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Extension of the standard Heisenberg Hamiltonian to multispin exchange interactions

An extension of the Heisenberg Hamiltonian is discussed, that allows to go beyond the standard bilinear spin Hamiltonian taking into account various contributions due to multispin interactions having both chiral and non-chiral character. The parameters of the extended Hamiltonian are calculated from first principles within the framework of the multiple scattering Green function formalism giving access to an explicit representation of these parameters in real space. The discussions are focused on the chiral interactions, i.e.\ biquadratic and three-spin Dzyaloshinskii-Moriya like vector interactions $\vec{\cal{D}}_{ijij}$ (BDMI) and $\vec{\cal{D}}_{ijkj}$ (TDMI), respectively, as well as the three-spin chiral interaction (TCI) $J_{ijk}$. Although all parameters are driven by spin-orbit coupling (SOC), some differences in their properties are demonstrated by calculations for real materials. In particular it is shown that the three-spin chiral interactions $J_{ijk}$ may be topology as well as SOC induced, while the TDMI is associated only with the SOC. As the magnitude of the chiral interactions can be quite sizable, they can lead to a stabilization of a noncollinear magnetic texture in some materials that is absent when these interactions are neglected.