Floquet topological phases emerge when systems are periodically driven out-of-equilibrium. They gained attention due to their external control, which allows to simulate a wide variety of static systems by just tuning the external field in the high frequency regime. However, it was soon clear that their relevance goes beyond that, as for lower frequencies, anomalous phases without a static counterpart are present and the bulk-to-boundary correspondence can fail. In this work we discuss the important role of resonances in Floquet phases. For that, we present a method to find analytical solutions when the frequency of the drive matches the band gap, extending the well-known high frequency analysis of Floquet systems. With this formalism, we show that the topology of Floquet phases with resonances can be accurately captured in analytical terms. We also find a bulk-to-boundary correspondence between the number of edge states in finite systems and a set of topological invariants in different frames of reference, which crucially do not explicitly involve the micromotion. To illustrate our results, we periodically drive a SSH chain and a $\pi$-flux lattice, showing that our findings remain valid in various two-band systems and in different dimensions. In addition, we notice that the competition between rotating and counter-rotating terms must be carefully treated when the undriven system is a semi-metal. To conclude, we discuss the implications to experimental setups, including the direct detection of anomalous topological phases and the measurement of their invariants.

中文翻译：

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异常的 Floquet 阶段。A 共振现象


当系统周期性地失去平衡时，就会出现 Floquet 拓扑相。它们因其外部控制而受到关注，只需在高频状态下调整外部磁场即可模拟各种静态系统。然而，很快就发现它们的相关性不止于此，因为对于较低的频率，存在没有静态对应物的异常相位，并且体与边界的对应可能会失败。在这项工作中，我们讨论了共振在 Floquet 相中的重要作用。为此，我们提出了一种在驱动频率与带隙匹配时找到解析解的方法，扩展了众所周知的 Floquet 系统的高频分析。通过这种形式主义，我们证明了具有共振的 Floquet 相的拓扑结构可以用解析术语准确捕获。我们还发现有限系统中的边缘状态数量与不同参考系中的一组拓扑不变量之间存在体积到边界的对应关系，关键是它们没有明确涉及微运动。为了说明我们的结果，我们定期驱动 SSH 链和 $\pi$-flux 晶格，表明我们的发现在各种双频系统和不同维度中仍然有效。此外，我们注意到，当非驱动系统是半金属时，必须仔细处理旋转项和反向旋转项之间的竞争。总而言之，我们讨论了对实验设置的影响，包括直接检测异常拓扑相位和测量它们的不变量。