The calculation of critical exponents is nontrivial, even for simple models such as the Ising model. Here, we will introduce an approximate technique known as mean field theory. The approximation that is made in the mean field theory (MFT) is that fluctuations can be neglected. Clearly, this is a severe approximation, the consequences of which we will see in the final results.
Consider the Hamiltonian for the Ising model:
The partition function is given by
Notice that we have written the partition function as an isothermal-isomagnetic partition function in analogy with the isothermal-isobaric ensemble. (Most books use the notation Q for the partition function and A for the free energy, which is misleading). This sum is nontrivial to carry out.
In the MFT approximation, one introduces the magnetization
explicitly into the partition function by using the identity
The last term is quadratic in the spins and is of the form , the average of which measures the spin fluctuations. Thus, this term is neglected in the MFT. If this term is dropped, then the spin-spin interaction term in the Hamiltonian becomes:
We will restrict ourselves to isotropic magnetic systems, for which is independent of i (all sites are equivalent). Define , where z is the number of nearest neighbors of each spin. This number will depend on the number of spatial dimensions. Since this dependence on spatial dimension is a trivial one, we can absorb the z factor into the coupling constant and redefine . Then,
where N is the total number of spins. Finally,
and the Hamiltonian now takes the form
and the partition function becomes
The free energy per spin g(h,T) = G(N,h,T)/N is then given by
The magnetization per spin can be computed from
Allowing , one finds a transcendental equation for m
which can be solved graphically as shown below:
Note that for , there are three solutions. One is at m=0 and the other two are at finite values of m, which we will call . For , there is only one solution at m=0. Thus, for , MFT predicts a nonzero magnetization at h=0. The three solutions coalesce onto a single solution at . The condition thus defines a critical temperature below which ( ) there is a finite magnetization at h=0. The condition defines the critical temperature, which leads to
To see the physical meaning of the various cases, consider expanding the free energy about m=0 at zero-field. The expansion gives
where c is a (possibly temperature dependent) constant with c;SPMgt;0. For , the sign of the quadratic term is negative and the free energy as a function of m looks like:
Thus, there are two stable minima at , corresponding to the two possible states of magnetization. Since a large portion of the spins will be aligned below the critical temperature, the magnetic phase is called an ordered phase. For , the sign of the quadratic term is positive and the free energy plot looks like:
i.e., a single minimum function at m=0, indicating no net magnetization above the critical temperature at h=0.
The exponent can be obtained directly from this expression for the free energy. For , the value of the magnetization is given by
From the equation for the magnetization at nonzero field, the exponent is obtained as follows:
where the second line is obtained by expanding the inverse hyperbolic tangent about m=0. At the critical temperature, this becomes
so that .
For the exponent , we need to compute the heat capacity at zero-field, which is either or . In either case, we have, for , where m=0,
from which is it clear that . For , approaches a different value as , however, the dependence on is the same, so that is still obtained.
Finally, the susceptibility, which is given by
but, near m=0,
As the critical temperature is approached, and
which implies .
The MFT exponents for the Ising model are, therefore
which are exactly the same exponents that the Van der Waals theory predict for the fluid system. The fact that two (or more) dissimilar systems have the same set of critical exponents (at least at the MFT level) is a consequence of a more general phenomenon known as universality, which was alluded to in the previous lecture.
Systems belonging to the same universality class will exhibit the same behavior about their critical points, as manifested by their having the same set of critical exponents.
A universality class is characterized by two parameters:
An order parameter is defined as follows:
The Ising system, for which is given by
is invariant under the group , which is the group that contains only two elements, an identity element and a spin reflection transformation: . Thus, under , the spins transform as
From the form of is can be seen that under both transformations of , so that it is invariant under . However, the magnetization
is not invariant under a spin reflection for , when the system is magnetized. In a completely ordered state, with all spins aligned, under a spin reflection . Thus, m is an order parameter for the Ising model, and, since it is a scalar quantity, its dimension is 1.
Thus, the Ising model defines a universality class known as the Ising universality class, characterized by d=3, n=1 in three dimensions. Note that the fluid system, which has the same MFT critical exponents as the Ising system, belongs to the same universality class. The order parameter for this system, by the analogy table defined in the last lecture, is the volume difference between the gas an liquid phases, , or equivalently, the density difference, . Although the solid phase is the truly ordered phase, while the gas phase is disordered, the liquid phase is somewhere in between, i.e., it is a partially ordered phase. The Hamiltonian of a fluid is invariant under rotations of the coordinate system. Ordered and partially ordered phases break this symmetry. Note also that a true magnetic system, in which the spins can point in any spatial direction, need an order parameter that is the vector generalization of the magnetization:
Since the dimension of the vector magnetization is 3, the true magnetic system belongs to the d=3, n=3 universality class.