Topological insulators and superconductors epitomize a large class of phases that fall outside Landau’s symmetry-breaking paradigm. By now it is clear that more than one phases can exist in a phase diagram even without spontaneous symmetry breaking. These are gapped phases at zero temperature distinguished by their topological properties and cannot be tuned to each other without breaking the symmetry or closing the gap. The topological properties manifest themselves in such forms as gapless boundary modes or anyonic excitations, which have garnered considerable interest due to their potential application to quantum computation.

In this talk, we will consider an interacting generalization of topological insulators and superconductors called symmetry protected topological (SPT) phases, which have been under intense investigation in recent years. I will first review the notion of SPT phases and efforts in the field to classify them, and then indicate how a seemingly abstract contruction in mathematics can unify various existing classification proposals. This construction, going under the name of “generalized cohomology,” is built upon a sequence of spaces …, F_{-1}, F_0, F_1, F_2, … that describe the topology of short-range entangled states in various physical dimensions. Adjacent spaces F_d and F_{d+1} are related by a looping procedure, which is believed to stem physically from a dimensional reduction process. I will show how the construction will give rise to relations between SPT phases across different dimensions and symmetry classes and thereby delegate the task of classifying SPT phases in one case

to another. We will derive the first known complete classification in the symmetry class of hourglass fermions KHgSb as an application.

References: 1701.00004 (theoretical framework), 1709.06998 (application to hourglass fermions KHgSb)