Kohn-Luttinger effect and the instability of a two-dimensional repulsive Fermi liquid at<i>T</i>=0

Abstract

We consider the possibility for a pairing in a two-dimensional (2D) repulsive Fermi liquid due to the singularity in the scattering amplitude \ensuremath{\Gamma}(q) at the momentum transfer q\ensuremath{\le}2${\mathit{p}}_{\mathit{F}}$ (Kohn-Luttinger effect). A common belief based on perturbative calculations up to second order in the s-wave scattering amplitude is that this effect is absent in two dimensions. I show that this is not the case. For an arbitrary Fermi liquid, \ensuremath{\Gamma}(q) is found to have a singular part, ${\mathrm{\ensuremath{\Gamma}}}^{\mathrm{sing}}$(q)\ensuremath{\sim} \ensuremath{\surd}1-${\mathit{q}}^{2}$/(2${\mathit{p}}_{\mathit{F}}$${)}^{2}$ , for q\ensuremath{\le}2${\mathit{p}}_{\mathit{F}}$. For large 2D orbital momentum l, this term gives a dominant attractive contribution to the scattering amplitude and leads to a pairing instability in a 2D Fermi liquid with arbitrary short-range repulsion. In the dilute limit, numerical studies show that the effect survives down to l=1 and gives rise to a p-wave pairing. The relevance of these results to experiments on $^{3}\mathrm{He}$ adsorbed on the free surface of $^{4}\mathrm{He}$ is discussed.