Advanced Solid State Physics

14.5 Landau Theory of Phase Transitions

Landau took a look at phase transitions and realized that each of these transitions is associated with
a break in symmetry (except for infinite-order phase transitions, which will not be approached here).
By cooling water to ice for example, there is a break in the translation symmetry, the transition
of a material to superconductivity leads to a break in the gauge symmetry and so on. Therefore,
Landau introduced an order parameter for each transition, which can be defined to be zero above the
phase transition and nonzero below the transition. For ferromagnetism, this order parameter is the
magnetization, for ferroelectricity the polarization, etc...

Phase transitions can be divided in first order and second order transitions. First order transitions
exhibit a discontinuity in the first derivate of the free energy, e.g. a latent heat. The phase transition
of water→ice is a typical example for a first order transition.
Second order phase transitions are continuous in the first derivate of the free energy but have a
discontinuity in the second derivate of the free energy, thus they have no latent heat and the order
parameter increases continuously. The transitions to ferromagnetism and ferroelectricity are second
order transitions for example.

14.5.1 Second Order Phase Transitions

We will now take a closer look at the second order phase transitions, because they can be treated
more easily with the Landau theory than first order transitions. Since the order parameter, in our
specific case the magnetization, is zero aboveTC and nonzero belowTC, it has to be small in the
closer neighbourhood ofTC. Because of that, we can expand the free energyf in terms of the order
parameterm.

f=f 0 +αm^2 +

1

2

βm^4 +... (258)

There are only even terms in this equation, because the odd ones cancel out for the magnetization
due to reasons concerning the symmetry. By minimizing the free energy

df

dm

= 2αm+ 2βm^3 +...

we get two solutions form:

m=±

√

−α

β

at T < TC and

m= 0 at T > TC

αmust be temperature dependent too, because it’s negative aboveTC and positive belowTC, so
we have another quantity that is small in the closer neighbourhood ofTC. Because of that, we
rewriteα=α 0 (T−TC)and are left with the following expressions for the free energy and the order
parameter:

f=f 0 +α 0 (T−TC)m^2 +

1

2

βm^4 +... ,m=±

√

−α 0 (TC−T)

β

at T < TC