Odd-parity superconductivity in the vicinity of inversion symmetry breaking in spin-orbit-coupled systems

We study superconductivity in spin-orbit-coupled systems in the vicinity of inversion symmetry breaking. We find that due to the presence of spin-orbit coupling, fluctuations of the incipient parity-breaking order generate an attractive pairing interaction in an odd-parity pairing channel, which competes with the $s$-wave pairing. We show that Coulomb repulsion or an external Zeeman field suppresses the $s$-wave pairing and promotes the odd-parity superconducting state. Our work provides a new mechanism for odd-parity pairing and opens a route to novel topological superconductivity.