Superconductivity from polar fluctuations in multi-orbital systems

Motivated by the superconductivity near paraelectric (PE) to ferroelectric (FE) quantum critical point (QCP) in polar metals, we study polar fluctuation mediated superconductivity in multi-orbital systems. The PE to FE QCP is approached by softening of a transverse optical (TO) phonon that is odd under inversion ($\mathcal{I}$). We show that the necessary and sufficient condition for the linear coupling between electron-TO phonon is the presence of multiple orbitals on the Fermi-surface, irrespective of the spin-orbital (SO) coupling, multiple electronic bands, or the vicinity to band crossings. We show that the linear coupling to the polar fluctuations (such as TO modes) can generally lead to superconductivity. We also show that irrespective of the strong $\vec{k}$ dependence of the effective electron-electron interaction in the BCS channel, quite generally the even-parity channel leads to the highest critical temperature. In the presence of additional repulsive electron-electron interactions, an odd-parity spin-triplet channel can become the leading BCS instability. Finally, we discuss our results in the context of the superconductivity in $\text{SrTiO}_3$ and $\text{KTaO}_3$ that highlights the importance of the underlying multi-orbital physics if the superconductivity is mediated by the polar fluctuations.