Topological quantum error correction based on the manipulation of the anyonic defects constitutes one of the most promising frameworks towards realizing fault-tolerant quantum devices. Hence, it is crucial to understand how these defects interact with external defects such as boundaries or domain walls. Motivated by this line of thought, in this work, we study the fusion events between anyons in the bulk and at the boundary in fixed-point models of 2 + 1-dimensional non-chiral topological order defined by arbitrary fusion categories. Our construction uses generalized tube algebra techniques to construct a bi-representation of bulk and boundary defects. We explicitly derive a formula to calculate the fusion multiplicities of a bulk-to-boundary fusion event for twisted quantum double models and calculate some exemplary fusion events for Abelian models and the (twisted) quantum double model of S3, the simplest non-Abelian group-theoretical model. Moreover, we use the folding trick to study the anyonic behavior at non-trivial domain walls between twisted S3 and twisted as well as models. A recurring theme in our construction is an isomorphism relating twisted cohomology groups to untwisted ones. The results of this work can directly be applied to study logical operators in two-dimensional topological error correcting codes with boundaries described by a twisted gauge theory of a finite group.
In fact, one can directly calculate the modular data of the anyons without deriving the R tensor first. For this, one considers the vector space defined by a cellulation of a torus and analyzes the endomorphism induced by the mapping class group of the torus, generated by S and T matrices. For a detailed derivation, see Ref. 6.
Note that the full Lagrangian algebra is not only described by an object in the UMTC, i.e. the set of condensable anyons, but also by an algebra morphism. This morphism can also be computed explicitly combining structures and techniques described in this manuscript. We plan to address this in future work.
N, M have to share a divisor in order for there to exist a non-trivial 2-coycle class, see Appendix B.
Note that here we think of anyons as defects, so “ground state with anyons” means ground states of a Hamiltonian which is altered at some points to enforce the existance of anyons.