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We show that Rashba spin-orbit interaction (RSOI) modifies electron-electroninteraction vertex giving rise to a spectrum of novel phenomena. First, thespin-orbit-modified Coulomb interactions induce $p$-wave superconducting order,without any need for other mediators of attraction. Remarkably, two distinctsuperconducting phases arise in 3D systems, mirroring the $\mathrm{A}$ or$\mathrm{B}$ phases of $^3\mathrm{He}$, depending on the sign of the SOIconstant. In contrast, 2D systems exhibit $p^x\pm i p^y$ order parameter,leading to time-reversal-invariant topological superconductivity. Second, asufficiently strong RSOI induces ferromagnetic ordering. It is associated witha deformation of the Fermi surface, which may lead to a Lifshitz transitionfrom a spherical to a toroidal Fermi surface, with a number of experimentallyobservable signatures. Finally, in sufficiently clean Rashba materials,ferromagnetism and $p$-wave superconductivity may coexist. This state resemblesthe $\mathrm{A}_1$ phase of $^3\mathrm{He}$, yet it may avoid nodal points dueto the toroidal shape of the Fermi surface.

Electron Interactions in Rashba Materials