Non-Abelian fusion rules from an Abelian system

Abstract We demonstrate the emergence of non-Abelian fusion rules for excitations of a two dimensional lattice model built out of Abelian degrees of freedom. It can be considered as an extension of the usual toric code model on a two dimensional lattice augmented with matter fields. It consists of the usual C ( Z p ) gauge degrees of freedom living on the links together with matter degrees of freedom living on the vertices. The matter part is described by a n dimensional vector space which we call H n . The Z p gauge particles act on the vertex particles and thus H n can be thought of as a C ( Z p ) module. An exactly solvable model is built with operators acting in this Hilbert space. The vertex excitations for this model are studied and shown to obey non-Abelian fusion rules. We will show this for specific values of n and p , though we believe this feature holds for all n > p . We will see that non-Abelian anyons of the quantum double of C ( S 3 ) are obtained as part of the vertex excitations of the model with n = 6 and p = 3 . Ising anyons are obtained in the model with n = 4 and p = 2 . The n = 3 and p = 2 case is also worked out as this is the simplest model exhibiting non-Abelian fusion rules. Another common feature shared by these models is that the ground states have a higher symmetry than Z p . This makes them possible candidates for realizing quantum computation.