We present a systematic derivation of effective lattice spin Hamiltonians derived from a rotationally invariant multiorbital Hubbard model including a term ensuring Hund's rule coupling. The Hamiltonians are derived down-folding the fermionic degrees of freedom of the Hubbard model into the proper low-energy spin sector using Löwdin partitioning, which will be outlined in detail for the case of two sites and two orbitals at each site. Correcting the ground state systematically up to fourth order in the hopping of electrons, we find, for spin $S\ge 1$, the biquadratic, three-spin, and four-spin interactions beyond the conventional Heisenberg term. Comparing the puzzling energy spectrum of the magnetic states for a single Fe monolayer on Ru(0001), obtained from density functional theory, with the spin Hamiltonians taken at the limit of classical spins, we show that the previously ignored three-spin interaction can be comparable in size to the conventional Heisenberg exchange.

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海森堡以外固体的实际自旋模型的系统推导

我们提出了一个有效的晶格自旋哈密顿量的系统推导，该自旋哈密顿量是从旋转不变的多轨道哈伯德模型中得出的，其中包括确保洪德法则耦合的项。使用Löwdin分区，将哈伯德模型的费米子自由度向下折叠成适当的低能自旋区，从而得出哈密顿量，这将在两个站点和每个站点两个轨道的情况下进行详细概述。我们发现，对于电子自旋，系统地将电子跃迁的基态校正至四阶$\mathrm{小号}\ge \mathrm{1个}$，超越了传统的海森堡一词的双二次，三轴和四轴相互作用。比较从密度泛函理论获得的Ru（0001）上单个Fe单层的磁性态的令人费解的能谱与在经典自旋的极限处采取的自旋哈密顿量，我们可以证明以前忽略的三自旋相互作用是大小可与传统的海森堡交易所媲美。