Results are presented for Floquet systems in two spatial dimensions where the Floquet driving breaks an effective time-reversal symmetry. The driving protocol also induces flat bands that correspond to anomalous Floquet phases where the Chern number is zero and yet chiral edge modes exist. Analytic expressions for the edge modes, Berry curvature, and orbital magnetization are derived for the flat bands. Results are also presented for the static Haldane model for parameters when the bands are flat. Floquet driving of the same model is shown to give rise to Chern insulators as well as anomalous Floquet phases. The orbital magnetization for these different topological phases are presented and are found to be enhanced at half filling by the broken particle-hole symmetry of the Haldane model.

中文翻译：

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具有平带的 Floquet 拓扑系统：边缘模式、贝里曲率和轨道磁化

在 Floquet 驱动破坏了有效的时间反转对称性的两个空间维度中呈现了 Floquet 系统的结果。驱动协议还引入了平带，这些平带对应于陈数为零但手性边缘模式存在的异常 Floquet 阶段。为平带导出了边缘模式、贝里曲率和轨道磁化强度的解析表达式。还给出了带平坦时参数的静态 Haldane 模型的结果。显示相同模型的 Floquet 驱动会产生 Chern 绝缘体以及异常的 Floquet 相位。呈现了这些不同拓扑相的轨道磁化强度，并且发现在半填充时通过 Haldane 模型的破碎粒子-孔对称性增强。