The classification of topological states of matter depends on spatial dimension and symmetry class. For non-interacting topological insulators and superconductors, the topological classification is obtained systematically and non-trivial topological insulators are classified by either integer or Z 2 . The classification of interacting topological states of matter is much more complicated and only special cases are understood. In this paper we study a new class of topological superconductors in (2 + 1) dimensions which has time-reversal symmetry and a spin conservation symmetry. We demonstrate that the superconductors in this class are classified by when electron interaction is considered, while the classification is without interaction.
The topology of an insulator can be defined even when all eigenstates of the system are localized-an extreme case of Anderson insulators that we call ultralocalized. We derive the classification of such ultralocalized insulators in all symmetry classes and dimensions. We clarify their bulk-boundary correspondence and show that ultralocalized systems are in many instances phases of matter not described by the known classification of topological insulators and superconductors. As a consequence, we clarify which conventional topological phases are Wannierizable, and which topological phases cannot exist without delocalized states.
Symmetry protected topological (SPT) phases are a minimal generalization of the concept of topological insulators to interacting systems. In this paper, we describe the classification and properties of such phases for three-dimensional (3D) electronic systems with a number of different symmetries. For symmetries representative of all classes in the famous 10-fold way of free-fermion topological insulators/superconductors, we determine the stability to interactions. By combining with results on bosonic SPT phases, we obtain a classification of electronic 3D SPT phases for these symmetries. In cases with a normal
U(1) subgroup we show that this classification is complete. We describe the nontrivial surface and bulk properties of these states. In particular, we discuss interesting correlated surface states that are not captured in a free-fermion description. We show that in many, but not all, cases, the surface can be gapped while preserving symmetry if it develops intrinsic topological order.