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

Abstract

We study the excitations in a three dimensional version of Kitaev's spin-1/2 model on the honeycomb lattice introduced by the present authors recently. The gapped phase of the system is analyzed using a low energy effective Hamiltonian which is defined on the diamond lattice and consists of plaquette operators. The excitations of the effective Hamiltonian form loops in an embedded lattice. The elementary excitations, which are the shortest loops, are fermions. Moreover, the excitations obey nontrivial braiding rules: when a fermion winds through a loop, the wave function acquires a phase $\pi$.