Proper and improper chiral magnetic interactions

Atomistic spin models are of great value for predicting and understanding the magnetic properties of real materials, and extensions of the existing models open routes to new physics and potential applications. The Dzyaloshinskii-Moriya interaction is the prototype for chiral magnetic interactions, and several recent works have uncovered or proposed various types of generalized chiral interactions. However, in some cases the proposed interactions or their interpretation do not comply with basic principles such as being independent of the magnetic configuration from which they are evaluated, or even obeying time-reversal invariance. In this Letter, we present a simple explanation for the origin of these puzzling findings and point out how to resolve them.

Figure 1

Dependence of the total energy of ${\mathrm{Mn}}_3\mathrm{Sn}$ on the magnetic configuration. (a) Top view of the unit cell of ${\mathrm{Mn}}_3\mathrm{Sn}$ with noncollinear magnetic structure viewed along the $z$ axis. The spheres indicate the Mn atoms on the $z=1/4$ plane (blue) and on the $z=3/4$ plane (green). Sn is not shown. (b) Total energy for the magnetic configurations shown in (a) fitted to the spin model of Eq. (1) including only isotropic two-sublattice ($J_{12}$) or also biquadratic and three-sublattice interactions ($B_{12}$ and $B_{123}$, respectively).

Figure 2

Dependence of the improper DMI on the magnetic configuration for a Cr trimer on Au(111). (a) Noncollinear magnetic structure. The blue spheres represent Cr atoms, and the golden spheres represent Au atoms. (b) Dependence of the improper DMI between atoms 1 and 2 due to isotropic four-spin interactions on the magnetic configuration, according to Eqs. (10a) and (10b) and the parameters $B_{12}=-4.42$ meV and $B_{123}=7.06$ meV from Ref. [22] and $B_{12}=-5.10$ meV and $B_{123}=8.06$ meV from Ref. [25].