Topological Order with a Twist: Ising Anyons from an Abelian Model

Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We study the realization of such defects in the toric code model, showing that a process where defects are braided and fused has the same outcome as if they were Ising anyons. These ideas can also be applied in the context of topological codes.

Figure 1

Anyon processes: time flows upwards. (a),(b) Two topologically distinct ways to exchange anyons. (c) A fusion of two anyons. (d) Two twists (crosses) connected by a line (dotted) across which $e$ charges become $m$ charges.

Figure 2

A square lattice with spins living at vertices. (a) Plaquette operators are products of four Pauli operators. String operators are products of Pauli operators on their edges. (b) A dislocation in the geometry of the Hamiltonian produced by shifting plaquettes. In the pentagon one can introduce the indicated plaquette operator, which commutes with the rest.

Figure 3

Several string operators (solid lines) and twists (crosses), in a top view of the system. The ordering of segments in the string dictates which is drawn over which. Dotted lines are fermion string operators, two parallel string operators of different type. (a) A process that creates or annihilates an $ϵ$. (b) Two inequivalent closed string operators winding around a given twist. We choose the one to the left to define ${\sigma }_{+}$ and ${\sigma }_{-}$. (c) Three equivalent string operators. The first equation uses property (ii) of string operators, the second uses (i) in addition to (ii). (d) The strings ${\gamma }^1,{\gamma }^2,{\gamma }^3,\dots$ that define Majorana operators. The stars mark their end points. (e) The effect on ${\gamma }^j$ strings of braiding adjacent twists as in Fig. 1a.