5Phase transitions
II Statistical Physics
5.4 Landau theory
What is it that made the Ising model behave so similarly to the liquid-gas
model? To understand this, we try to fit these into a “general” theory of phase
transitions, known as Landau theory. We will motivate Landau theory by trying
to capture the essential properties of the mean-field Ising model that lead to the
phase transition behaviour. Consequently, Landau theory will predict the same
“wrong” critical exponents, so perhaps it isn’t that satisfactory. Nevertheless,
doing Landau theory will help us understand better “where” the phase transition
came from.
In the mean field approximation for the Ising model, the free energy was
given by
F (T, B; m) =
1
2
JNqm
2
−
N
B
log (2 cosh(βB + βJqm)) ,
where m = m(T, B) is determined by the equation
m =
1
Nβ
∂ log Z
∂B
T
. (∗)
It is convenient to distinguish between the function
F
we wrote above, and the
value of F when we set m = m(T, B). So we write
˜
F (T, B; m) =
1
2
JNqm
2
−
N
B
log (2 cosh(βB + βJqm))
F (T, B) =
˜
F (T, B, m(T, B)).
The idea of Landau theory is to see what happens when we don’t impose (
∗
), and
take
˜
F
seriously. There are some questions we can ask about
˜
F
. For example,
we can try to minimize
˜
F . We set
∂
˜
F
∂m
!
T,B
= 0.
Then we find
m = tanh(βB + βJqm),
which is exactly the same thing we get by imposing (
∗
)! Thus, another way to
view the mean-field Ising model is that we have a system with free parameters
m, T, B
, and the equilibrium systems are those where
m
minimizes the free
energy.
To do Landau theory, we need a generalization of
m
in an arbitrary system.
The role of
m
in the Ising model is that it is the order parameter — if
m 6
= 0,
then we are ordered, i.e. spins are aligned; if m = 0, then we are disordered.
After we have identified an order parameter
m
, we need to have some function
˜
F
such that the equilibria are exactly the minima of
˜
F
. In the most basic set up,
˜
F
is an analytic function of
T
and
m
, but in general, we can incorporate some
other external parameters, such as
B
. Finally, we assume that we have a
Z/
2
Z
symmetry, namely
F
(
T, m
) =
F
(
T, −m
), and moreover that
m
is sufficiently
small near the critical point that we can analyze
˜
F using its Taylor expansion.
Since
˜
F is an even function in m, we can write the Taylor expansion as
˜
F (T, m) = F
0
(T ) + a(T )m
2
+ b(T )m
4
+ ··· .
Example. In the Ising model with B = 0, we take
˜
F
Ising
(T, m) = −NkT log 2 +
NJq
2
(1 − JqB)m
2
+
Nβ
3
J
4
q
4
24
m
4
+ ··· .
Now of course, the value of
F
0
(
T
) doesn’t matter. We will assume
b
(
T
)
>
0.
Otherwise, we need to look at the
m
6
terms to see what happens, which will be
done in the last example sheet.
There are now two cases to consider. If a(T ) > 0 as well, then it looks like
m
˜
F
However, if
a
(
T
)
<
0, then now
m
= 0 is a local maximum, and there are two
other minimum near m = 0:
m
˜
F
m
0
(T )
We call the minima ±m
0
(T ).
Thus, we expect a rather discrete change in behaviour when we transition
from
a
(
T
)
>
0 to
a
(
T
)
<
0. We let
T
C
be the temperature such that
a
(
T
C
) = 0.
This is the critical temperature. Since we are only interested in the behaviour
near the critical point, We may wlog assume that
a(T )
> 0 if T > T
C
= 0 if T = T
C
< 0 if T < T
C
.
In other words,
a
(
T
) has the same sign as
T
. We will further assume that this is
a simple zero at T = T
C
.
Example. In the Ising model, we have a(T ) = 0 iff kT = Jq.
We can work out what the values of
m
0
(
T
) are, assuming we ignore
O
(
m
6
)
terms. This is simply given by
m
0
(T ) =
r
−a
2b
.
Having determined the minimum, we plug it back into
˜
F
to determine the free
energy F (T ). This gives
F (T ) =
(
F
0
(T ) T > T
C
F
0
(T ) −
a
2
4b
T < T
C
.
