Fermions and nontrivial loop-braiding in a three-dimensional toric code

Abstract

We study an exactly solvable toric code type of Hamiltonian in three dimensions, defined on the diamond lattice with spin-$1/2$ degrees of freedom at each site. The Hamiltonian is a sum of mutually commuting plaquette operators ${B}_{p}$, all of which have eigenvalue +1 in the ground state. The excitations are ``fluxes,'' which are plaquettes with ${B}_{p}=\ensuremath{-}1$. Due to certain local kinematic constraints, fluxes form loops. The elementary flux-loop excitations are fermions, in contrast to other solvable spin-$1/2$ models in three dimensions, where the excitations are bosons. Furthermore, the flux loops braid nontrivially, giving rise to Abelian anyonlike statistics.