Critical behavior of the two-dimensional anisotropic Heisenberg antiferromagnet: A numerical test of spin-wave theory

We present results from a numerical study of the spin-excitation energy gap of the two-dimensional anisotropic Heisenberg antiferromagnet. Our study finds a nonzero gap for Ising-like (easy-axis) anisotropy which approaches zero at the isotropic point. With XY-like (easy-plane) anisotropy the system is planar antialigned and remains gapless, which is consistent with the nearly gapless spin-wave dispersion relation observed in the high-$T_c$ precursor insulators. Near isotropy the gap is not well described by the g<1 spin-wave-theory prediction $E_{\mathrm{gap}=2(1\mathrm{}\mathrm{-}{g}^2{)}^{1/2}}$. Possible sources of this discrepancy are discussed; one such effect, ‘‘softening’’ of the mean antiferromagnetic background by quantum fluctuations, provides corrections to spin-wave theory that bring it into agreement with our Monte Carlo results. This effect may require reinterpretation of neutron scattering and far-infrared absorption data in the determination of the precursor insulator spin Hamiltonian. Our results also clarify the physical mechanisms underlying the Mermin-Wagner theorem and allow a higher-dimensional generalization of the Haldane conjecture.