In this work, we report a ubiquitous presence of topological Floquet time crystal ($\mathrm{TFTC}$) in one-dimensional periodically driven systems. The rigidity and realization of spontaneous discrete time-translation symmetry ($\mathrm{DTS}$) breaking in our $\mathrm{TFTC}$ model require necessarily coexistence of anomalous topological invariants (0 modes and $\pi$ modes), instead of the presence of disorders or many-body localization. We found that in a particular frequency range of the underlying drive, the anomalous Floquet phase coexistence between 0 and $\pi$ modes can produce the period doubling ($2T$, two cycles of the drive) that breaks the $\mathrm{DTS}$ spontaneously, leading to the subharmonic response ($\omega /2$, half the drive frequency). The rigid period-$2T$ oscillation is topologically protected against perturbations due to both nontrivially opening of 0 and $\pi$ gaps in the quasienergy spectrum, thus, as a result, can be viewed as a specific “Rabi oscillation” between two Floquet eigenstates with certain quasienergy splitting $\pi /T$. Our modeling of the time-crystalline “ground state” can be easily realized in experimental platforms such as topological photonics and ultracold fields. Also, our work can bring significant interest to explore topological phase transition in Floquet systems and to bridge the gap between Floquet topological insulators and photonics, and period-doubled time crystals.

中文翻译：

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一维Floquet拓扑绝缘体中的时晶相和周期倍增振荡

在这项工作中，我们报告了拓扑Floquet时间晶体（$\mathrm{薄膜晶体管}$）在一维周期性驱动的系统中。自发离散时间平移对称性的刚性和实现$\mathrm{DTS}$）打破我们 $\mathrm{薄膜晶体管}$ 模型需要异常拓扑不变变量的共存（0模式和 $\pi$模式），而不是存在障碍或多体定位。我们发现，在基础驱动器的特定频率范围内，异常的Floquet相位在0和0之间共存。$\pi$ 模式可以使周期倍增（$2Ť$，驱动器的两个周期） $\mathrm{DTS}$自发地导致亚谐波响应（$\omega /2$，是驱动频率的一半）。刚性时期$2Ť$ 振荡在拓扑上受到保护，不会因0的平凡打开和 $\pi$ 因此，可以将准能谱中的缺口视为具有特定准能分裂的两个Floquet本征态之间的特定“拉比振荡” $\pi /Ť$。我们的时晶“基态”建模可以在诸如拓扑光子学和超冷场等实验平台上轻松实现。同样，我们的工作可能会引起人们极大的兴趣来探索Floquet系统中的拓扑相变，并弥合Floquet拓扑绝缘体和光子学与周期倍增时间晶体之间的差距。