Critical properties of the two-dimensional anisotropic Heisenberg model

Abstract

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 $H=\ensuremath{-}J\ensuremath{\Sigma}[(1\ensuremath{-}\ensuremath{\Delta})({S}_{i}^{x}{S}_{j}^{x}+{S}_{i}^{y}{S}_{j}^{y})+{S}_{i}^{z}{S}_{j}^{z}]$ on $N\ifmmode\times\else\texttimes\fi{}N$ square lattices with periodic boundary conditions is investigated for $N\ensuremath{\le}100$ and $\ensuremath{\Delta}$ 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 ${T}_{c}(\ensuremath{\Delta})$ 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 ${T}_{c}(\ensuremath{\Delta})\ensuremath{\propto}{|\mathrm{ln}(\frac{1}{\ensuremath{\Delta}})|}^{\ensuremath{-}1}$. The asymptotic critical exponent for the order parameter is $\ensuremath{\beta}=\frac{1}{8}$ for all $\ensuremath{\Delta}$. The susceptibility data show crossover from $\ensuremath{\gamma}=1.75$ near ${T}_{c}(\ensuremath{\Delta})$ to higher values farther from ${T}_{c}(\ensuremath{\Delta})$. It is shown that finite-size rounding may lead to erroneously large estimates for $\ensuremath{\beta}$ and erroneously small estimates for $\ensuremath{\gamma}$. 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 ${T}_{c}(\ensuremath{\Delta})\ensuremath{-}{T}_{c}^{\mathrm{SK}}\ensuremath{\propto}{\ensuremath{\Delta}}^{\frac{1}{\ensuremath{\varphi}}}$ with $\mathrm{phi}\ensuremath{\approx}4$. Assuming power-law behavior we show that our data obey crossover scaling with two-dimensional Heisenberg exponents ${\ensuremath{\gamma}}_{H}\ensuremath{\approx}3$, ${\ensuremath{\alpha}}_{H}\ensuremath{\approx}\ensuremath{-}2$ 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.