Superconductivity in a two-dimensional repulsive Rashba gas at low electron density

We study the superconducting instability and the resulting superconducting states in a two-dimensional repulsive Fermi gas with Rashba spin-orbit coupling at low electron density (namely the Fermi energy $E_F$ is lower than the energy $E_R$ of the Dirac point induced by Rashba coupling). We find that superconductivity is enhanced as the dimensionless Fermi energy $\epsilon_F$ ($\epsilon_F\equiv E_F/E_R$) decreases, due to the following two reasons. First, the density of states at $\epsilon_F$ increases as $1/\sqrt{\epsilon_F}$. Second, the particle-hole bubble becomes more anisotropic, resulting in an increasing effective attraction. The superconducting state is always in the total angular momentum $j_z=+2$ (or $j_z=-2$) channel with Chern number $C=4$ (or $C=-4$), breaking time reversal symmetry spontaneously. Although a putative Leggett mode is expected due to the two-gap nature of the superconductivity, we find that it is always damped. More importantly, once a sufficiently large Zeeman coupling is applied to the superconducting state, the Chern number can be tuned to be $\pm1$ and Majorana zero modes exist in the vortex cores.