Interatomic spin-orbit coupling in atomic orbital based tight-binding models

Spin-dependent interatomic hopping gives rise to exotic phenomena such as topological insulators and cross-correlated effects. The authors systematically investigate interatomic spin-orbit coupling (SOC) and derive an explicit expression for the SOC hopping integrals. Using a multipole technique, they discuss how spin and orbital degrees of freedom couple in several specific tight-binding models on several distinct lattices. In particular, they reveal that the spin splitting, unique to chiral systems, is induced by the electric toroidal quadrupole.