Interacting anyonic fermions in a two-body color code model

We introduce a two-body quantum Hamiltonian model of spin 1/2 on a two-dimensional spatial lattice with exact topological degeneracy in all coupling regimes. There exists a gapped phase in which the low-energy sector reproduces an effective color code model. High-energy excitations fall into three families of anyonic fermions that turn out to be strongly interacting. The model exhibits a Z(2) x Z(2) gauge-group symmetry and string-net integrals of motion, which are related to the existence of topological charges that are invisible to moving high-energy fermions.