Chern_simons Theory of the Anisotropic Quantum Heisenberg Antiferromagnet on a Square Lattice
We consider the anisotropic quantum Heisenberg antiferromagnet (with anisotropy $λ$) 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 Ne{è}l state for $λ>λ_c$ and a flux phase for $λ<λ_c$ separated by a second order transition at $λ_c<1$. 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 $λ^* > λ_c$. We identify this transition with the isotropic Heisenberg point. It has a non-vanishing Ne{\` e}l order parameter, which drops to zero discontinuously for $λ<λ^*$.