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.
Robust gapless surface state and Rashba-splitting bands upon surface deposition of magnetic Cr on Bi$_2$Se$_3$
磁性杂质与无带隙表面态之间的相互作用对于在拓扑绝缘体中实现新的量子现象和新的功能是至关重要的。通过结合角度分辨光电子能谱实验和密度泛函理论计算，我们发现Cr原子在Bi $_2$ Se $_3$表面的沉积不会导致狄拉克点处表面态的能隙打开，表明在我们的测量温度15 K下不存在长程面外铁磁性.这与体Cr掺杂形成鲜明对比，并且起源归因于不同的Cr占据位点。这些结果突出了掺杂磁性杂质的纳米级结构在决定拓扑绝缘体的电子和磁性方面的重要性。
