Synthetic non-Abelian statistics by Abelian anyon condensation

Topological degeneracy is the degeneracy of the ground states in a many-body system in the large-system-size limit. Topological degeneracy cannot be lifted by any local perturbation of the Hamiltonian. The topological degeneracies on closed manifolds have been used to discover/define topological order in many-body systems, which contain excitations with fractional statistics. In this paper, we study a new type of topological degeneracy induced by condensing anyons along a line in two-dimensional topological ordered states. Such topological degeneracy can be viewed as carried by each end of the line defect, which is a generalization of Majorana zero modes. The topological degeneracy can be used as a quantum memory. The ends of line defects carry projective non-Abelian statistics even though they are produced by the condensation of Abelian anyons, and braiding them allows us to perform fault tolerant quantum computations.

Figure 1

The lattice is partitioned into even and odd plaquettes, which, respectively, host electric and magnetic excitations. (a) The on-site action of the ${\sigma }^y$ operator creates, annihilates, or moves fermion excitations. (b) Turn on ${\sigma }^y$ term along a line of sites (marked by ) to condense the fermions into a Majorana chain. The chain (branch-cut) region $\mathcal{C}$ is outlined by the dashed line. (c) The second-order perturbation paths leading to the double-plaquette operators. Excited sites in the intermediate states are marked by . (d) The double-plaquette operators on the branch-cut and around the synthetic dislocation.

Figure 2

represent the ${\mathbb{Z}}_N$ charges. (a) The on-site action of link operator $I$. (b) Turn on $I$ term along a series of links (marked by ) to condense the anyons. (c) The second-order perturbation paths leading to the triple-plaquette operators. Excited links in the intermediate states are marked by . (d) The triple-plaquette operators on the branch-cut and around the synthetic dislocation.

Figure 3

Noncontractible Wilson loops are chosen according to the homology basis, following Ref. 2.

Figure 4

Noncontractible Wilson loops for four dislocations on a sphere in the case of (a) $e$-$m$ duality dislocations and (b) charge conjugate dislocations.