Driven topological phases with space and time symmetries

拓扑绝缘体具有带隙的体积，但主机保护无隙边界状态，这是负责鲁棒的，量化的可观测量。它们可以发生在一维、二维和三维系统中，它们的边界状态通常需要一个或多个对称性才能保持保护。最近，越来越多的新类型的时间无关的拓扑相位出现在理论和实验上，考虑到空间对称性的作用：对称的原子配置，因为它们被发现在晶体中。相比之下，拓扑相位出现由于系统参数的周期性调制的时间少得多studied.We将研究几乎完全未开发的系统，“Floquet晶体绝缘体”（FCI）的类。这些是时间周期系统，从空间对称性的存在中继承了它们的拓扑特征。我们的目标是发展一个理论框架，可以处理静态和时间依赖的拓扑相位在同一个立足点，然后将其应用到FCI。使用这个框架，我们将设计新类型的FCI，并研究其对无序和不可避免的实验缺陷的鲁棒性。最后，这种方法将被扩展到描述FCI中的拓扑特征出现由于时空对称性。例如，这些对称性可以被认为是空间旋转和时间平移的综合效应。由于目前还没有一致的方法来分析这种拓扑相的行为，我们的工作将是他们研究的一个里程碑。

Topological insulators have a gapped bulk, but host protected gapless boundary states, which are responsible for robust, quantized observables. They can occur in one-, two-, and three-dimensional systems, and their boundary states often require one or more symmetries in order to remain protected. Recently, a growing number of new types of time-independent topological phases have emerged both theoretically and experimentally, by taking into account the role of space symmetries: the symmetric configurations of atoms as they are found in crystals. In contrast, topological phases which appear due to the periodic modulation of system parameters in time are much less studied.We will study an almost entirely unexplored class of systems, "Floquet crystalline insulators" (FCI). These are time-periodic systems that inherit their topological features from the presence of spatial symmetries. The goal is to develop a theoretical framework which can treat both static and time-dependent topological phases on the same footing, and then apply it to FCI. Using this framework, we will design new types of FCI and study their robustness against disorder and unavoidable experimental imperfections. Finally, this approach will be extended to describe FCI in which topological features appear due to space-time symmetries. These symmetries can be thought of, for instance, as the combined effect of a rotationof space followed by a translation of time. Since there is currently no consistent method of analyzing the behavior such topological phases, our work will represent a milestone in their study.