Skip to Content

Duality between symmetry protected topological order and intrinsic topological order

Symmetry protected topological(SPT) phase is a generalization of topological insulator(TI). Different from the intrinsic topological phase, e.g., the fractional quantum hall(FQH) phase, SPT phase is only distinguishable from a trivial disordered phase when certain symmetry is preserved. Indeed, SPT phase has a long history in 1D, and it has been shown that the well known Haldane phase of S=1 Heisenberg chain belongs to this class. However, in higher dimensions, most of the previous studies focus on free electron systems. Until very recently, it was realized that SPT phase also exists in interacting boson/spin systems in higher dimensions. In this talk, I will show that there is an interesting duality map between intrinsic topological phase and SPT phase. I will focus on a simple model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. The duality provides us 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, these braiding statistics directly implies the existence of protected edge modes. The duality map also provides us a complementary way to derive topological terms of SPT phases, which classify different SPT phases in bosonic systems. Finally, I will discuss the possible generalization for interacting fermion/electron systems.