Statistical Mechanics - Physics at Oregon State University

8.2. BASIC MEAN FIELD THEORY. 163
This can be rewritten by using the number of nearest neighbors q for each site
and the total number N of sites.
H = −J
(Siz − m)(Sjz − m) − Jm 1
q Siz − Jm
2 1
q
2
i
j
2 1
Sjz + Jm
2 Nq
(8.26)
where the factors 1
2 occur because we are counting nearest neighbor bonds. For
each site we include only half of the neighbors, in agreement with the requirement
i < j. The two terms in the middle are now combined, and we end with:
H = −J
(Siz − m)(Sjz − m) − Jmq
2 1
Siz + Jm Nq (8.27)
2
The internal energy is the thermodynamical average of the Hamiltonian.
Hence we find
U = 〈H〉 = −J
〈(Siz − m)(Sjz − m)〉 − Jmq〈
2 1
Siz〉 + Jm Nq (8.28)
2
U = −J
2 1
〈(Siz − m)(Sjz − m)〉 − Jm Nq (8.29)
2
The second term on the right hand side is the energy of a system where each
spin has its average value. The first term is a correction to this simple expression
related to fluctuations in the spin variables. In mean field theory we
ignore these fluctuations. We assume that the fluctuations on different sites are
independent, they are uncorrelated, and write
〈(Siz − m)(Sjz − m)〉 = 〈(Siz − m)〉〈(Sjz − m)〉 = 0 (8.30)
We approximate the expression for the Hamiltonian operator by ignoring all
terms containing products of differences of the form Si − m. Hence the meanfield
Hamiltonian is
i
H mf = −Jmq
2 1
Siz + Jm Nq (8.31)
2
i
The name mean field (or average field) is derived from the physical interpretation
of this Hamiltonian. The energy of a spin at a given site i is determined
only by the average of the spins on the neighboring sites. Another way of obtaining
the mean-field is the following. We would like to write the Hamiltonian
8.21 in a linearized form:
H lin = −heff
i
Si + H0
i
(8.32)