Statistical Mechanics - Physics at Oregon State University

162 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE.
to
Z(T, h, N) =
σ1,···,σN
e β(J σiσj+h i σi)
(8.18)
The magnetic Gibbs energy G follows from the partition function according
G(T, h, N) = −kBT log (Z(T, h, N)) (8.19)
The expectation value of the Hamiltonian 8.9 for a given quantum state is a
quadratic function in the variables σi. This is the source of all calculational
problems. In mean field theory we approximate this Hamiltonian by a form
that is linear in the spin-variables σi.
The spin-variables σi are pure numbers, and it will be useful to connect them
with operators representing them. We have
and
1
2
Szi|σ1, · · · , σN >= σi|σ1, · · · , σN > (8.20)
H = −J
SizSjz
(8.21)
Note that these are not quite the usual spin operators, we have taken a factor
out. The average value of the spin per site is given by
m(T, h, N) = 1
N
N
i=1
〈Si〉T,h
where we have defined the thermodynamical average by
〈Si〉T,h =
1
−β(H−hM)
Tr Sie
Z(T, h, N)
(8.22)
(8.23)
If it is obvious which type of ensemble we are using, we will drop the labels on
the average. In the remainder of this section it is understood that averages are
at a given value of the temperature T and magnetic field h. We now consider
an infinite solid and ignore surface effects. Also we assume that there is only
one type of atomic site. The spin averages are in that case independent of the
atomic position i and we have
m(T, h, N) = 〈Si〉 (8.24)
We now rewrite the energy of a given quantum state using the average of
the spin variables and deviations from average. We find
H = −J
(Siz −m)(Sjz −m)−J
Sizm−J
mSjz +J
m 2 (8.25)