This thesis is concerned with phases of matter, one of the central notions in condensed matter physics. Traditionally, condensed matter physics has been concerned with phases of matter in thermal equilibrium, which means it is coupled to a heat bath. The main interest of this thesis, however, is isolated systems, in which the system is allowed to reach a steady state on its own, without interacting with a heat bath. In such a context it is possible for the steady state to be non-thermal in character, leading to many new phenomena.

A main interest of this thesis will be Floquet systems, which are systems that are periodically driven, for example by a time-oscillatory electric field. In this thesis, we will identify and charcterize phases of matter occuring in Floquet systems that are entirely new, in the sense that they have no analog in equilibrium.

We introduce a “Floquet equivalence principle”, which states that Floquet topological phases with symmetry G are in one-to-one correspondence with stationary topological phases with additional symmetry. This allows us to leverage the existing literature on topological phases with symmetries to understand Floquet topological phases. Such phases can be stabilized in driven strongly disordered systems through the phenomenon of “many-body localization” (MBL). We discuss properties of Floquet phases such as the “pumping” of lower-dimensional topological phases onto the boundary at each time cycle.

We then turn to spontaneous symmetry-breaking phases. We show that in Floquet systems, there is a striking new kind of such phase: the Floquet time crystal, in which the symmetry that is spontaneously broken is discrete time-translation symmetry. Such systems, though driven at frequency \omega, respond at a fractional frequency \omega/n. We show using analytical arguments and numerical evidence that such phases can be stabilized in driven strongly disordered systems through the phenomenon of “many-body localization” (MBL).

Next, we show that both Floquet time crystals and Floquet topological phases can be stabilized even without disorder. We establish a new scenario for “pre-thermalization”, a phenomenon where the eventual thermalization of the system takes place at a rate that is exponentially small in a parameter. In the intermediate regime, before pre-thermalization, there is a quasi-stationary pre-thermal regime in which Floquet phases can be stabilized.

In a slight digression, we then develop a systematic theory of stationary topological phases with discrete spatial symmetries (as opposed to the discrete temporal symmetry characterizing Floquet phases), showing that they also satisfy a “crystalline

equivalence principle” relating phases of matter with spatial symmetry to phases of matter with internal symmetry. Our arguments are based on notions of “gauging spatial symmetries” as well as a viewpoint based on topological quantum field theory (TQFT).

Finally, we put the Floquet equivalence principle on a systematic footing, and unify it with the crystalline equivalence principle for stationary topological phases, by invoking a powerful homotopy-theoretic viewpoint on phases of matter. The end result is a general theory of strongly correlated phases of matter with space-time symmetries.