Anyons

In this paper the hard core boson gas in (2 + 1)-dimensions in the presence of a Chern-Simons term is considered, the authors find that the long wave description is given in terms of Coulomb (longitudinal) and transversal electrodynamics. The longitudinal part is described in terms of electric charges which carry vorticity. When binding of electrical charges occurs the transversal part is described by a photon field and corresponds to a superfluid. Using the same language the authors find that a spinless fermion is described by a massive photon. The authors use these concepts to understand the relation between the negative-U Hubbard model in the presence of a magnetic field, and vortices which carry electric charge induced by doping in the repulsive Hubbard model