Critical properties of the two-dimensional anisotropic Heisenberg model

While a nonzero spontaneous magnetization m cannot exist in a $d=2$ Heisenberg spin system, it is possible that a phase transition associated with a divergent susceptibility occurs at the Stanley-Kaplan temperature TcSK. The crossover from this special isotropic case to the anisotropic (Ising) behavior is studied using a Monte Carlo technique. The classical model with Hamiltonian H=-JΣ[(1-Δ)(SᵢˣSⱼˣ+SᵢʸSⱼʸ)+SᵢᶻSⱼᶻ] on N×N square lattices with periodic boundary conditions is investigated for N≤100 and Δ varying from 0.005 to 1. The spontaneous magnetization m, energy, specific heat, longitudinal and transverse susceptibilities, and the self-correlation are determined over a wide temperature range. For weak anisotropy dm/dT decreases nonmonotonically with increasing temperature and deviations from simple spin-wave theory occur at surprisingly low temperatures. Transition temperatures Tc(Δ) exceed the isotropic value TcSK predicted by series expansions although the values would also be consistent with TcSK=0 if Tc(Δ)∝|ln(1Δ)|^-1. The asymptotic critical exponent for the order parameter is β=1/8 for all Δ. The susceptibility data show crossover from γ=1.75 near Tc(Δ) to higher values farther from Tc(Δ). It is shown that finite-size rounding may lead to erroneously large estimates for β and erroneously small estimates for γ. These effects may invalidate some conclusions drawn from experiments on planar systems, too. Accepting the series estimate kBTcSKJ=0.588 we find that the data are consistent with Tc(Δ)-TcSK∝Δ^1φ with phi≈4. Assuming power-law behavior we show that our data obey crossover scaling with two-dimensional Heisenberg exponents γH≈3, αH≈-2 in the isotropic limit. The data are also consistent with scaling theory based on the assumption of singularities which are stronger than any power law.