Carsten Timm: Theory of superconductivity

But now the right-hand side is always negative. Consequently, there is no non-trivial solution and thus no
superconductivity for a k-independent repulsion.
Now let us look at a strong interaction between nearest-neighbor sites. We consider a two-dimensional square
lattice for simplicity and since it is thought to be a good model for the cuprates. In momentum space, a nearestneighbor
interaction is written as
V kk ′ = 2V 1 [cos (k x − k ′ x)a + cos (k y − k ′ y)a], (12.6)
where V 1 > 0 (V 1 < 0) for a repulsive (attractive) interaction. For our discussion of momentum-dependent
interactions it is crucial to realized which terms in the k ′ sum in the gap equation
∆ k = − 1 N
∑
k ′ V kk ′
1 − n F (E k ′)
2E k ′
∆ k ′ (12.7)
are most important. The factor [1−n F (E k ′)]/2E k ′ is largest on the normal-state Fermi surface, where E k = |∆ k |,
and is exponentially suppressed on an energy scale of k B T away from it. Thus only the vicinity of the Fermi
surface is important.
Let us first consider the repulsive case V 1 > 0. Then V kk ′ is most strongly repulsive (positive) for momentum
transfer k − k ′ → 0. But ∆ k should be a smooth function of k, thus for k and k ′ close together, ∆ k and ∆ k ′ are
also similar. In particular, ∆ k will rarely change its sign between k and k ′ . Consequently, the right-hand side
of the gap equation always contains a large contribution with sign opposite to that of ∆ k , coming from the sum
over k ′ close to k. Hence, a repulsive nearest-neighbor interaction is unlikely to lead to superconductivity.
For the attractive case, V 1 < 0, V kk ′ is most strongly attractive for k − k ′ → 0 and most strongly repulsive
for k − k ′ → (π/a, π/a) and equivalent points in the Brillouin zone. The attraction at small q = k − k ′ is always
favorable for superconductivity. However, we also have an equally strong repulsion around q ≈ (π/a, π/a). A
critical situation thus arises if both k and k ′ lie close to the Fermi surface and their difference is close to (π/a, π/a).
The central insight is that this can still help superconductivity if the gaps ∆ k and ∆ k ′ at k and k ′ , respectively,
have opposite sign. In this case the contribution to the right-hand side of the gap equation from such k ′ has the
same sign as ∆ k since V kk ′ > 0 and there is an explicit minus sign.
This effect is crucial in the cuprates, which do have an effective attractive nearest-neighbor interaction and
have a large normal-state Fermi surface shown here for a two-dimensional model:
k y
Q
0
k x
The vector Q in the sketch is Q = (π/a, π/a). Following the previous discussion, ∆ k close to the Fermi surface
should have different sign between points separated by Q. On the other hand, the small-q attraction favors gaps
∆ k that change sign “as little as possible.” By inspection, these conditions are met by a gap changing sign on the
diagonals:
121