CHERN-SIMONS THEORY OF THE ANISOTROPIC QUANTUM HEISENBERG-ANTIFERROMAGNET ON A SQUARE LATTICE

We consider the anisotropic quantum Heisenberg antiferromagnet (with a nisotropy lambda) on a square lattice using a Chern-Simons (or Wigner- Jordan) approach. We show that the average field approximation (AFA) y ields a phase diagram with two phases: a Neel state for lambda > lambd a(c) and a flux phase for lambda < lambda(c) separated by a second-ord er transition at lambda(c) < 1. We show that this phase diagram does n ot describe the XY regime of the antiferromagnet. Fluctuations around the AFA induce relevant operators which yield the correct phase diagra m. We find an equivalence between the antiferromagnet and a relativist ic field theory of two self-interacting Dirac fermions coupled to a Ch ern-Simons gauge field. The field theory has a phase diagram with the correct number of Goldstone modes in each regime and a phase transitio n at a critical coupling lambda > lambda(c). We identify this transit ion with the isotropic Heisenberg point. It has a nonvanishing Neel or der parameter, which drops to zero discontinuously for lambda < lambda .