Braiding statistics approach to symmetry-protected topological phases

Abstract

We construct a 2D quantum spin model that realizes an Ising paramagnet with\ngapless edge modes protected by Ising symmetry. This model provides an example\nof a "symmetry-protected topological phase." We describe a simple physical\nconstruction that distinguishes this system from a conventional paramagnet: we\ncouple the system to a Z_2 gauge field and then show that the \\pi-flux\nexcitations have different braiding statistics from that of a usual paramagnet.\nIn addition, we show that these braiding statistics directly imply the\nexistence of protected edge modes. Finally, we analyze a particular microscopic\nmodel for the edge and derive a field theoretic description of the low energy\nexcitations. We believe that the braiding statistics approach outlined in this\npaper can be generalized to a large class of symmetry-protected topological\nphases.\n