Driven topological phases with space and time symmetries

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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.