Lattice Realization of Twist Defects in a $\mathbb{Z}_2\times \mathbb{Z}_2$ Topological Order

TL;DR

This paper presents a microscopic lattice model realizing twist defects in a $ obreak bZ_2 imes bZ_2$ topological order, analyzing their properties and fusion rules through a layered Wen plaquette setup.

Contribution

It introduces a lattice construction of twist defects in a double toric code model, characterizing their quantum dimensions and fusion rules.

Findings

Identified localized twist defects via dislocations in stacked Wen models.

Calculated quantum dimensions of the defects.

Determined non-Abelian fusion rules for defects with different permutations.

Abstract

In this work, we explore a microscopic realization of three types of anyonic symmetries in a $\mathbb{Z}_2\times\mathbb{Z}_2$ topological order, corresponding to a double toric code. These symmetries act as nontrivial permutations on the anyon labels of the parent state. We consider a setup consisting of two decoupled Wen plaquette models stacked on top of each other and introduce dislocations that modify the Hamiltonian, giving rise to localized twist defects, eventually inducing interactions between the layers. In this context, branch cuts act as sources of anyon permutations when they cross it. We characterize the defects by calculating their quantum dimensions, and we also consider double loop operators around them that allow us to determine the non-Abelian fusion rules between the defects, including when they carry different anyon permutations.