Recall that
a
passes through 0 when
T
=
T
C
, so this is in fact a continuous
function.
Now we want to determine the order of this phase transition. Since we
assumed
F
is analytic, we know in particular
F
0
,
a
and
b
are smooth functions
in T . Then we can write
a(T ) = a
0
(T − T
C
), b(T ) = b
0
(T
C
) > 0
near T = T
C
, where a
0
> 0. Then we get
F (T ) =
(
F
0
(T ) T > T
C
F
0
(T ) −
a
2
0
4b
0
(T − T
C
)
2
T − T
C
.
So we see that
S = −
dF
dT
is continuous. So this is not a first-order phase transition, as
S
is continuous.
However, the second-order derivative
C = T
dS
dT
is discontinuous. This is a second-order phase transition. We can calculate
critical exponents as well. We have
m
0
≈
r
a
0
2b
0
(T
C
− T )
1/2
.
So we have
β
=
1
2
, which is exactly what we saw for the mean field approximation
and van der Waals. So Landau theory gives us the same answer, which are still
wrong. However, it does give a qualitative description of a phase transition. In
particular, we see where the non-analyticity of the result comes from.
One final comment is that we started with this
˜
F
which has a reflection
symmetry in
m
. However, below the critical temperature, the ground state does
not have such symmetry. It is either +
m
0
or
−m
0
, and this breaks the symmetry.
This is known as spontaneous symmetry breaking.
Non-symmetric free energy
In our Ising model, we had an external parameter
B
. We saw that when
T < T
C
,
then we had a first-order phase transition when
B
passes through 0. How can
we capture this behaviour in Landau theory?
The key observation is that in the Ising model, when
B 6
= 0, then
˜
F
is not
symmetric under m ↔ −m. Instead, it has the symmetry
˜
F (T, B, m) =
˜
F (T, −B, −m).
Consider a general Landau theory, where
˜
F
has an external parameter
B
with
this same symmetry property. We can expand
˜
F = F
0
(T, B) + F
1
(T, B)m + F
2
(T, B)m
2
+ F
3
(T, B)m
3
+ F
4
(T, B)m
4
+ ··· .
In this case,
F
n
(
T, B
) is odd/even in
B
if
m
is odd/even (resp.). We assume
F
4
>
0, and ignore
O
(
m
5
) terms. As before, for any fixed
B
, we either have a
single minimum, or has a single maxima and two minima.
As before, at high temperature, we assume we have a single minimum. Then
the
˜
F looks something like this:
m
˜
F
At low temperature, we assume we go to a situation where
˜
F
has multiple
extrema. Then for B > 0, we have
m
˜
F
M
U
G
So we have a ground state
G
; a metastable state
M
and an unstable state
U
.
When we go to B = 0, this becomes symmetric, and it now looks like
m
˜
F
m
0
(T )
Now we have two ground states. When we move to
B <
0, the ground state now
shifts to the left:
m
˜
F
M
U
G
Now we can see a first-order phase transition. Our
m
is discontinuous at
B
= 0.
In particular,
lim
B→0
+
m(T, B) = m
0
(T ), lim
B→0
−
m(T, B) = −m
0
(T ).
This is exactly the phenomena we observed previously.
Landau–Ginzburg theory*
The key idea of Landau theory was that we had a single order parameter
m
that
describes the whole system. This ignores fluctuations over large scales, and lead
to the “wrong” answers. We now want to understand fluctuations better.
Definition (Correlation function). We define the correlation function
G
ij
= hs
i
s
j
i − hs
i
ihs
j
i.
Either numerically, or by inspecting the exact solution in low dimensions, we
find that
G
ij
∼ e
−r
ij
/ξ
,
where
r
ij
is the separation between the sites, and
ξ
is some function known as
the correlation length. The key property of this function is that as we approach
the critical point
T
=
T
C
,
B
= 0, we have
ξ → ∞
. So we have correlations over
arbitrarily large distance.
In this case, it doesn’t make much sense to use a single
m
to represent the
average over all lattice sites, as long-range correlations means there is some
significant variation from site to site we want to take into account. But we
still want to do some averaging. The key idea is that we averaging over short
distances. This is sometimes called course graining.
