# Colloquium: Topological phases in periodically driven systems

"Topological phases in periodically driven systems"

Lukasz Fidkowski (University of Washington, Seattle)

Topological insulators can be understood at a free fermion (i.e. single particle) level as band insulators with interesting topological twists in their band structures. Recently it has been found that introducing a periodic Floquet drive (i.e. periodic time dependence) into such a free fermion system allows one to realize more interesting twisted band structures, leading to a new class of so-called `Floquet' topological insulators. However, it is also known that when Coulomb interactions between electrons dominate the electron kinetic energy, novel exotic quantum phases, such as the fractional quantum Hall states, can be realized, with a richer phenomenology than what is possible in free fermion systems, including fractional charge and exotic quasiparticle statistics. It is natural to wonder whether one can combine a periodic drive together with interactions to generate even more exotic phases. One immediate obstacle is the so-called `heating problem': a driven interacting system will generically heat up to infinite temperature, obliterating any distinction between phases. In this talk I will discuss one possible way around this heating problem, involving recent ideas from quantum statistical mechanics in the form of `many-body localization', as well as some progress in identifying new non-equilibrium interacting Floquet phases.