Carsten Timm: Theory of superconductivity

At lower temperatures, details become resolved that are obscured by thermal broadening at high T . The RPA
and also more advanced approaches are very sensitive to the electronic bands close to the Fermi energy; states
with |ξ k | ≫ k B T have exponentially small effect on the susceptibility. Therefore, the detailed susceptibility at low
T strongly depends on details of the model Hamiltonian. Choosing nearest-neighbor and next-nearest-neighbor
hopping in such a way that a realistic Fermi surface emerges, one obtains a spin susceptiblity with incommensurate
q
0
π a
peaks at π a (1, 1 ± δ) and π a (1 ± δ, 1). y
π a
q x
These peaks are due to nesting: Scattering is enhanced between parallel portions of the Fermi surface, which in
turn enhances the susceptibility [see M. Norman, Phys. Rev. B 75, 184514 (2007)].
k y
π
a
π
a (1− δ , 1)
π
a (1, 1− δ )
π
a
k x
The results for the spin susceptibility are in qualitative agreement with neutron-scattering experiments. However,
the RPA overestimates the tendency toward magnetic order, which is reduced by more advanced approaches.
Spin-fluctuation exchange
The next step is to construct an effective electron-electron interaction mediated by the exchange of spin fluctuations.
The following diagrammatic series represents the simplest way of doing this, though certainly not the only
one:
V eff :=
U
+ + + · · · (12.35)
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