An operator algebraic criterion for extracting symmetry defect and fractionalisation data

Topological orders are characterised by the presence of long range quasi-particle excitations called anyons. There is a rich mathematical structure underlying anyons due to the possibility of braiding and fusion. Symmetry enriched topological (SET) order is a type of topological order having a symmetry. In SETs, the mathematical structure is even richer. For instance, the symmetry may 'fractionalise' onto the anyons, and the action of such a symmetry may also permute the anyon types. The landmark paper by BBCW in '14 started from macroscopic first principles and described the algebraic theory of anyons in the presence of such a symmetry.

In this talk I will start from microscopic models and give a 'selection criterion' to extract the symmetry data out of SETs in the bulk and without gauging the system. Such a selection criterion relies on procedurally constructing symmetry defect states starting from the ground-state. The data selected by this criterion is an invariant of the phase and is thus robust against small perturbations. I will also show that the mathematical structure of the symmetry data matches the literature for specific commuting projector models. This approach lends itself nicely to be able to compute SPT cocycles in the bulk.