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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; 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. In 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'. On a square lattice we 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 $ J_{c1}$ and a second transition 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 $α$ and $β$ phases proposed by Wu and Zhang in the Fermi Liquid formalism and (ii) experimental observations of spin-orbit splitting in \emph{Au}(111) surface states.