A microscopic understanding of magnetic exchange parameters beyond the conventional Heisenberg and Dzyaloshinskii-Moriya interaction (DMI) is crucial for $3d$ difluorides, particularly for ${\mathrm{NiF}}_2$, where spins are tilting about $0.{43}^{\circ }$ along the $z$ axis, which cannot be explained by DMI. It was proposed by Moriya Phys. Rev. 117, 635 (1960) that this canting is related to single-site anisotropy, but the anisotropic contribution to magnetic exchange remains insufficiently addressed. It is unclear what role anisotropic exchange plays in comparison with single-site anisotropy, which can also induce a spin canting. Motivated by Moriya’s work, we have calculated from first principles the full anisotropic exchange matrix for $3d$ rutile difluorides and performed a systematic comparison of ${\mathrm{MnF}}_2, {\mathrm{FeF}}_2, {\mathrm{CoF}}_2$, and ${\mathrm{NiF}}_2$. Our approach integrates symmetry-based spin-Hamiltonian construction and first-principles calculations, enabling the determination of the minimal form of the magnetic exchange matrix in the presence of spin-orbit coupling.

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• Electronic structure
• Exchange interaction
• Spin-orbit coupling
• Altermagnets
• Density functional calculations