This simple result correctly predicts for a wide class of crystals found in nature, assuming the same level of anisotropy, that is, . The high- superconductor exhibits a Néel transition at . With , our results give quite a close estimate: . Similar close predictions hold for other systems, such as superconductor and insulator . For the high- material , [Ding:90g]. This material undergoes a Néel transition at . Our prediction of is in the same range of , and much better than the naive expectation that . In this crystal, there is some degree of frustration (see below), so the actual transition is pushed down. These examples clearly indicate that the in-plane anisotropy could be quite important to bring the system to the Néel order for these high- materials. For the S=1 system, , our results predict a , quite close to the observed .
These results have direct consequences regarding the critical exponents. The onset of transition is entirely due to the Ising-like anisotropy. Once the system becomes Néel-ordered, different layers in the three-dimensional crystals will order at the same time. Spin fluctuations, in different layers, are incoherent so that the critical exponents such as , , and will be the two, rather than three-dimensional Ising exponents. and show such behaviors clearly. However, the interlayer coupling, although very small (much smaller than the in-plane anisotropy), could induce coherent correlations between the layers, so that the critical exponents will be somewhere between the two and three-dimensional Ising exponents. and seem to belong to this category.
Whether the ground state of the spin- antiferromagnet spins has the long-range Néel order, is a longstanding problem [Anderson:87a]. The existence of the Néel order is vigorously proved for . In the most interesting case , numerical calculations on small lattices suggested the existence of the long-range order. Our simulation establishes the long-range order for .
The fact that near , the spin system is quite sensitive to the tiny anisotropy could have a number of important consequences. For example, the correlation lengths measured in are systematically smaller than the theoretical prediction [Ding:90g] near . The weaker correlations probably indicate that the frustrations, due to the next to nearest neighbor interaction, come into play. This is consistent with the fact that is below the suggested by our results.