For the large anisotropy system, h=1, the specific heat are shown for several spin systems in Figure 6.15(a). The peak becomes sharper and higher as the system size increases, indicating a divergent peak in an infinite system, similar to the two-dimensional Ising model. Defining the transition temperature at the peak of for the finite system, the finite-size scaling theory [Landau:76a] predicts that relates to through the scaling law
Setting , the Ising exponent, a good fit with , is shown in Figure 6.15(b). A different scaling with the same exponent for the correlation length,
is also satisfied quite well, resulting in . The staggered magnetization drops down near , although the behaviors are rounded off on these finite-size systems. All the evidence clearly indicates that an Ising-like antiferromagnetic transition occurs at , with a divergent specific heat. In the smaller anisotropy case, , similar behaviors are found. The scaling for the correlation length is shown in Figure 6.16, indicating a transition at . However, the specific heat remains finite at all temperatures.
Figure 6.15: (a) The Specific Heat for Different Size Systems of h=1. (b) Finite Size Scaling for .
Figure 6.16: The Inverse Correlation Lengths for System (), System (), and h=0 System () for the Purpose of Comparison. The straight lines are the scaling relation: . From it we can pin down .
The most interesting case is (or , very close to those in [Birgeneau:71a]). Figure 6.17 shows the staggered correlation function at compared with those on the isotropic model [Ding:90g]. The inverse correlation length measured, together with those for the isotropic model (h=0), are shown in Figure 6.16. Below , the Ising behavior of a straight line becomes clear. Clearly, the system becomes antiferromagnetically ordered around . The best estimate is
Figure 6.17: The Correlation Function on the System at for system. It decays with correlation length . Also shown is the isotropic case h=0, which has .