0
= 0: (6.50)
If c <0 this minimum will be at 1 which is unphysical. If indeedc <0one needs to take a term 6 into account and see what happens. In what follows we will always assume c > 0. In the absence of an external …eld should hold that f( ) =f( ), implyingh=b= 0. Whether or not there is a minimum Since is expected to vanish atT =Tc we conclude thata(T)changes sign at Tc suggesting the simple ansatz
a(T) =a0(T Tc) (6.52)
with a0 > 0 being at most weakly temperature dependent. This leads to a temperature dependence of the order parameter
qa0(Tc T)
It will turn out that a powerlaw relation like
(Tc T) (6.54)
is valid in a much more general context. The main change is the value of . The prediction of the Landau theory is =12.
Next we want to study the e¤ect of an external …eld (= magnetic …eld in case characterizes the magnetization of an Ising ferromagnet). This is done by keeping the termh in the expansion forf. The actual external …eld will be proportional toh. Then we …nd thatf is minimized by
a 0+c 30=h (6.55)
Right at the transition temperature wherea= 0this gives
3
0 h1= (6.56)
where the Landau theory predicts = 3. Finally we can analyze the change of the order parameter with respect to an external …eld. We introduce the susceptibility
= @ 0
@h h!0 (6.57)
and …nd from Eq.6.55
a + 3c 20(h= 0) = 1 (6.58) with exponent = 1.
Next we consider the speci…c heat where we insert our solution for 0 into the free energy density.
f = a(T) This yields for the speci…c heat per volume
c= T@2f
6.5. LANDAU THEORY OF PHASE TRANSITIONS 69 The speci…c heat is discontinuous. As we will see later, the general form of the speci…c heat close to a second order phase transition is
c(T) (T Tc) +const (6.62)
where the result of the Landau theory is
= 0: (6.63)
So far we have considered solely spatially constant solutions of the order parameter. It is certainly possible to consider the more general case of spatially varying order parameters, where the free energy
F = Z
ddrf[ (r)] (6.64)
is given as
f[ (r)] = a
2 (r)2+c
4 (r)4 h(r) (r) + b
2(r (r))2 (6.65) where we assumed that it costs energy to induce an inhomogeneity of the order parameter (b >0). The minimum ofFis now determined by the Euler-Lagrange equation
@f
@ r @f
@r = 0 (6.66)
which leads to the nonlinear partial di¤erential equation
a (r) +c (r)3=h(r) +br2 (r) (6.67) Above the transition temperature we neglect again the non-linear term and have to solve
a (r) br2 (r) =h(r) (6.68) It is useful to consider the generalized susceptibility
(r) = Z
ddr0 (r r0) h(r0) (6.69) which determines how much a local change in the order parameter is a¤ected by a local change of an external …eld at a distancer r0. This is often written as
(r r0) = (r)
h(r0): (6.70)
We determine (r r0)by Fourier transforming the above di¤erential equation with
(r) = Z
ddkeikr (k) (6.71)
which gives
a (k) +bk2 (k) =h(k) (6.72)
In addition it holds for (k):
(k) = (k) h(k): (6.73)
This leads to
(k) = b 1
2+k2 (6.74)
where we introduced the length scale
= rb
a = r b
a0(T Tc) 1=2 (6.75)
This result can now be back-transformed yielding (r r0) =
r r0
d 1 2
exp jr r0j (6.76)
Thus, spins are not correlated anymore beyond the correlation length . In general the behavior of close toTc can be written as
(T Tc) (6.77)
with =12.
A similar analysis can be performed in the ordered state. Starting again at a (r) +c (r)3=h(r) +br2 (r) (6.78) and assuming (r) = 0+ (r)where 0is the homogeneous, h= 0, solution, it follows for small (r):
a+ 3c 20 (r) =h(r) +br2 (r) (6.79) and it holdsa+ 3c 20= 2a >0. Thus in momentum space
(k) =d (k)
dh(k) = b 1
2
< +k2 (6.80)
with
= r b
2a = r b
2a0
(Tc T) 1=2 (6.81)
We can now estimate the role of ‡uctuations beyond the linearized form used.
This can be done by estimating the size of the ‡uctuations of the order parameter compared to its mean value 0. First we note that
(r r0) =h( (r) 0) ( (r0) 0)i (6.82)
6.5. LANDAU THEORY OF PHASE TRANSITIONS 71 Thus the ‡uctuations of (r)in the volume d is
2 = 1
The last integral can be evaluated by substitutingza =k leading to Z
This must be compared with the mean value of the order parameter
2
Thus, ford > 4 ‡uctuations become small as ! 1, whereas they cannot be neglected for d <4. Ind= 4, a more careful analysis shows that h 2i
2
0 /log . Role of an additional term 6:
f = 1
2a'2+c 4'4+w
6'6 (6.90)
The minimum is at
@E
Ifc >0 we can exclude the two solutions' which are purely imaginary. If a >0, then the'+ are imaginary as well and the only solution is'= 0. Close to the transition temperature at a= 0holds for r! 0 : '=
q a
c , i.e. the behavior is not a¤ected byw.
