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Physics Open

Boundary time crystals through the lens of nonequilibrium thermodynamics: A thermodynamic framework for emergent temporal order

Saeed Shahsavari

Published Sep 29, 2026• 1 citation
Publisher

Abstract

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.

DOI
10.1016/j.physo.2026.100495

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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.
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.
Boundary time crystals (BTCs) in dissipative collective spin systems have been extensively studied using numerical, mean-field, and perturbative approaches. However, an explicit Liouvillian description governing the long-time dynamics deep within the time crystal phase has remained elusive. Here, we derive an effective Liouvillian that analytically captures the extreme BTC regime, where dissipation is parametrically weak and oscillatory order is maximally robust. By introducing a superspin representation of Liouville space, we obtain closed-form expressions for the Liouvillian eigenvalues to first order in the dissipation strength, providing direct access to decay rates, oscillation frequencies, and their thermodynamic scaling. Applying this framework to the canonical BTC model we analytically recover spontaneous breaking of continuous time-translation symmetry and persistent oscillations in the thermodynamic limit. In contrast, we show that other dissipative spin models exhibit only single-frequency oscillatory dynamics and therefore do not support genuine BTC phases. Our results establish a controlled analytical framework for the long-time dynamics in the extreme BTC regime.
Quantum synchronization (QS) in dissipative systems is often inferred from smooth phase locking, leaving open whether its breakdown constitutes a genuine nonequilibrium transition. Here we introduce a Liouvillian framework that classifies driven-dissipative dynamics by the structure of the undriven dissipative background and show that QS breaks down via a Hopf-type dynamical phase transition into a boundary time crystal (BTC). The character of this transition is determined by the background attractor: systems with a self-sustained oscillator (SSO) support robust non-resonant BTCs, whereas those with a polar fixed point (PFP) sustain BTCs only at resonance and lose them under detuning. We identify sharp dynamical and spectral signatures of the QS-BTC transition and thereby establish, within U(1)-symmetric collective-spin Lindbladians driven by a single coherent tone, a background-based allowed/forbidden criterion that unifies QS, its breakdown, and time-crystalline order within a single Liouvillian framework.
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.
Boundary time crystals (BTCs) provide a route to persistent symmetry-breaking oscillations in open quantum systems without an external clock. We study finite-size BTC-like dynamics in collectively driven and dissipative spin ensembles using a hardware-motivated spin–cavity description and an autonomous collective-spin Liouvillian. A unified diagnostic pipeline identifies late-time oscillatory pockets, quantifies their metrological resource through the long-time averaged quantum Fisher information (QFI), and tests their compatibility with complex Liouvillian slow modes. In the autonomous regime, the QFI displays near-Heisenberg scaling over the accessible sizes. The corresponding Liouvillian modes are oscillatory, but their real decay rates do not show reliable finite-size closing; the spectrum is therefore used as a consistency check rather than proof of a thermodynamic BTC phase. Local dephasing simultaneously suppresses the oscillatory response and metrological gain. The results establish a finite-size connection among autonomous oscillations, slow modes, and persistent metrological resources while leaving the thermodynamic gap-closing question open.
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.