Chern-Simons theory of the anisotropic quantum Heisenberg antiferromagnet on a square lattice

We consider the anisotropic quantum Heisenberg antiferromagnetic (with anistropy λ) on a square lattice using a Chern-Simons (or Wigner-Jordan) approach. We show that the average field approximation (AFA) yields a phase diagram with two phases: a Neèl state for λ>${\lambda }_{\mathit{c}}$ and a flux phase for λ<${\lambda }_{\mathit{c}}$ separated by a second-order transition at ${\lambda }_{\mathit{c}}$<1. We show that this phase diagram does not describe the XY regime of the antiferromagnet. Fluctuations around the AFA induce relevant operators which yield the correct phase diagram. We find an equivalence between the antiferromagnet and a relativistic field theory of two self-interacting Dirac fermions coupled to a Chern-Simons gauge field. The field theory has a phase diagram with the correct number of Goldstone modes in each regime and a phase transition at a critical coupling ${\lambda }^{\mathrm{*}}$>${\lambda }_{\mathit{c}}$. We identify this transition with the isotropic Heisenberg point. It has a nonvanishing Neèl order parameter, which drops to zero discontinuously for λ<${\lambda }^{\mathrm{*}}$.