If c < 0 then the solutions ' might become relevant if c2 > 4aw. For a = 4wc2 new solitions at' =
q c
2w occur for the …rst time. fora < a these solutions are given by
'=
24w2 of this solution equals the one for ' = 0. The order parameter at this point is 'c =
q 3c
4w. Finally, for a = 0the solution at'= 0which was metastable fora < a < ac disappears whereas it vanishes fora >0.
= 1=
d
d = 1= 1= 1 (6.95)
Statistical mechanics motivation of the Landau theory:
We start from the Ising model
H[Si] = X
ij
JijSiSj (6.96)
in an spatially varying magnetic …eld. The partition function isZ=P
fSige H[Si]. In order to map this problem onto a continuum theory we use the identity
Z YN which can be shown to be correct by rotating the variablesxi into a represen-tation whereV is diagonal and using
Z
dxexp x2
4v +sx = 2p
4 veV s2 (6.98)
6.5. LANDAU THEORY OF PHASE TRANSITIONS 73 This identity can now be used to transform the Ising model (use Vij = Jij) according to The last term is just the partition function of free spins in an external …eld xi
and it holds Transforming i=p1
2 It is useful to go into a momentum representation
i= (Ri) =
with
u(k1; k2; k3) =
4
3 Jk1Jk2Jk3J k1 k2 k3 (6.108) UsingJij =J for nearest neighbors and zero otherwise gives for a cubic lattice
Jk= 2J X
=x;y;:::
cos (k a)'2J d a2k2 =J0 1 a2
dk2 (6.109) Here a is the lattice constant and we expanded Jk for small momenta (wave length large compared toa)
Jk This is precisely the Landau form of an Ising model, which becomes obvious if one returns to real space
He [ ] = 1
From these considerations we also observe that the partition function is given as
Z = Z
D exp ( He [ ]) (6.114)
and it is, in general, not the minimum of He [ ] w.r.t. which is physically realized, instead one has to integrate over all values of to obtain the free energy. Within Landau theory we approximate the integral by the dominating contribution of the integral, i.e. we write
Z
D exp ( He [ ])'exp ( He [ 0]) (6.115) where H
= 0= 0.
6.5. LANDAU THEORY OF PHASE TRANSITIONS 75 Ginzburg criterion
One can now estimate the range of applicability of the Landau theory. This is best done by considering the next order corrections and analyze when they are small. If this is the case, one can be con…dent that the theory is controlled.
Before we go into this we need to be able to perform some simple calculations with these multidimensional integrals.
First we consider for simplicity a case where He [ ] has only quadratic contributions. It holds
It follows for the free energy
F = kBT 2
Z
ddklog (k) (6.118)
One can also add to the Hamiltonian an external …eld He [ ]!He [ ]
Z
ddkh(k) (k) (6.119)
Then it is easy to determine the correlation function
(k) = k k h ki k (6.120)
This can again be done explicitly for the case with u= 0:
Z[h] =
Performing the second derivative oflogZ gives indeed k k =a+bk1 2. Thus, we obtain as expected
(k) = k h k
. (6.123)
Let us analyze the speci…c heat related to the free energy F = kBT
2 Z
ddklog (k) (6.124)
It holds for the singular part of the speci…c heat c @2F Thus, as ! 1follows that there is no singular (divergent) contribution to the speci…c heat ifd >4just as we found in the Landau theory. However, ford <4 the speci…c heat diverges and we obtain a behavior di¤erent from what Landau theory predicted.
Another way to see this is to study the role of inhomogeneous ‡uctuations as caused by the
Hinh=d 2 Z
ddr(r )2 (6.126)
Consider a typical variation on the scale r
q a u
1 and integrate those over a volume of size d gives
Hinh b d 2a u
b2 u
d 4 (6.127)
Those ‡uctuations should be small compared to temperature in order to keep mean …eld theory valid. If their energy is large compared tokBT they will be rare and mean …eld theory is valid. Thus we obtain again that mean …eld theory breaks down for d < 4. This is called the Ginzburg criterion. Explicitly this criterion is
Note, if b is large for some reason, ‡uctuation physics will enter only very close to the transition. This is indeed the case for many so called conventional superconductors.