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.

中文翻译：

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边界时间晶体的量子热力学

时间平移对称性破缺是马尔可夫开放量子系统中出现非平稳多体相（即所谓的时间晶体）的机制。近年来，时间晶体的动力学方面得到了广泛的探索。然而，由于这些相固有的非平衡性质，人们对它们的热力学性质知之甚少。在这里，我们考虑有限温度环境中的范式边界时间晶体系统，并证明时间晶体相在任何温度下的持久性。此外，我们分析了模型的热力学方面，特别是热流、能量交换和不可逆熵产生。我们的工作揭示了维持非平衡时间晶体相的热力学成本，并提供了一个框架来表征时间晶体作为可能的资源，例如量子传感。我们的结果可以在实验中得到验证，例如使用捕获离子或超导电路，因为我们将热力学量与集体（磁化）算子的平均值和协方差联系起来。