Crystalline topological phases as defect networks

Abstract

A crystalline topological phase is a topological phase with spatial symmetries. In this work, we give a very general physical picture of such phases: A topological phase with spatial symmetry G (with internal symmetry G[subscript int] ≤ G) is described by a defect network, a G-symmetric network of defects in a topological phase with internal symmetry G[subscript int]. The defect network picture works both for symmetry-protected topological (SPT) and symmetry-enriched topological (SET) phases, in systems of either bosons or fermions. We derive this picture both by physical arguments and by a mathematical derivation from the general framework of Thorngren and Else [Phys. Rev. X 8, 011040 (2018)2160-330810.1103/PhysRevX.8.011040]. In the case of crystalline SPT phases, the defect network picture reduces to a previously studied dimensional reduction picture, thus establishing the equivalence of this picture with the general framework of Thorngren and Else applied to crystalline SPTs.