Effect of electron-electron interactions on Rashba-like and spin-split systems

The role of electron-electron interactions is analyzed for Rashba-like and spin-split systems within a tight-binding single-band Hubbard model with on-site and all nearest-neighbor matrix elements of the Coulomb interaction. By Rashba-like systems we refer to the Dresselhaus and Rashba spin-orbit-coupled phases while spin-split systems have spin-up and spin-down Fermi surfaces shifted relative to each other. Both systems break parity but preserve time-reversal symmetry. They belong to a class of symmetry-breaking ground states that satisfy: (i) electron crystal momentum is a good quantum number, (ii) these states have no net magnetic moment, and (iii) their distribution of polarized spin in momentum space breaks the lattice symmetry. For all members of this class, the relevant Coulomb matrix elements are found to be nearest-neighbor exchange J, pair hopping J', and nearest-neighbor repulsion V. These ground states lower their energy most effectively through J, hence we name them class J states. The competing effects of V-J' on the direct and exchange energies determine the relative stability of class J states. We show that the spin-split and Rashba-like phases are the most favored ground states within class J because they have the minimum anisotropy in polarized spin. We analyze these two states on a square lattice and find that the spin-split phase is always favored for near-empty bands; above a critical filling, we predict a transition from the paramagnetic to the Rashba-like phase at a critical J(J(c1)) and a second transition from the Rashba-like to the spin-split state at J(c2)> J(c1). An energetic comparison with ferromagnetism highlights the importance of the role of V in the stability of class J states. We discuss the relevance of our results to (i) the alpha and beta phases proposed by Wu and Zhang in the Fermi-liquid formalism and (ii) experimental observations of spin-orbit splitting in Au (111) surface states.