Mean field theory for the Blume-Emery-Griffiths model

The Blume-Emery-Griffiths model (often shortened in "BEG" model) is a lattice gas model used to describe the properties of the superfluid transition of He, which when it is cooled under approximately undergoes a continuous phase transition[1] from fluid to superfluid; superfluids exhibit several interesting properties, like having zero viscosity. The BEG model is used to describe what happens when we add some He to the system; it does not consider quantum effects, but only the "messing up" due to the He impurities. Experimentally when He is added to He the temperature of the fluid-superfluid transition decreases. For small concentration of He the mixture remains homogeneous, and the only effect is the change of ; however, when the concentration of He reaches the critical value , He and He separate into two phases (just like oil separates from water) and the transition becomes first-order (namely, discontinuous). The transition point where the system shifts from a continuous transition to a discontinuous one is that where the phase separation starts and is called tricritical point. The BEG model was introduced to describe such a situation.

As we have anticipated, it is a lattice gas model and so it is based on an Ising-like Hamiltonian (see Ising model and fluids). On the sites of this lattice we define a variable which can assume the values , and : we decide that when an He atom is present in a lattice site then , while when it means that the site is occupied by an He atom. We then define our order parameter to be ; in the Ising model can only be equal to 1, while in this case it can be either 0 or 1: we can thus interpret as the concentration of He atoms, and as the fraction of He. We also define to be the difference of the chemical potentials of He and He; since this parameter is related to the number of He and He atoms, we expect that when (namely, there is only He) , while if then .

We consider the following Hamiltonian for the system[2]:

where is the total number of lattice sites. Since we want to apply the second variational method that we have seen, we write the mean field probability density as:
and the free energy:
The mean value of the Hamiltonian is:
and since (it's the fundamental hypothesis of mean field theories) we get[3]:
where is the coordination number of the lattice. Therefore, the free energy of the system is:
We now must minimize this expression with respect to , with the constraint . Since we have:


which leads to:

where we have reabsorbed into the normalization constant . From the constraint we find:
Substituting this expression of into , after some mathematical rearrangement we get:
In order to find the equilibrium state for any and , we must minimize this expression with respect to . If we expand for small , keeping in mind the Taylor expansions and we get:
and ; turns out to be always positive. Note that unlike the Ising model in the Weiss approximation (see Weiss mean field theory for the Ising model) in this case both the quadratic and the quartic terms, and , can change sign when the parameters assume particular values. Let us also note that the order parameter of the system, namely the concentration of He, is:

Therefore, in the disordered phase (both He and He are present) we have and the concentration of He becomes:

This way we can determine how the temperature of the transition depends on ; in fact, the critical temperature will be the one that makes change sign, so we can determine it from the condition :

Since as we have just seen , we have:
where . The other transition (from the continuous to the discontinuous one) will occur when the quartic term changes sign, and so we can determine the critical value of at which it occurs from the condition :
which is in astonishingly good agreement with the experimental result of .

  1. This is generally called transition, because the plot of the specific heat as a function of the temperature has a shape that resembles a .
  2. We do not justify why it has this form; furthermore, the Hamiltonian we are considering is in reality a simplification of the original one.
  3. We are introducing the factor for later convenience. As we have already stated, this is perfectly possible if we change the convention of the sum on nearest neighbours.