It may seem a little surprising that a very small anisotropy can lead to a substantially high . This may be explained by the following argument. At low T, the spins are highly correlated in the isotropic case. Since no direction is preferred, the correlated spins fluctuate in all directions, resulting in zero net magnetization. Adding a very small anisotropy into the system introduces a preferred direction, so that the already highly correlated spins will fluctuate around this direction, leading to a global magnetization.
More quantitatively, the crossover from the isotropic Heisenberg behavior to the Ising behavior occurs at , where the correlation length is of order of some power of the inverse anisotropy. From the scaling arguments [Riedel:69a], where is the crossover exponent. In the two-dimensional model, both and are infinite, but the ratio is approximately 1/2. For , this relation indicates that the Ising behavior is valid for , which is clearly observed in Figure 6.16. Similar crossover around for is also observed in Figure 6.16. At low T, for the isotropic quantum model, the correlation length behaves as [Ding:90g] where . Therefore, we expect
where is spin-S dependent constant of order one. Therefore, even a very small anisotropy will induce a phase transition at a substantially high temperature (). This crude picture, suggested a long time ago to explain the observed phase transitions, is now confirmed by the extensive quantum Monte Carlo simulations for the first time. Note that this problem is an extreme case both because it is an antiferromagnet (more difficult to become ordered than the ferromagnet), and because it has the largest quantum fluctuations (spin-). Since varies slowly with h, we can estimate at :