Zeeman-field-induced three-spin interactions and relativistic coupling to the scalar spin chirality

Magnetic fields are commonly incorporated in atomistic spin models through the Zeeman energy of individual magnetic moments. Starting from a microscopic electronic Hamiltonian, we show that the Zeeman coupling of conduction electrons also generates field-linear three-spin interactions, the lowest-order corrections allowed by time-reversal symmetry. Using a Green-function expansion of the grand potential, we identify two classes of interactions. Isotropic terms of the form ( Sᵢ· Sⱼ)( Sₖ· B) exist without spin-orbit coupling, and their total strength obeys a sum rule involving the three-site electronic loop at the Fermi energy. Relativistic terms require spin-orbit coupling and include Kᵢⱼₖ· B,χᵢⱼₖ, where χᵢⱼₖ= Sᵢ·( Sⱼ× Sₖ) is the scalar spin chirality. We show that this coupling is forbidden in the spin channel without spin-orbit coupling, whereas the orbital channel recovers the orbital-flux mechanism of Mott insulators. Exact diagonalization of the Hubbard trimer confirms the independence and additivity of both channels. For magnetic trimers in Rashba electron gases, we derive analytical expressions showing that Kᵢⱼₖ lies in-plane at first order in spin-orbit coupling but vanishes for C₃ᵥ-symmetric trimers, where an out-of-plane component emerges at second order. These interactions can shift magnetization-step fields and modify field-induced noncoplanarity. In the continuum limit, the chirality coupling splits skyrmions from antiskyrmions, while the relativistic terms generate a field-dependent Dzyaloshinskii-Moriya interaction.