Carsten Timm: Theory of superconductivity

9
Cooper instability and BCS ground state
In this chapter we will first show that the attractive effective interaction leads to an instability of the normal
state, i.e., of the Fermi sea. Then we will dicuss the new state that takes its place.
9.1 Cooper instability
Let us consider the scattering of two electrons due to the effective interaction.
represented by the diagram
k’
, i ’
ω n
k’ + q, iω’ n + iν n
A single scattering event is
k,iω n
k − q, iω n − iν n
Electrons can also scatter multiple times:
· · · + + + + · · · (9.1)
An instability occurs if this series diverges since then the scattering becomes infinitely strong. In this case the
perturbative expansion in the interaction strength V 0 represented by the diagrams breaks down. This means that
the true equilibrium state cannot be obtained from the equilibrium state for V 0 = 0, namely the noninteracting
Fermi gas, by perturbation theory. A state that is perturbatively connected to the free Fermi gas is called a
Landau Fermi liquid. It is an appropriate description for normal metals. Conversely, a scattering instability
signals that the equilibrium state is no longer a Fermi liquid.
Like in the RPA, it turns out to be sufficient to consider the dominant diagrams at each order. These are
the ladder diagrams, which do not contain crossing interaction lines. Moreover, the instability occurs first for
the scattering of two electrons with opposite momentum, frequency, and spin. We thus restrict ourselves to the
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