3Functional derivatives and integrals
III Theoretical Physics of Soft Condensed Matter
3.2 Functional integrals
Given a coarsegrained ψ, we have can define the total partition function
e
−βF
TOT
= Z
TOT
=
Z
e
−βF [ψ]
D[ψ],
where D[
ψ
] is the “sum over all field configurations”. In mean field theory, we
approximate this
F
TOT
by replacing the functional integral by the value of the
integrand at its maximum, i.e. taking the minimum value of
F
[
ψ
]. What we
are going to do now is to evaluate the functional integral “honestly”, and this
amounts to taking into account fluctuations around the minimum (since those
far away from the minimum should contribute very little).
To make sense of the integral, we use the fact that the space of all
ψ
has a
countable orthonormal basis. We assume we work in [0
, L
]
q
of volume
V
=
L
q
with periodic boundary conditions. We can define the Fourier modes
ψ
q
=
1
√
V
Z
ψ(r)e
−iq·r
dr,
Since we have periodic boundary conditions,
q
can only take on a set of discrete
values. Moreover, molecular physics or the nature of coarsegraining usually
implies there is some “maximum momentum”
q
max
, above which the wavelengths
are too short to make physical sense (e.g. vibrations in a lattice of atoms cannot
have wavelengths shorter than the lattice spacing). Thus, we assume
ψ
q
= 0 for
q > q
max
. This leaves us with finitely many degrees of freedom.
The normalization of ψ
q
is chosen so that Parseval’s theorem holds:
Z
ψ
2
dr =
X
q
ψ
q

2
.
We can then define
D[ψ] =
Y
q
dψ
q
.
Since we imposed a
q
max
, this is a finite product of measures, and is welldefined.
In some sense,
q
max
is arbitrary, but for most cases, it doesn’t really matter
what
q
max
we choose. Roughly speaking, at really short wavelengths, the
behaviour of
ψ
no longer depends on what actually is going on in the system,
so these modes only give a constant shift to
F
, independent of interesting,
macroscopic properties of the system. Thus, we will mostly leave the cutoff
implicit, but it’s existence is important to keep our sums convergent.
It is often the case that after doing calculations, we end up with some
expression that sums over the
q
’s. In such cases, it is convenient to take the
limit V → ∞ so that the sum becomes an integral, which is easier to evaluate.
An infinite product is still bad, but usually molecular physics or the nature
of coarse graining imposes a maximum
q
max
, and we take the product up to
there. In most of our calculations, we need such a
q
max
to make sense of our
integrals, and that will be left implicit. Most of the time, the results will be
independent of
q
max
(for example, it may give rise to a constant shift to
F
that
is independent of all the variables of interest).
Before we start computing, note that a significant notational annoyance is
that if
ψ
is a real variable, then
ψ
q
will still be complex in general, but they will
not be independent. Instead, we always have
ψ
q
= ψ
∗
−q
.
Thus, we should only multiply over half of the possible
q
’s, and we usually
denote this by something like
Q
+
q
.
In practice, there is only one path integral we are able to compute, namely
when βF is a quadratic form, i.e.
βF =
1
2
Z
φ(r)G(r − r
0
)φ(r
0
) dr dr
0
−
Z
h(r)φ(r) dr.
Note that this expression is nonlocal, but has no gradient terms. We can think
of the gradient terms we’ve had as localizations of firstorder approximations to
the nonlocal interactions. Taking the Fourier transform, we get
βF [ψ
q
] =
1
2
X
q
G(q)φ
q
φ
−q
−
X
q
h
q
φ
q
.
Example. We take Landau–Ginzburg theory and consider terms of the form
βF [φ] =
Z
n
ξφ
2
− hφ +
κ
2
(∇φ)
2
+
γ
2
(∇
2
φ)
2
o
dr
The
γ
term is new, and is necessary because we will be interested in the case
where κ is negative.
We can now take the Fourier transform to get
βF {φ
q
} =
1
2
X
q
+
(a + κq
2
+ γq
4
)φ
q
φ
−q
−
X
q
+
h
q
φ
q
.
=
X
q
+
(a + κq
2
+ γq
4
)φ
q
φ
−q
−
X
q
+
h
q
φ
q
.
So our G(q) is given by
G(q) = a + kq
2
+ γq
4
.
To actually perform the functional integral, first note that if
h 6
= 0, then we
can complete the square so that the
h
term goes away. So we may assume
h
= 0.
We then have
Z
TOT
=
Z
"
Y
q
+
dφ
q
#
e
−βF {φ
q
}
=
Y
q
+
Z
dφ
q
e
−φ
q

2
G(q)
Each individual integral can be evaluated as
Z
dφ
q
e
−φ
q

2
G(q)
=
Z
ρ dρ dθ e
−G(q)ρ
2
=
π
G(q)
,
where φ
q
= ρe
iθ
. So we find that
Z
TOT
=
Y
q
+
π
G(q)
,
and so
βF
T
= −log Z
T
=
X
q
+
log
G(q)
π
.
We now take large V limit, and replace the sum of the integral. Then we get
βF
T
=
1
2
V
(2π)
d
Z
q
max
dq log
G(q)
π
.
There are many quantities we can compute from the free energy.
Example. The structure factor is defined to be
S(k) = hφ
k
φ
−k
i =
1
Z
T
Z
φ
k
φ
−k
e
−
P
+
q
φ
q
φ
−q
G(q)
Y
q
+
dφ
q
.
We see that this is equal to
1
Z
T
∂Z
T
∂G(k)
= −
∂ log Z
T
∂G(k)
=
1
G(k)
.
We could also have done this explicitly using the product expansion.
This
S
(
k
) is measured in scattering experiments. In our previous example,
for small k and κ > 0, we have
S(q) =
1
a + κk
2
+ γk
4
≈
a
−1
1 + k
2
ξ
2
, ξ =
r
κ
a
.
where ξ is the correlation length. We can return to real space by
hφ
2
(r)i =
1
V
Z
φ(r)
2
dr
=
1
V
X
q
hφ
q

2
i
=
1
(2π)
d
Z
q
max
dq
a + κq
2
+ γq
4
.