Phase transition
A lot happens at T
c
m
Tc
T
Both m(T) and (T) are non-analytic at Tc
This is actually a sign of a phase transition
Quantum treatment
So far, we treated the ferromagnet in mean
eld theory
This is approximate. Usually qualitatively
corr
Magnetic order
In a crystal with a periodic lattice of spins,
exchange interactions typically induce an
ordered state at low temperature
For example, the ferromagnetic Heisenberg
model:
H=
J
X
hiji
Si Sj
J >0
Wants every spin parallel to its neighbor,
Meissner effect
To estimate the critical eld, we need to
compare the Gibbs free energy
G=
Z
ns e 2
d3 r
|A|2 + a(ns
2m
|B|2
neq )2 +
s
20
HB
In the SC state, ns=nseq, A=B=0
Gsc = 0
In the normal state, nh=0, B= H
s
Gn = V
Equality G
sc=Gn
eq 2
a (ns )