Symmetry protected topological (SPT) phases are generalizations of topological band insulators; they are quantum phases of matter with a bulk energy gap and characteristic edge or surface properties. Over the past few years, exciting progress has been made in the theory of SPT phases with strong interactions, and, separately, SPT phases with crystalline symmetry. The intersection of these two directions — strongly interacting crystalline SPT phases — has potential experimental relevance but has remained rather poorly understood. In this talk, I will explain how to classify and characterize SPT phases protected by crystalline symmetries, without regard to the strength of interactions. Surprisingly, this problem turns out to be simpler than for SPT phases with internal symmetries, and admits a physically transparent approach that I will explain, where crystalline SPT states are constructed from lower-dimensional building blocks.
This leads to a simple and general picture of crystalline SPT phases as "topological crystals," i.e. as crystalline patterns of lower-dimensional topological liquids. The same picture also applies to topological phases with bulk fractional excitations, where the possibilities are even richer and largely unexplored. This way of thinking provides a means of understanding at least some of the so-called fracton topological orders, in which fractional excitations move in constrained geometries.