Driven topological phases with space and time symmetries

拓扑绝缘体具有带隙的整体，但宿主保护的无间隙边界态，负责提供健壮的、量子化的可观测数据。它们可以出现在一维、二维和三维系统中，它们的边界态通常需要一个或多个对称性才能保持保护。最近，通过考虑空间对称性的作用，越来越多的新类型的与时间无关的拓扑相在理论和实验上都出现了，空间对称性是原子在晶体中发现的对称构型。相比之下，由于系统参数在时间上的周期性调制而出现的拓扑相的研究要少得多。我们将研究一类几乎完全未被探索的系统--Floket晶体绝缘体(FCI)。这些是时间周期系统，它们从空间对称性的存在继承了它们的拓扑特征。我们的目标是开发一种理论框架，可以在相同的基础上处理静态和时间相关的拓扑相，并将其应用于FCI。利用这个框架，我们将设计新型的FCI，并研究它们对无序和不可避免的实验缺陷的稳健性。最后，将这种方法扩展到描述由于时空对称性而出现的拓扑特征的FCI。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阶段的行为，我们的工作将是他们研究中的一个里程碑。

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.