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 $T_c^{\mathrm{SK}}$. The crossover from this special isotropic case to the anisotropic (Ising) behavior is studied using a Monte Carlo technique. The classical model with Hamiltonian $H=-J\Sigma \left[\right(1-\Delta \left)\right(S_i^x{S}_j^x+S_i^y{S}_j^y)+S_i^z{S}_j^z]$ on $N\times N$ square lattices with periodic boundary conditions is investigated for $N\le 100$ and $\Delta$ 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 $\frac{\mathrm{dm}}{\mathrm{dT}}$ decreases nonmonotonically with increasing temperature and deviations from simple spin-wave theory occur at surprisingly low temperatures. Transition temperatures $T_c\left(\Delta \right)$ exceed the isotropic value $T_c^{\mathrm{SK}}$ predicted by series expansions although the values would also be consistent with $T_c^{\mathrm{SK}}=0$ if $T_c\left(\Delta \right)\propto {\left|\ln \right(\frac{1}{\Delta }\left)\right|}^{-1\mathrm{}}$. The asymptotic critical exponent for the order parameter is $\beta =\frac{1}{8}$ for all $\Delta$. The susceptibility data show crossover from $\gamma =1.75\mathrm{}$ near $T_c\left(\Delta \right)$ to higher values farther from $T_c\left(\Delta \right)$. It is shown that finite-size rounding may lead to erroneously large estimates for $\beta$ and erroneously small estimates for $\gamma$. These effects may invalidate some conclusions drawn from experiments on planar systems, too. Accepting the series estimate $\frac{k_B{T}_c^{\mathrm{SK}}}{J}=0.588$ we find that the data are consistent with $T_c\left(\Delta \right)-T_c^{\mathrm{SK}}\propto {\Delta }^{\frac{1}{\phi }}$ with $\mathrm{phi}\approx 4$. Assuming power-law behavior we show that our data obey crossover scaling with two-dimensional Heisenberg exponents ${\gamma }_H\approx 3$, ${\alpha }_H\approx -2\mathrm{}$ 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.

• Received 14 July 1975

DOI:https://doi.org/10.1103/PhysRevB.13.1140