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Motivated by the proposed topological state in Cu$_x$Bi$_2$Se$_3$, we studythe possibility of phonon-mediated odd-parity superconductivity in spin-orbitcoupled systems with time-reversal and inversion symmetry. For such systems, weshow that, in general, pure electron-phonon coupling can never lead to atriplet state with a higher critical temperature than the leading singletstate. The Coulomb pseudopotential, which is the repulsive part of theelectron-electron interaction and is typically small in weakly correlatedsystems, is therefore critical to stabilizing the triplet state. We introduce achirality quantum number, which identifies the electron-phonon vertexinteractions that are most favorable to the triplet channel as those thatconserve chirality. Applying these results to Cu$_x$Bi$_2$Se$_3$, we find thata phonon-mediated odd-parity state may be realized in the presence of weakelectronic correlations if the chirality-preserving electron-phonon verticesare much stronger than the chirality-flipping vertices.

Odd-parity superconductivity from phonon-mediated pairing: Application to Cu$_x$Bi$_2$Se$_3$