Parafermions and Z₃ Charge-Flux attachment

A recent construction (Ann. Phys. 393, 234 (2018)) has introduced an interesting route to bosonization of fermions in two spatial dimensions by implementing a precise lattice version of flux-charge binding. The idea is to modify Kitaev's Toric code so that the electric charge (e) and magnetic flux (m) are always created in a tight ``dipolar" pair ε (=e x m) which is a fermion, providing a way to represent any local fermionic Hamiltonian as a local Hamiltonian of spins. We discuss an extension of these ideas to the case of a Z_N toric which allows a bosonization of anyons with more general statitistical angles. We have focused particularly in the case of N=3. We wil prove that ground states in the torus in this case are at least three-fold degenerate, and, discuss a model featuring a topological phase transition between a 3-fold degenerate state and a 9-fold degenerate ground state featuring parafermionic excitations.