Topological degeneracy and vortex manipulation in Kitaev's honeycomb model

TL;DR

This paper analyzes the topological degeneracy in Kitaev's honeycomb model, deriving a comprehensive effective Hamiltonian, and shows how vortex-based fermions can be manipulated without energy cost, confirming robustness in the thermodynamic limit.

Contribution

It provides a perturbative low-energy effective Hamiltonian valid to all orders and configurations, clarifying topological degeneracy lifting and vortex fermion dynamics.

Findings

Topological degeneracy is lifted at finite size but remains robust in the thermodynamic limit.

Loop symmetries correspond to fermion creation, propagation, and annihilation.

Vortex pairs can be moved without additional energy cost.

Abstract

The classification of loop symmetries in Kitaev's honeycomb lattice model provides a natural framework to study the abelian topological degeneracy. We derive a perturbative low-energy effective Hamiltonian, that is valid to all orders of the expansion and for all possible toroidal configurations. Using this form we demonstrate at what order the system's topological degeneracy is lifted by finite size effects and note that in the thermodynamic limit it is robust to all orders. Further, we demonstrate that the loop symmetries themselves correspond to the creation, propagation and annihilation of fermions. Importantly, we note that these fermions, made from pairs of vortices, can be moved with no additional energy cost.