Statistical Mechanics - Physics at Oregon State University

8.2. BASIC MEAN FIELD THEORY. 161
is a monoatomic periodic lattice. We are also able to study surfaces in such a
limit by assuming that the atomic positions only fill half of space. Even in the
limit N → ∞ this still leaves us with a surface. Due to the reduced symmetry
atoms at the surface do not have the same environment as atoms far inside, and
calculations are in general harder to perform.
8.2 Basic Mean Field theory.
The easiest way to obtain an approximate solution for the thermodynamics
of almost all models is called mean field theory. In this section we discuss the
basic mean field theory in the context of the Ising model, and from this it will be
clear how to generalize this approximation for arbitrary physical models. In later
sections we extend the idea of mean field theory to obtain better approximations.
First, we need to cast the energy function 8.9) in the form of an operator.
Assume that the quantum state in which atom i has spin σi is given by
|σ1, · · · , σN >. The Hamiltonian operator representing the energy of the system
and acting on the space of all possible quantum states |σ1, · · · , σN > is
H =
σ1,···,σN
|σ1, · · · , σN > E {σ1, · · · , σN } < σ1, · · · , σN | (8.13)
The operator for the magnetic moment is
M =
σ1,···,σN
|σ1, · · · , σN > M {σ1, · · · , σN } < σ1, · · · , σN| (8.14)
At this point we have to make a choice whether to sum over quantum states
with a given value of the total moment or to introduce a magnetic field and
sum over all states. The latter is much easier. The change is similar to what we
did when we replaced the canonical (fixed N) ensemble by the grand canonical
(fixed µ) ensemble. We will evaluate probabilities for a given temperature and
magnetic field. The thermodynamic energy function we want to get is the
magnetic Gibbs free energy
G(T, h, N) = U − T S − hM (8.15)
where M is the thermodynamic average of the magnetization density M.
The partition function to calculate is
Z(T, h, N) = Tr e −β(H−hM)
The trace is best evaluated in the basis of eigenstates, and we have
Z(T, h, N) =
σ1,···,σN
(8.16)
< σ1, · · · , σN|e −β(H−hM) |σ1, · · · , σN > (8.17)