Anyon-Permuting Symmetry

Abstract

Abstract A theory with anyon permuting symmetry is unchanged under permutation of the names of the particle types. The toric code is an example of a theory with such a symmetry. The toric code, and all theories with such symmetries can be viewed as arising as a condensation of anyons from some more complicated parent theory. Given that such a symmetry exists, we consider defects of this symmetry which generalize the mathematical structure of the TQFT to a so-called G-crossed extension. To clarify how this works we use a particularly tranparent example of the toric code where the symmetry of particle types is reduced to a lattice symmetry, and the defects correspond to dislocations in the lattice. It can be shown that these defects harbor Majorana zero modes. We discuss the general structure of a parent theory condensing via a G-crossed theory to a theory with permutation symmetry. Finally we mention theories with both on-site symmetries and anyons, which are known as symmetry enriched topological phases.