilibrium phases of matter, such as time crystals. In these phases, the state of the system spontaneously breaks the time-translation symmetry of the dynamical generator, which typically manifests through persistent oscillations of an order parameter. A paradigmatic model displaying such a symmetry breaking is the boundary time crystal (BTC), which has been extensively analyzed experimentally and theoretically. Despite the broad interest in these nonequilibrium phases, their thermodynamics and their fluctuating behavior remain largely unexplored, in particular for the case of coupled time crystals. In this work, we consider two interacting BTCs and derive a consistent interpretation of their thermodynamic behavior. We fully characterize their average dynamics and the behavior of their quantum fluctuations, which allows us to demonstrate the presence of quantum and classical correlations in both the stationary and the time-crystal phases displayed by the system. We furthermore exploit our theoretical derivation to explore possible applications of time crystals as quantum batteries, demonstrating their ability to efficiently store energy.
Abstract Open many-body quantum systems can exhibit intriguing nonequilibrium phases of matter, such as time crystals. In these phases, the state of the system spontaneously breaks the time-translation symmetry of the dynamical generator, which typically manifests through persistent oscillations of an order parameter. A paradigmatic model displaying such a symmetry breaking is the boundary time crystal (BTC), which has been extensively analyzed experimentally and theoretically. Despite the broad interest in these nonequilibrium phases, their thermodynamics and their fluctuating behavior remain largely unexplored, in particular for the case of coupled time crystals. In this work, we consider two interacting BTCs and derive a consistent interpretation of their thermodynamic behavior. We fully characterize their average dynamics and the behavior of their quantum fluctuations, which allows us to demonstrate the presence of quantum and classical correlations in both the stationary and the time-crystal phases displayed by the system. We furthermore exploit our theoretical derivation to explore possible applications of time crystals as quantum batteries, demonstrating their ability to efficiently store energy.
Abstract Time-translation symmetry breaking is a mechanism for the emergence of non-stationary many-body phases, so-called time-crystals, in Markovian open quantum systems. Dynamical aspects of time-crystals have been extensively explored over the recent years. However, much less is known about their thermodynamic properties, also due to the intrinsic nonequilibrium nature of these phases. Here, we consider the paradigmatic boundary time-crystal system, in a finite-temperature environment, and demonstrate the persistence of the time-crystalline phase at any temperature. Furthermore, we analyze thermodynamic aspects of the model investigating, in particular, heat currents, power exchange and irreversible entropy production. Our work sheds light on the thermodynamic cost of sustaining nonequilibrium time-crystalline phases and provides a framework for characterizing time-crystals as possible resources for, e.g. quantum sensing. Our results may be verified in experiments, for example with trapped ions or superconducting circuits, since we connect thermodynamic quantities with mean value and covariance of collective (magnetization) operators.
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.
In this paper we study genuine multipartite correlations (GMC's) in a boundary time crystal (BTC). Boundary time crystals are nonequilibrium quantum phases of matter in contact to an environment, for which a macroscopic fraction of the many-body system breaks the time translation symmetry. We analyze both (i) the structure (orders) of GMC's among the subsystems, as well as (ii) their build-up dynamics for an initially uncorrelated state. We find that, in the thermodynamic limit (and only in such a limit), multipartite correlations of all orders grow indefinitely in time in the BTC phase, further displaying a persistent oscillatory behavior around their mean growth. The orders of the correlations show a power-law decaying hierarchy among its k partitions. Moreover, in the long-time limit the GMC's are shown extensive with the system size, contrasting to the subextensive scaling in the non time crystals (ferromagnetic) phase of the model. We also discuss the classical and quantum nature of these correlations with basis on multipartite entanglement witnesses, specifically, the analysis of the quantum Fisher information (QFI). Both GMC and QFI are able to capture and distinguish the different phases of the model. Our paper highlights the genuine many-body properties of these peculiar nonequilibrium phases of matter.
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.
Driven-dissipative many-body systems support nontrivial quantum phases absent in equilibrium. As a prominent example, the interplay between coherent driving and collective dissipation can lead to a dynamical quantum phase that spontaneously breaks time-translation symmetry. This so-called boundary time crystal (BTC) is fragile in the presence of local dissipation, which can easily relax the system to a stationary state. In this Letter, we demonstrate a robust BTC that is intrinsically induced by local dissipation. We provide extensive numerical evidence to support the existence of the BTC and study its behaviors in different regimes. In particular, with decreasing interaction range, we identify a transition from mean-field limit cycles to correlated BTCs featuring sizable quantum correlations. Our studies significantly broaden the scope of nonequilibrium phases and shed new light on the experimental search for dynamical quantum matter.
Boundary time crystals (BTC's) are nonequilibrium phases of matter occurring in quantum systems in contact to an environment, for which a macroscopic fraction of the many-body system breaks time translation symmetry. We study BTC's in collective d-level systems, focusing on the cases with d=2, 3, and 4. We find that BTC's appear in different forms for the different cases. We first consider the model with collective d=2-level systems [Phys. Rev. Lett. 121, 035301 (2018)], whose dynamics is described by a Gorini-Kossakowski-Sudarshan-Lindblad master equation, and perform a throughout analysis of its phase diagram and Jacobian stability for different interacting terms in the coherent Hamiltonian. In particular, using perturbation theory for general (non-Hermitian) matrices, we obtain analytically how a specific Z2 symmetry-breaking Hamiltonian term destroys the BTC phase in the model. Based on these results we define a d=4 model composed of a pair of collective two-level systems interacting with each other. We show that this model support richer dynamical phases, ranging from limit cycles, period-doubling bifurcations, and a route to chaotic dynamics. The BTC phase is more robust in this case, not annihilated by the former symmetry-breaking Hamiltonian terms. The model with collective d=3-level systems is defined similarly, as competing pairs of levels, but sharing a common collective level. The dynamics can deviate significantly from the previous cases, supporting phases with the coexistence of multiple limit cycles, closed orbits and a full degeneracy of zero Lyapunov exponents.