Lacing topological orders in two dimensions: Exactly solvable models for Kitaev's sixteen-fold way

Abstract

A family of two-dimensional (2D) spin-1/2 models have been constructed to realize Kitaev’s sixteen-fold way of anyon theories. Defining a one-dimensional (1D) path through all the lattice sites, and performing the Jordan-Wigner transformation with the help of the 1D path, we find that such a spin-1/2 model is equivalent to a model with \nuν species of Majorana fermions coupled to a static \mathbb{Z}_2ℤ2 gauge field. Here each species of Majorana fermions gives rise to an energy band that carries a Chern number \mathcal{C}=1𝒞=1, yielding a total Chern number \mathcal{C}=\nu𝒞=ν. It has been shown that the ground states are three (four)-fold topologically degenerate on a torus, when \nuν is an odd (even) number. These exactly solvable models can be achieved by quantum simulations.