appears as b in the axis label.)
At infinite temperature, 
,
every state is equally probable, and the entropy/spin reaches its maximum
value of log(2).
For ferromagnetic, or positive, spin-spin coupling constant Q
> 0 the behavior of the entropy of any order and any number of spins is
simple. For both decreasing temperature (
)
and increasing external field R the entropy decreases. This is seen
in the plots of the first and second order entropy of the two spin system
in figures 8.2
and 8.3
which show the typical ferromagnetic behavior. When the external field
is zero, the entropy approaches a nonzero value at zero temperature, determined
by the degeneracy of the lowest energy level. The entropy 
in zero field for 2,3, and 4 spins is graphed in figure 8.4.
The temperature at which the entropy ``transitions'' from its infinite
temperature value to its zero temperature value is given approximately
by the mean-field theory value of the temperature, which is different for
each system depending upon the number of spins in the system. Note that
the ``transition'' temperature is increasing as the number of spins increases.
This is in accord with the mean field theory result in the thermodynamic
limit of an infinite number of spins that 
,
where d is the coordination number of each spin and the critical
is 
[30].
Thus we expect for a system of N spins, where because the spins
are fully coupled to each other d = N-1, that 
.
All of the plots are with Q=1, so that this is 
here, decreasing for increasing number of spins in the system. The decreasing
behavior is seen. That the transition is not sharp, nor properly given
by the mean field theory value, may be attributed to the fact that a finite
number of spins has been considered.
Figure 8.2: The ferromagnetic side of the first order entropy 
for the 2 spin case showing relatively little structure.
Figure 8.3: The ferromagnetic side of the second order entropy
for the 2 spin case showing relatively little structure.
Figure 8.4: Entropy of the density matrix
for the 2 (narrowest dashes), 3 and 4 spin (widest dashes) cases. Note
the increasing temperature at which the larger spin system transitions
on the ferromagnetic side of the plots (positive )
in approximate agreement with the mean field theory result that the transition
temperature is proportional to the coordination number.
Antiferromagnetic, or negative spin-spin coupling Q, also effectively
occurs for positive Q and 
.
The graphs of
in the 2,3, and 4 spin cases in figures 8.5-8.13
show more complex structure in this case than in the ferromagnetic case,
with several entropy maxima occurring as the external field varies. These
maxima correspond to states having higher energy in zero external field
becoming energetically favorable in nonzero fields. Consider three probabilities
of energy eigenstates, each having a different coupling to the external
field given by 
,
and let 
for now. For 
they are ordered as 
.
For R=1/2 there is a crossing, 
,
and then for 1/2< R < 2/3 we have 
.
For R=2/3 there is another crossing, and 
.
For 2/3<R<1 we have 
.
Finally, for R=1 we find 
and for R>1 we have 
.
In this manner there can be phase transitions where the dominant behavior
switches between different eigenstates. At zero temperature, such transitions
are sharp. At nonzero temperature, the entropy increases at such transitions
because there are more states then contributing on the average. This is
seen clearly in a cross section of the four-spin system, where there are
two entropy local maxima as the external field changes, figure 8.14.
Figures 8.15
and 8.16
show the probabilities of states and the transition behavior just described
for the three spin system. Figure 8.15
shows the energy eigenstate probabilities, while figure 8.16
shows the probabilites of measuring 
values.
Figure 8.14 also shows the emergence of the structure in the entropy. Plotted here are the entropies of orders one through four. Note the increasing amount of structure as the order increases. Note also that the complete structure does not make itself apparent until the fourth order, though the third order entropy does show a hint that there are two transition regions in the phase space. The entropies have been normalized by the number of spins involved in the summation for the entropy, and they are ordered consistent with the entropy per degree of freedom relationship theorem of chapter 2.
Figure 8.5: First order entropy
for the 2 spin case. Detail in this and the next eight plots is of the
antiferromagnetic phase.
Figure 8.6: Second order entropy
for the 2 spin case.
Figure 8.7: First order entropy
for the 3 spin case.
Figure 8.8: Second order entropy
for the 3 spin case.
Figure 8.9: Third order entropy
for the 3 spin case.
Figure 8.10: First order entropy
for the 4 spin case.
Figure 8.11: Second order entropy
for the 4 spin case.
Figure 8.12: Third order entropy
for the 4 spin case.
Figure 8.13: Fourth order entropy
for the 4 spin case.
Figure 8.14: Entropies per spin for the 4 spin case of orders
one thru four. Note the two entropy maxima, occurring at transitions between
dominant energy eigenstates. The top graph is the first order entropy,
the bottom is the fourth order entropy. Note that the full transition structure
does not make itself apparent until the higher orders have been explored.
Note also the ordering of the entropies per degree of freedom here, consistent
with the entropy reduction per degree of freedom theorem.
Figure 8.15: Probabilities of the two spin energy eigenstates.
Note the transition between the eigenstates occurring at R=1.
Figure 8.16: Probabilities of the two spin z-comonent
states. Note the transition between the eigenstates occurring at R=1.
Two of the eigenstates are degenerate.
Figure 8.17: Cross section of the eighth order entropy of eight
Heisenberg spins. Note that there are eight regions where the slope of
the region has a sign different from the neighboring regions.
Consider the sequence of graphs of figures 8.10-8.13 showing the reduced entropies and the full entropy for the four spin system. The reduced entropy graphs show much less structure. It is thus important to consider the full entropy for such systems in order to understand their behavior. For instance, the local entropy maxima seen in the order four plot indicate that there are two distinct states that might be set with an external field. The information carrying capacity of the system at these fields is greater than it is at others. These indications of complex state behavior open the the possibility that significant applications of these systems will arise. Systems like these might serve as a way to prepare information for another system or an interface to another system, perhaps a system performing quantum computations directly based on the states of magnetic clusters like these. In section 8.12 we will see that the entropy changes are associated with changes in the magnetic moment of the cluster.
Finally, the amount of structure in the spin system of order k is reflected in the fact that there are k regions where the slope has a sign different from the neighboring regions, as seen in the graph of the eighth order entropy for eight spins, figure 8.17, i.e. there are k/2 bumps.