Momentum dependence of the spin susceptibility in two dimensions: Nonanalytic corrections in the Cooper channel

We consider the effect of rescattering of pairs of quasiparticles in the Cooper channel resulting in the strong renormalization of second-order corrections to the spin susceptibility in a two-dimensional electron system. We use the Fourier expansion of the scattering potential in the vicinity of the Fermi surface to find that each harmonic becomes renormalized independently. Since some of those harmonics are negative, the first derivative of the spin susceptibility is bound to be negative at small momenta, in contrast to the lowest order perturbation theory result, which predicts a positive slope. We present in detail an effective method to calculate diagrammatically corrections to the spin susceptibility to infinite order.

Figure 1

The building block (on the left) of any ladder diagram (on the right). Of special interest is the limit of correlated momenta $\mathbf{p}=-\mathbf{k}$, leading to the Cooper instability.

Figure 2

The nonvanishing second-order diagrams contributing to the nonanalytic behavior of the electron spin susceptibility.

Figure 3

Labeling of the $\delta {\chi }_1^{\left(2\right)}$ diagram, as in Eq. (18).

Figure 4

The series of diagrams contributing to $\delta {\chi }_1\left(Q\right)$.

Figure 5

(Color online) An example of diagram contributing to $\delta {\chi }_2\left(Q\right)$. The maximally crossed diagram (left) is topologically equivalent to its untwisted counterpart (right) in which the particle-particle ladder appears explicitly.

Figure 6

A maximally crossed diagram (left) and its untwisted equivalent (right) contributing to $\delta {\chi }_4\left(Q\right)$.

Figure 7

The series of diagrams contributing to $\delta {\chi }_3\left(Q\right)$. At the top, the second- and third-order order diagrams. Two equivalent third-order diagrams arise from the addition of a parallel interaction line to either the upper or the lower part of the second-order diagram. At the bottom, three fourth-order diagrams.

Figure 8

First-order diagrams contributing to the spin susceptibility. These are renormalized by the leading logarithmic terms of the higher order diagrams (see Figs. 4, 5, 6). However, they do not produce a nonanalytic correction and can be neglected in the limit of small $Q$.