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(Jc1) and a second transition from the Rashba-like to the spin-split state at Jc2>Jc1. 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 α and β phases proposed by Wu and Zhang in the Fermi-liquid formalism and (ii) experimental observations of spin-orbit splitting in Au(111) surface states.