Unconventional Superconductivity Mediated by Nematic Fluctuations in a Multi-Orbital System -- Application to doped FeSe

We analyze superconductivity in a multi-orbital fermionic system near the onset of a nematic order, using doped FeSe as an example. We associate nematicity with a spontaneous polarization between $d_{\text{xz}}$ and $d_{\text{yz}}$ orbitals (a Pomeranchuk-type order) and analyze the pairing mediated by soft nematic fluctuations. Such a pairing gives rise to a highly anisotropic gap function whose structure strongly varies with temperature, and leads to strongly non-BCS behavior in thermodynamics, spectroscopy and transport. We compute the specific heat and its directional variation with a magnetic field, magnetic susceptibility, density of states, tunneling conductance, Raman intensity, superfluid stiffness and penetration depth without and with impurity scattering and for the latter computed also optical conductivity and $T_c$ variation. We find good agreement with the existing data for FeSe$_{1-x}$S$_x$ and FeSe$_{1-x}$Te$_x$ and suggest new experiments.