  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 $\alpha$ and $\beta$ 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.
