Notes on Mean Field Theory

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