The Landau-Ginzberg (LG) theory is a phenomenological theory meant to
be used only near the critical point. Thus, it is formulated
as a macroscopic theory. The basic idea of LG theory is to introduce
a spin density field variable 
defined by
Then the total magnetization is given by
Since the free energy at zero field is A=A(N,M,T), there should
be a corresponding free energy density 
such that
It is assumed that 
can be represented as a power series
according to
such that 
and 
are both positive for 
and vanishes at 
. These conditions are
analogous to those exhibited by the total free energy
in MFT above and near the critical point (see previous lecture).
By symmetry, all odd terms vanish and are, therefore, not explicitly
included.
In the presence of a magnetic field 
, we have a Gibbs
free energy density given by
In addition, a term 
with 
is added to
the free energy in order to damp out local fluctuations. If there
are significant local fluctuations, then 
becomes
large, so these field configurations contribute negligibly to the
partition function, which is defined by
Thus, the LG theory is a field theory. The form of this field theory is well known in quantum field theory, and is known as a d-dimensional scalar Klein-Gordon theory. In terms of Z, a correlation function at zero field can be defined by
Thus, by studying the behavior of the correlation function,
the exponents 
and 
can be determined.
For the choice 
, the theory can be solved exactly analytically.
This is known as the Gaussian model, for which
which leads to values of and of 0 and 1/2, respectively.
These values are known as the ``classical'' exponents.
They are independent of the number of spatial dimensions d.
Dependence on d comes in at higher orders in S.
Comparing these to the exact exponents from the Onsager solution,
which gives 
and 
, it can be seen that
the classical exponents are only qualitatively correct.
Going to higher orders in the theory leads to improved results, although the
theory cannot be solved exactly analytically. It can either be solved
numerically using path integral Monte Carlo or Molecular Dynamics or
analytically perturbatively using Feynman diagrams.