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 $\mathbb{C}({\mathbb{Z}}_p)$ 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 ${\mathbb{Z}}_p$ gauge particles act on the vertex particles and thus $H_n$ can be thought of as a $\mathbb{C}({\mathbb{Z}}_p)$ module. An exactly solvable model is built with operators acting in the corresponding 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 $\mathbb{C}(S_3)$ 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 ${\mathbb{Z}}_p$. This makes them possible candidates for realizing quantum computation.

12 pages, 1 figure

References in corpus (11)

• Non-Abelian Anyons and Topological Quantum Computation
• Topological Order with a Twist: Ising Anyons from an Abelian Model
• A classification of symmetry enriched topological phases with exactly solvable models
• A Family of Non-Abelian Kitaev Models on a Lattice: Topological Confinement and Condensation
• Theory of defects in Abelian topological states
• Qudit surface codes and gauge theory with finite cyclic groups
• Non-abelian statistics from an abelian model
• The Spin-Statistics Theorem for Anyons and Plektons in d=2+1
• Exactly soluble lattice models for non-abelian states of matter in 2 dimensions
• 2D Quantum Double Models From a 3D Perspective
• Quantum loop models and the non-abelian toric code

Cited by in corpus (3)

• Non-Abelian fusion rules from Abelian systems with SPT phases and graph topological order
• Finite-group gauge theories on lattices as Hamiltonian systems with constraints
• Quantum Double Models coupled to matter fields: a detailed review for a dualization procedure