Superconductivity near a nematic quantum critical point: Interplay between hot and lukewarm regions

We present a strong-coupling dynamical theory of the superconducting transition in a metal near a quantum-critical point toward Q=0 nematic order. We use a fermion-boson model, in which we treat the ratio of effective boson-fermion coupling and the Fermi energy as a small parameter λ. We solve, both analytically and numerically, the linearized Eliashberg equation. Our solution takes into account both strong fluctuations at small momentum transfers ∼λkF and weaker fluctuations at large momentum transfers. The strong fluctuations determine Tc, which is of order λ2EF for both s- and d-wave pairing. The weaker fluctuations determine the angular structure of the superconducting order parameter F(θk) along the Fermi surface, separating between hot and lukewarm regions. In the hot regions F(θk) is largest and approximately constant. Beyond the hot region, whose width is θh∼λ1/3, F(θk) drops by a factor λ4/3. The s- and d-wave states are not degenerate but the relative difference (Tcs-Tcd)/Tcs∼λ2 is small.