One-dimensional scattering of two-dimensional fermions near quantum criticality

Forward scattering and backscattering play an exceptional role in the physics of two-dimensional interacting fermions. In a Fermi liquid, both give rise to a nonanalytic ω2 ln (ω) form of the fermionic scattering rate at second order in the interaction. Here we argue that higher powers of ln (ω) appear in the backscattering contribution at higher orders. We show that these terms come from "planar" processes, which are effectively one-dimensional. This is explicitly demonstrated by extending a Fermi liquid to the limit of N ≫ 1 fermionic flavors, when only planar processes survive. We sum the leading logarithms for the case of a two-dimensional Fermi liquid near a nematic transition, and we obtain an expression for the scattering rate at T = 0 to all orders in the interaction. For a repulsive interaction, the resulting rate is logarithmically suppressed, and the result is valid down to ω = 0. For an attractive interaction, the ground state is an s -wave superconductor with a gap Δ0. We show that in this case the scattering rate increases as ω is reduced toward Δ0. At ω ≥ Δ0, the behavior of the scattering rate is rather unconventional as many pairing channels compete near a nematic critical point, and the s -wave wins only by a narrow margin. We take superconductivity into consideration and obtain the scattering rate also at smaller ω ≃ Δ0.