Exact results of the quantum phase transition for the topological order

Abstract

In this paper a duality between the d=2 Wen-plaquette model in a transverse field and the d=1 Ising model in a transverse field is used to learn the nature of the quantum phase transition (QPT) between a spin-polarized phase and a topological ordered state with the string-net condensation. The QPT is not induced by spontaneous symmetry breaking and there are no conventional Landau-type local order parameters. Instead the string-like non-local order parameters are introduced to describe the QPT. In particular, the duality character between the open-string and closed string for the QPT are explored.