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The mechanism behind the three-spin chiral interaction (TCI) included in theextended Heisenberg Hamiltonian and represented by an expression worked outrecently (Phys.\ Rev.\ B, {\bf 101}, 174401 (2020)) is discussed. It isstressed that this approach provides a unique set of the multispin exchangeparameters which are independent of each other either due to their differentorder of perturbation or due to different symmetry. This ensures in particularthe specific properties of the TCI that were demonstrated previously via fullyrelativistic first principles calculations, and that result from the commoninfluence of several issues not explicitly seen from the expression for the TCIparameters. Therefore, an interpretation of the TCI is suggested, showingexplicitly its dependence on the relativistic spin-orbit coupling and on thetopological orbital susceptibility (TOS). This is based on an expression forthe TOS that is worked out on the same footing as the expression for the TCI.Using first-principles calculations we demonstrate in addition numerically thecommon topological properties of the TCI and TOS. To demonstrate the role ofthe relativistic spin-orbit coupling (SOC) for the TCI, a so-calledtopological' spin susceptibility (TSS) is introduced. This quantitycharacterizes the SOC induced spin magnetic moment on the atom in the presenceof non-collinear magnetic structure, giving a connection between the TOS andTCI. Numerical results again support our conclusions.

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