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

TL;DR

This paper constructs exactly solvable 2D spin-1/2 models that realize all phases of Kitaev's sixteen-fold way of anyon theories, linking Majorana fermions, topological degeneracy, and potential for quantum simulation.

Contribution

It introduces a family of 2D models that realize Kitaev's sixteen-fold way, connecting Majorana fermions, Chern numbers, and topological degeneracies in an exactly solvable framework.

Findings

Models realize all sixteen phases of Kitaev's classification.

Ground state degeneracy depends on the parity of .

Models are suitable for quantum simulation.

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 $\nu$ species of Majorana fermions coupled to a static $\mathbb{Z}_2$ gauge field. Here each specie of Majorana fermions gives rise to an energy band that carries a Chern number $\mathcal{C}=1$, yielding a total Chern number $\mathcal{C}=\nu$. It has been shown that the ground states are three (four)-fold topologically degenerate on a torus, when $\nu$ is an odd (even) number. These exactly solvable models can be achieved by quantum simulations.