Chern_simons Theory of the Anisotropic Quantum Heisenberg Antiferromagnet on a Square Lattice

Abstract

We consider the anisotropic quantum Heisenberg antiferromagnet (with anisotropy $\lambda$) 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 $\lambda >{\lambda }_c$ and a flux phase for $\lambda <{\lambda }_c$ separated by a second order transition at ${\lambda }_c<1$. 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 ${\lambda }^{\ast }>{\lambda }_c$. 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 $\lambda <{\lambda }^{\ast }$.

53 pages, one figure available upon request, Revtex

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• Mott Transition in An Anyon Gas
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• Intersections forms and the geometry of lattice Chern-Simons theory
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