Braiding statistics approach to symmetry-protected topological phases

We construct a two-dimensional (2D) quantum spin model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. This model provides an example of a ``symmetry-protected topological phase. '' We describe a simple physical construction that distinguishes this system from a conventional paramagnet: We couple the system to a Z₂ gauge field and then show that the -flux excitations have different braiding statistics from that of a usual paramagnet. In addition, we show that these braiding statistics directly imply the existence of protected edge modes. Finally, we analyze a particular microscopic model for the edge and derive a field theoretic description of the low energy excitations. We believe that the braiding statistics approach outlined in this paper can be generalized to a large class of symmetry-protected topological phases.