Abstract

The Toric Code is a celebrated toy model that gives us an explicit construction of $\mathbb{Z}_2$ topological order on a lattice. In this model, one can create states with anyons, and move them around using string operators. For all its strengths, it is still a toy model with a completely flat spectrum (the Hamiltonian is made up of commuting projectors). There is a generalisation of this model to abelian finite groups $G$, known as the Quantum Double models. This is the class of models we consider in this talk. To make this model more physical, we introduce a family of perturbations to this Hamiltonian that correspond to anyon hoppings, in the same spirit as the Hubbard model. These perturbations respect the important symmetries of the original model. We then solve for the resulting spectrum and find interesting features in it such as a massless generalised Dirac Cone which becomes massive after duality symmetry breaking, and the creation of a stable bound-pair for a large range of perturbation parameters.