Odd-Parity Superconductivity near an Inversion Breaking Quantum Critical Point in One Dimension

We study how an inversion-breaking quantum critical point affects the ground state of a one-dimensional electronic liquid with repulsive interaction and spin-orbit coupling. We find that regardless of the interaction strength, the critical fluctuations always lead to a gap in the electronic spin sector. The origin of the gap is a two-particle backscattering process, which becomes relevant due to renormalization of the Luttinger parameter near the critical point. The resulting spin-gapped state is topological and can be considered as a one-dimensional version of a spin-triplet superconductor. Interestingly, in the case of a ferromagnetic critical point, the Luttinger parameter is renormalized in the opposite manner, such that the system remains nonsuperconducting.