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

TL;DR

This paper uses a Chern-Simons approach to analyze the anisotropic quantum Heisenberg antiferromagnet on a square lattice, revealing a phase diagram with Neel and flux phases and identifying a transition point with the isotropic Heisenberg model.

Contribution

It introduces a novel Chern-Simons field theory framework to describe the phase diagram and critical behavior of the anisotropic quantum Heisenberg antiferromagnet.

Findings

Identifies a phase transition at a critical coupling * > _c.

Shows the phase diagram includes Neel and flux phases separated by a second order transition.

Establishes an equivalence between the antiferromagnet and a relativistic Dirac fermion field theory.

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{\`e}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 $\lambda^* > \lambda_c$. 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 $\lambda<\lambda^*$.