Bragg-Williams approximation for the Ising model

Again, we consider a system with the following Hamiltonian:

In this case our random variables are the spins: , and our order parameter is (we will later see that this is exactly the local magnetization also in the mean field approximation). For what we have previously stated we must build a single-particle probability density in terms of this order parameter, such that and . For example, since we have two constraints on we could use a linear expression for the probability density involving two parameters[1]:
where and are, respectively, the probability that and . Inserting this expression of in the two constraints we get:
and therefore:
We are now able to compute the two terms that contribute to the free energy of the system.

First of all:

However, since (namely the degrees of freedom are independent, as we have already seen), then . Furthermore, if is a generic function of then:
and so if we have : our order parameter is (as we expected) the local magnetization of the system. Therefore:

Then, for the other term of we have:

The total energy of the system will be:

We now have to minimize with respect to :
where means that the -th site is a nearest neighbour of the -th one. Now, recalling that:
(and surely ) we can write:
and inverting the hyperbolic tangent:
We have again found the self-consistency equation for the magnetization that we have already encountered in the Weiss mean field theory for the Ising model! This is again a confirmation that all mean field theories are equivalent.

We can thus see that in the Bragg-Williams approximation also the and exponents are the same of the Weiss mean field theory; in fact, we have seen in Critical exponents of Weiss mean field theory for the Ising model that they come from the expansion of the free energy density for small values of the magnetization when . If we now set into , so that , we get:

If we now expand the logarithm for small , we get:
Therefore we see that the behaviour of near is the same of equation Weiss mean field theory for the Ising model , and so both the exponents and turn out to be equal to what we have determined[2] in Critical exponents of Weiss mean field theory for the Ising model.

We can also say something[3] in the case . Supposing that our system is uniform, i.e. , we can rewrite as:

and since we are looking for the absolute minimum of :
which gives the self-consistency equation we already know:
As we can see from the figures below the number of the possible solutions depends on the temperature of the system:

Solutions of the self-consistency equation for
Solutions of the self-consistency equations for

In particular if (i.e. ), expanding the right hand side of the self-consistency equation we get a positive linear term (), so that the behaviour of is as shown in the first figure, and the equation has only one solution. On the other hand if then the linear term changes sign and behaves as in the second figure: in this case if is small enough there are three possible solutions, which we have called , and . These are all extrema of , but how can we understand which is a minimum or a maximum? And above all, which of them is the global minimum. If we suppose to be large there will be only one solution, , and as decreases also and will appear; we can therefore argue that for the continuity of the solution is still a minimum also when and are present. Similarly, if we take we can conclude that also is a minimum; therefore will necessarily be a maximum. Now, in order to see which between and is the global minimum of let us take and compute:

From the second figure we see that this is equal to the area enclosed by the the straight line and the graph of , which is clearly negative if . Therefore and so we conclude that is always the global minimum of . Similarly, when the global minimum of is .

This means that as soon as changes sign the global minimum of changes abruptly from to . We are thus obtaining the phenomenology that is indeed observed for a magnet when we change the external field . In other words the sets of points are exactly the graphs of the phase diagram we have seen in Phase transitions and phase diagrams, i.e. the graphs of the magnetization seen as a function of the external field.

  1. This form of is very general, and does not depend on the fact that the degrees of freedom of the system can only assume two values: if there is a different number of possible states, say , then can be written in the same form, but will be the probability of one state while the probability of the remaining ones. We will shortly see this when we will apply the Bragg-Williams approximation to the Potts model.
  2. Note that also the temperature of the transition is still the same, considering also the factor we have already mentioned.
  3. These considerations apply in general also in the other mean field theories considered, but we show them now.