Lay summary

One of the greatest triumphs of Dirac was to predict the value taken by the Land\'e $g$ factor that controls the Zeeman interaction between the spin of a free electron and an applied magnetic field.Another consequence of the Dirac equation is the prediction of a spin-orbit coupling between the spin of an electron and its angular momentum, when it orbits around a central potential.The closer the orbit of the electron is to the origin of the central potential, the stronger this effect.The spin-orbit coupling is, of course, essential to explain atomic spectra, in particular when the atomic number $Z$ is large.Electrons confined to a quasi-two-dimensional geometry as occurs at the interfaces between semiconductors, Mott insulators, or as occurs on the surface of three-dimensional bulk crystals can be exposed to a strong spin-orbit coupling, since there is no inversion symmetry perpendicular to the plane.The low-dimensionality also favors interaction effects and fluctuation-driven phenomena. Compared to the literature dedicated to the instabilities of a two-dimensional metal without spin-orbit coupling, there is relatively little theoretical work done on the instabilities of a \textit{correlated two-dimensional metal with strong spin-orbit coupling}.The goal of this project is to go beyond the single-particle approximation and to explore the competing instabilities of noncentrosymmetric quasi-two-dimensional metals, with an emphasis on the superconducting and magnetic instabilities that are driven by the subtle interplay between \textit{strong} spin-orbit couplings and \textit{strong} fluctuations induced by electron-phonon and electron-electron interactions.