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.