Interatomic spin-orbit interaction in a $p$-orbital helical atomic chain

We derive the interatomic spin-orbit interaction (SOI) from a helical atomic chain composed of $p$-orbitals with intra-atomic SOI, which exhibits a helical state--a potential origin of the chiral-induced spin selectivity (CISS) effect. In this model, a strong crystal field in the tangential direction of the helix leads to the formation of energetically separated $σ$- and $π$-bands. In the second-order process, a spin in the $σ$-orbital virtually hops to the $π$-orbital, flips its direction due to intra-atomic SOI, and then hops back to the $σ$-orbital in the neighboring atom due to the misalignment of $p$-orbitals along the helix. This process induces an interatomic SOI in the $σ$-band, which takes the form of a Rashba-type SOI generated by an electric field normal to the helical axis. The magnitude of the SOI is proportional to the curvature, the hopping energy, the intra-atomic SOI energy, and inversely proportional to the crystal field strength. The second-order process also induces long-range second-nearest-neighbor hoppings. We analytically derive the spin-split band structure in the zero-torsion limit.