Modified toric code models with flux attachment from Hopf algebra gauge theory

TL;DR

This paper generalizes the toric code model by incorporating flux attachment via non-trivial R-matrices in Hopf algebra gauge theory, leading to new topological phenomena and particle permutation effects.

Contribution

It introduces a novel class of Hamiltonian models based on Hopf algebra gauge theories with non-trivial R-matrices, extending the toric code framework.

Findings

Non-commutative algebra of functions on the gauge group

Particle permutation depending on R-matrix

Flux attachment in Z_N gauge theory

Abstract

Kitaev's toric code is constructed using a finite gauge group from gauge theory. Such gauge theories can be generalized with the gauge group generalized to any finite-dimensional semisimple Hopf algebra. This also leads to generalizations of the toric code. Here we consider the simple case where the gauge group is unchanged but furnished with a non-trivial quasitriangular structure (R-matrix), which modifies the construction of the gauge theory. This leads to some interesting phenomena; for example, the space of functions on the group becomes a non-commutative algebra. We also obtain simple Hamiltonian models generalizing the toric code, which are of the same overall topological type as the toric code, except that the various species of particles created by string operators in the model are permuted in a way that depends on the R-matrix. In the case of $\mathbb{Z}_{N}$ gauge theory, we find that the introduction of a non-trivial R-matrix amounts to flux attachment.