Symmetry-Protected Topological Phases of Quantum Matter

We describe recent progress in our understanding of the interplay between interactions, symmetry, and topology in states of quantum matter. We focus on a minimal generalization of the celebrated topological band insulators (TBIs) to interacting many-particle systems known as symmetry-protected topological (SPT) phases. As with the TBIs, these states have a bulk gap and no exotic excitations but have nontrivial surface states that are protected by symmetry. We describe the various possible phases and their properties in three-dimensional systems with realistic symmetries. We develop many key ideas for the theory of these states using simple examples. The emphasis is on physical rather than mathematical properties. We survey insights obtained from the study of SPT phases for a number of other theoretical problems.