Notes on Mean Field Theory

Mean Field Theory
In a ferromagnet, interaction between neighboring spins due to a combination
of electrostatic interactions and the exclusion principle can bring
about a phase transition between a state with no magnetization in the absence
of a field and a state which is spontaneously magnetized. In the unmagnetized
state, the paramagnetic state, the average spin is zero and the
existence of elementary magnetic moments associated with the spin is shown
by the fact that the spins can be aligned and the system magnetized by an
external field. In the spontaneously magnetized state, the ferromagnetic
state, no external field is needed to magnetize the material.
A model for the energy of a ferromagnet that gives good qualitative
results and shows the existence of a phase transition is the following:
E = −mB ∑ σ n − J ∑ nm
σ n σ m , (1)
where B is the external magnetic field, m is the magnetic moment of the
electron, and J represents the interaction between spins. The interaction
is of very short range, implying that the second sum should be taken over
nearest neighbor pairs of spins, each distinct pair appearing once. The spin
σ appearing here is actually the z-component of the spin of the electron
normalized so that it takes on values ±1. The model based on Eq.(1) is
called the Ising model, and is a simplified version of the somewhat more
realistic Heisenberg model in which all three of the components of the spin
appear.
Despite its simplicity, the Ising model is very difficult to solve exactly.
There is, however, an approximation, the mean field theory, which gives
a good qualitative description of ferromagnetism. The idea of the mean
field theory is to focus on a single spin σ n , and replace the spins σ m of
its neighbors by their averages. We consider only the case in which all the
spins have identical surroundings, so that all spins will have the same average
〈σ〉 = s. In this way, we obtain the mean field energy for a single spin
E mf = −mBσ − Jνsσ, (2)
where ν is the number of neighboring spins that interact with σ. From this
energy we find the mean field partition function
( )
mB + Jνs
Z mf = e (mB+Jνs)/τ + e −(mB+Jνs)/τ = 2 cosh
. (3)
τ
1

Recalling that the first term is the relative probability of σ = +1 and the
second is the relative probability of σ = −1, we easily find the average spin
to be
( )
mB + Jνs
s = tanh
. (4)
τ
This equality is a self-consistency condition for the average spin s.
Figure 1:
Solutions of the mean field equation for (i) τ > Jν, and (ii)τ < Jν.
Consider this condition in the absence of an external field. The equation
s = tanh Jνs/τ can be solved graphically by plotting the left hand side and
the right hand side on the same graph as functions of s. Where the plots
intersect, there is a solution. There are two possibilities: (i) if Jν/τ < 1,
there is there is a single solution s = 0, and (ii) if Jν/τ > 1, there are three
solutions, s = 0, ±s 1 (τ). See Figure (). Since the two states with non-zero s
have the same free energy, they are the stable states because the state with
s = 0 has a higher free energy. The mean field free energy can be calculated
from the standard equation F = −τ log Z, giving
F mf = −Nτ log(2 cosh Jν/τ) (5)
if there a total of N spins. Thus the theory predicts that at high temperature
τ > Jν, the average spin, and hence the total magnetization M = Nms, is
2

zero, while at low temperature τ < Jν there is a magnetization of magnitude
Nms 1 (τ), which can be in either direction. The transition takes place at a
critical temperature τ c = Jν. This is the paramagnetic-ferromagnetic phase
transition.
Magnetization near τ c . We can find the magnetization as a function
of the temperature near the critical temperature from the self consistency
equation, which can be written s = tanh(τ c s/τ), by expanding the hyperbolic
tangent to third order in s. The resulting equation is
τ 3 c
s = τ c
τ s − 1 3 τ 3 s3 . (6)
Solving this for s, we obtain the magnetization
√
M = Nms ≈ Nm 3 τ c − τ
, (7)
τ c
correct to first order in the temperature difference τ c − τ. Thus the magnetization
increases very rapidly below the critical temperature, having a
behavior something like the one shown in Figure ().
Figure 2:
Magnetization as a function of temperature
Susceptibility above τ c . Just above the critical temperature, there is
no spontaneous magnetization, but we can see the effect of an external field
3

y expanding the full equation s = tanh(mB + Jνs)/τ assuming that both
s and B are small. The result is
M = Nms = Nm2
τ − τ c
B (8)
The coefficient of B is a quantity called the magnetic susceptibility. Its
divergence at the critical temperature is a sign that the system is on the
verge of being ordered: a very small magnetic field will produce a large
magnetization.
Energy and heat capacity. The energy U of the system is best obtained
directly from Eq. (1), since the standard method of obtaining it from
Z mf would overcount the interaction terms. If the number of spins is N,
the number of nearest neighbor pairs is Nν/2. Therefore, Eq. (1) gives
The heat capacity is
U = −NmBs − 1 2 NνJs 2 . (9)
C = ∂U
∂τ
= −NmB
ds
dτ −1 2 NνJ ds2
dτ . (10)
Since the derivatives of s and s 2 with respect to τ are both negative, as
indicated in Figure (), the heat capacity is positive. When the external field
B is absent, C is zero above the critical temperature where s = 0. Eq. (10)
shows that C is discontinuous at τ c , since ds/dτ is negative and finite at
this point (see Eq. (7). The discontinuity of C at the critical temperature is
characteristic of this type of phase transition and shows that the two phases
have different physical properties.
4