Topologically-driven three-spin chiral exchange interactions treated from first principles

Abstract

The mechanism behind the three-spin chiral interaction (TCI) included in the extended Heisenberg Hamiltonian and represented by an expression worked out recently (Phys. Rev. B, 101, 174401 (2020)) is discussed. It is stressed that this approach provides a unique set of the multispin exchange parameters which are independent of each other either due to their different order of perturbation or due to different symmetry. This ensures in particular the specific properties of the TCI described before. An interpretation of the TCI is suggested, showing explicitly its dependence on the relativistic spin-orbit coupling and on the topological orbital susceptibility (TOS). This is based on an expression for the TOS that is worked out on the same footing as the expression for the TCI. Using first-principles calculations we demonstrate in addition numerically the common topological properties of the TCI and TOM. To demonstrate the role of the relativistic spin-orbit coupling (SOC) for the TCI, a so-called 'topological' spin susceptibility (TSS) is introduced. This quantity characterizes the SOC induced spin magnetic moment on the atom in the presence of non-collinear magnetic structure, giving a connection between the TOS and TCI. Numerical results again support our conclusions.