Unconventional superconductivity from electronic dipole fluctuations

We study electron-electron Coulomb interactions in electronic systems whose Fermi surfaces possess a finite electric dipole density. Although there is no net dipole moment, we show that electric monopole-dipole interactions can become sufficiently strong in quasi-2D Dirac metals with spin-orbit coupling to induce unconventional odd-parity superconductivity, similar to the Balian-Werthamer state of ${}^{3}$He-B. Hence materials with spin-orbit-induced band inversion, such as the doped topological insulators Bi$_2$Se$_3$, Bi$_2$Te$_3$, and SnTe, are natural candidate materials where our theory could be relevant. We discuss the conditions for an electric dipole density to appear on the Fermi surface and develop the formalism to describe its coupling to the plasmon field which mediates the Coulomb interaction. A mechanism for the enhancement of dipolar coupling is then provided for quasi-2D Dirac systems. Within a large-$N$ renormalization group treatment, we show that the out-of-plane ($z$-axis) dipole coupling is marginally relevant, in contrast to the monopole coupling which is marginally irrelevant. For physically realistic parameters, we find that dipole fluctuations can get sufficiently enhanced to result in Cooper pairing. In addition, we establish that the proposed pairing glue is directly measurable in the $z$-axis optical conductivity.