Symmetry fractionalization and twist defects

Abstract

Topological order in two dimensions can be described in terms of deconfined quasiparticle excitations - anyons - and their braiding statistics. However, it has recently been realized that this data does not completely describe the situation in the presence of an unbroken global symmetry. In this case, there can be multiple distinct quantum phases with the same anyons and statistics, but with different patterns of symmetry fractionalization - termed symmetry enriched topological (SET) order. When the global symmetry group $G$, which we take to be discrete, does not change topological superselection sectors - i.e. does not change one type of anyon into a different type of anyon - one can imagine a local version of the action of $G$ around each anyon. This leads to projective representations and a group cohomology description of symmetry fractionalization, with $H^2(G,{\cal A})$ being the relevant group. In this paper, we treat the general case of a symmetry group $G$ possibly permuting anyon types. We show that despite the lack of a local action of $G$, one can still make sense of a so-called twisted group cohomology description of symmetry fractionalization, and show how this data is encoded in the associativity of fusion rules of the extrinsic `twist' defects of the symmetry. Furthermore, building on work of Hermele, we construct a wide class of exactly solved models which exhibit this twisted symmetry fractionalization, and connect them to our formal framework.