Carsten Timm: Theory of superconductivity

N s is the total number of particles in the condensate. They then chose the complex amplitude ψ of Ψ s (r) as the
order parameter, with the normalization
|ψ| 2 ∝ n s . (6.4)
Thus the order parameter in this case is a complex number. They thereby neglect the spatial variation of Ψ s (r)
on an atomic scale.
The free energy must not depend on the global phase of Ψ s (r) because the global phase of quantum states is
not observable. Thus we obtain the expansion
F = α |ψ| 2 + β 2 |ψ|4 + O(|ψ| 6 ). (6.5)
Only the absolute value |ψ| appears since F is real. Odd powers are excluded since they are not differentiable at
ψ = 0. If β > 0, which is not guaranteed by symmetry but is the case for superconductors and superfluids, we
can neglect higher order terms since β > 0 then makes sure that F (ψ) is bounded from below. Now there are two
cases:
• If α ≥ 0, F (ψ) has a single minimum at ψ = 0. Thus the equilibrium state has n s = 0. This is clearly a
normal metal (n n = n) or a normal fluid.
• If α < 0, F (ψ) has a ring of minima with equal amplitude (modulus) |ψ| but arbitrary phase. We easily see
√
∂F
∂ |ψ| = 2α |ψ| + 2β |ψ|3 = 0 ⇒ |ψ| = 0 (this is a maximum) or |ψ| = − α (6.6)
β
Note that the radicand is positive.
F
α > 0
α = 0
α < 0
0
Re ψ
Imagine this figure rotated around the vertical axis to find F as a function of the complex ψ. F (ψ) for α < 0
is often called the “Mexican-hat potential.” In Landau theory, the phase transition clearly occurs when α = 0.
Since then T = T c by definition, it is useful to expand α and β to leading order in T around T c . Hence,
Then the order parameter below T c satisfies
√
|ψ| =
ψ
α/
β
α ∼ = α ′ (T − T c ), α ′ > 0, (6.7)
β ∼ = const. (6.8)
− α′ (T − T c )
β
=
√
α ′ √
T − Tc . (6.9)
β
T c
T
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