Explicit illustration of non-abelian fusion rules in a small spin lattice

We exactly solve a four-site spin model with site-dependent Kitaev's coupling in a tetrahedron by means of an analytical diagonalization. The non-abelian fusion rules of eigen vortex excitations in this small lattice model are explicitly illustrated in real space by using Pauli matrices. Comparing with solutions of Kitaev models on large lattices, our solution gives an intuitional picture using real space spin configurations to directly express zero modes of Majorana fermions, non-abelian vortices and non-abelian fusion rules. We generalize the single tetrahedron model to a chain model of tetrahedrons on a torus and find the non-abelian vortices become well-defined non-abelian anyons. We believe these manifest results are very helpful to demonstrate the nonabelian anyon in laboratory.