To do this, we pick some length scale that is large relative to the lattice
separation, but small relative to the correlation scale. In other words, we pick a
b such that
ξ b a = lattice spacing.
Let m
i
be the average of all s
j
such that |r
i
− r
j
| < b.
Now the idea is to promote these numbers to a smooth function. Given some
set of values {m
i
}, we let m(r) be a smooth function such that
m(r
i
) = m
i
,
with
m
slowly varying over distances
< b
. Of course, there is no unique way of
doing this, but let’s just pick a sensible one.
We now want to regard this
m
as an order parameter, but this is more general
than what we did before. This is a spatially varying order parameter. We define
a functional
˜
F [T, m] by
e
−B
˜
F [T ,m]
=
X
{s
i
}
e
−βE[{s
i
}]
,
but this time we sum only over the {s
i
} giving {m
i
} such that m
i
= m(r
i
).
In principle, this determines the Landau functional
˜
F
. In practice, we can’t
actually evaluate this.
To get the full partition function, we want to sum over all
m
(
r
). But
m
(
r
)
is a function! We are trying to do a sum over all possible choices of a smooth
function. We formally write this sum as
Z =
Z
Dm e
−β
˜
F [T ,m(r)]
.
This is an example of a functional integral, or a path integral. We shall not
mathematically define what we mean by this, because this is a major unsolved
problem of theoretical physics. In the context we are talking about here, it
isn’t really a huge problem. Indeed, our problem was initially discrete, and we
started with the finitely many points
m
i
. It made sense to sum over all possible
combinations of
{m
i
}
. It’s just that for some reason, we decided to promote
this to a function
m
(
r
) and cause ourselves trouble. However, writing a path
integral like this makes it easier to manipulate, at least formally, and in certain
other scenarios in physics, such path integrals are inevitable.
Ignoring the problem, for small
m
, if we have reflection symmetry, then we
can expand
˜
F =
Z
d
d
r (a(t)m
2
+ b(T )m
4
+ c(T )(∇m)
2
+ ···).
To proceed with this mathematically fearsome path integral, we make an approx-
imation, namely the saddle point approximation. This says the path integral
should be well-approximated by the value of
˜
F at the minimum, so
Z ≈ e
−β
˜
F [T ,m]
,
where m is determined by minimizing
˜
F , i.e.
δ
˜
F
δm
= 0.
To solve this, we have to vary
m
. A standard variational calculus manipulation
gives
δ
˜
F =
Z
d
d
r
2amδm + 4bm
3
δm + 2c∇m · ∇(δm) + ···
=
Z
d
d
r (2am + 4bm
2
− 2c∇
2
m + ···)δm.
So the minimum point of
˜
F is given by the solution to
c∇
2
m = am + 2bm
2
+ ··· . (∗)
Let’s first think about what happens when
m
is constant. Then this equation
just reproduces the equation for Landau theory. This “explains” why Landau
theory works. But in this set up, we can do more than that. We can study
corrections to the saddle point approximation, and these are the fluctuations.
We find that the fluctuations are negligible if we are in
d ≥
4. So Landau theory
is reliable. However, this is not the case in
d <
4. It is an unfortunate fact that
we live in 3 dimensions, not 4.
But we can also go beyond Landau theory in another way. We can consider
the case where
m
is not constant. Really, we have been doing this all along for
the liquid-gas system, because below the critical temperature, we can have a
mixture of liquid and gas in our system. We can try to describe it in this set up.
Non-constant solutions are indeed possible for appropriate boundary condi-
tions. For example, suppose
T < T
C
, and assume
m → m
0
(
T
) as
x → ∞
and
m → −m
0
(
T
) as
x → −∞
. So we have gas on one side and liquid on the other.
We can take
m
=
m
(
x
), and then the equation (
∗
) becomes the second-order
ODE
d
2
m
dx
2
=
am
c
+
2bm
3
c
,
and we can check that the solution is given by
m = m
0
(T ) tanh
r
−a
2c
x
!
,
and
m
0
=
r
−a
2b
.
This configuration is known as the domain wall that interpolates between the
two ground states. This describes an interface between the liquid and gas phases.
x
m
m
0
−m
0
To learn more about this, take Part III Statistical Field Theory.