Projective non-Abelian statistics of dislocation defects in a ${\mathbb{Z}}_N$ rotor model

Non-Abelian statistics is a phenomenon of topologically protected non-Abelian geometric phases as we exchange quasiparticle excitations. Here we construct a ${\mathbb{Z}}_N$ rotor model that realizes a self-dual ${\mathbb{Z}}_N$ Abelian gauge theory. We find that lattice dislocation defects in the model produce topologically protected degeneracy. Even though dislocations are not quasiparticle excitations, they resemble non-Abelian anyons with quantum dimension $\sqrt{N}$. Exchanging dislocations can produce topologically protected projective non-Abelian geometric phases. Therefore, we discover a kind of (projective) non-Abelian anyon that appears as the dislocations in an Abelian ${\mathbb{Z}}_N$ rotor model. These types of non-Abelian anyons can be viewed as a generalization of the Majorana zero modes.

Figure 1

$\mathrm{Even}\times \mathrm{even}$ periodic lattice with plaquettes colored in a checkerboard pattern: The red and darker (blue and lighter) plaquette will be called even (odd). Each directed string represents a product of $U_i$ and/or $V_i$ operators on the sites along the string. The operator on each site is specified by the string direction (see text).

Figure 2

(a) Lattice with a pair of dislocations, marked out by $⊣$ and $⊢$. Plaquettes on the branch cut are colored in violet. Periodic boundary conditions are assumed in both directions by sticking the dashed edges with the solid edges on the opposite side. The ring operators $O_p$ are redefined around the pentagonal plaquette. $C$ operators denote large closed-strings looping around the lattice. (b) Plaquette to site mapping. The site that is not mapped to is marked by a black dot.

Figure 3

Diffeomorphism of the torus with a pair of dislocations at $c$ and $d$. Expand the branch cut (violet dashed line) between $c$ and $d$ into a hole, with two edges marked by $\alpha$ and $\beta$. Separate the $e$- and $m$ layers. Unwrap both layers by cutting along the large loops around the torus. Rotate one layer to glue the $\beta$ edges together along the marked direction. Glue the other edges and rewrap into a double torus.

Figure 4

Diffeomorphism of string operators on the torus (a) with a pair of dislocations, or (b) with three pairs of dislocations.