# 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]}:

^{[3]}:

then:

which leads to:

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 :

- ↑ This is generally called
*transition*, because the plot of the specific heat as a function of the temperature has a shape that resembles a . - ↑ We do not justify why it has this form; furthermore, the Hamiltonian we are considering is in reality a simplification of the original one.
- ↑ 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.