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 .

Wed Mar 1 10:19:35 EST 1995