13.4 Phase Transitions and Critical Phenomena 433
interaction between two spins at sitesiandj, respectively, by adding and subtracting the
mean value of the spin〈S〉, that is
SiSj= (Si−〈S〉+〈S〉)(Si−〈S〉+〈S〉)≈〈S〉^2 +〈S〉(Si−〈S〉)+〈S〉(Sj−〈S〉),where we have ignored terms of the order(Si−〈S〉)(Si−〈S〉). These are the terms which lead
to correlations between neighbouring spins. In mean field theory we ignore correlations.
This means that we can rewrite the Hamiltonian
E=−JN
∑
<i j>SkSl−BN
∑
iSi,as
E=−J∑
〈S〉^2 +〈S〉(Si−〈S〉)+〈S〉(Sj−〈S〉)−B∑
iSi,resulting in
E=−(B+zJ〈S〉)∑
i
Si+zJ〈S〉^2 ,withzthe number of nearest neighbours for a given sitei. We have included the external
magnetic fieldBfor completeness.
We can then define an effective field which all spins see, namely
Beff= (B+zJ〈S〉).To obtain the vaue of〈S〉)we employ again our results from the canonical ensemble. The
partition function reads in this case
Z=e−NzJ〈S〉(^2) /kT
( 2 cosh(Beff/kT))N,
with a free energy
F=−kT lnZ=−NkT ln( 2 )+NzJ〈S〉^2 −NkT ln(cosh(Beff/kT))
and minimizingFwith respect to〈S〉we arrive at
〈S〉=tanh( 2 cosh(Beff/kT)).
Close to the phase transition we expect〈S〉to become small and eventually vanish. We can
then expandFin powers of〈S〉as
F=−NkT ln( 2 )+NzJ〈s〉^2 −NkT−BN〈s〉+NkT
(
1
2
〈s〉^2 +1
12
〈s〉^4 +...)
,
and using〈M〉=N〈S〉we can rewrite Helmholtz free energy as
F=F 0 −B〈M〉+1
2 a〈M〉(^2) +^1
4 b〈M〉
(^4) +...
Let〈M〉=mand
F=F 0 +
1
2
am^2 +1
4
bm^4 +1
6
cm^6Fhas a minimum at equilibriumF′(m) = 0 andF′′(m)> 0
F′(m) = 0 =m(a+bm^2 +cm^4 ),