Robustness of the Kohn-Luttinger mechanism against symmetry breaking

We investigate how strongly broken spatial symmetries affect the Kohn--Luttinger (KL) mechanism, in which superconductivity emerges purely from repulsive interactions. While the original KL argument assumes continuous rotational symmetry, real materials possess only discrete point-group symmetries, raising a central question: can sufficiently strong symmetry breaking suppress or eliminate KL superconductivity? Using controlled perturbation theory and explicit two-dimensional models with Ising and Rashba spin--orbit coupling (SOC), we find that KL superconductivity is broadly robust and exhibits qualitatively universal behavior across models: the transition temperature $T_c$ is nonmonotonic in the symmetry-breaking field, shows a pronounced maximum at scales of the order of the Fermi energy, and decays exponentially toward zero at asymptotically large fields. However, the physical mechanisms determining this suppression may differ between models. Overall, these results demonstrate that KL-type superconductivity can persist across a wide class of spin--orbit-coupled systems.