Anyonization of lattice Chern-Simons theory

We formulate Hamiltonian lattice Chern-Simons theory which has the property that the Chern-Simons gauge fields of the theory can be eliminated by making matter fields multivalued operators with anyonic statistics. We prove that, when the statistics parameter is an odd integer so that the anyons are bosons, the ground state, which consists of a condensate of bound pairs of flux tubes and fermions, breaks phase invariance. The ensuing long-range order implies that the system is an unconventional superfluid. We formulate a condition which may be useful as a numerical signal for symmetry breaking in the ground state for any statistics parameter. We also discuss an exotic lattice Chern-Simons theory, which makes explicit the relation of anyons to framed knot invariants. We discuss various lattice representations of the Chern-Simons term and find the unique local lattice Chern-Simons term with the appropriate naive continuum limit, which permits anyonization.