The interplay of symmetry and topology in quantum condensed matter

Many quantum phases of matter can be understood on the basis of symmetry. For example, the universal properties magnets, crystals, and even superfluids are a consequence of their spontaneous breaking of rotational, translational, and charge conservation symmetries, respectively. The fractional quantum Hall effect, discovered in 1982, is the first striking example of a phase that transcends this symmetry breaking framework, and instead exhibits a non-local `topological’ order, which does not rely on symmetry at all, but instead is characterized by emergent fractionalized excitations and chiral edge modes. More recently, with the discovery of topological insulators, it has been realized that in general, the interplay between symmetry and topology is more subtle, and a much richer phenomenology than previously thought can result. In this talk, I will discuss the new `symmetry protected’ and `symmetry enriched’ topological orders that can arise in this way, including their theoretical classification and potential physical signatures. I will also discuss exotic strongly interacting surface states that are available to such systems, and their connections to anomalies in discrete gauge theories. Finally, I will also talk about ongoing work on symmetry protected topological phenomena away from equilibrium.​