Various multi-spin magnetic exchange interactions (MEI) of chiral nature have been recently unveiled. Owing to their potential impact on the realisation of twisted spin-textures, their implication in spintronics or quantum computing is very promising. Here, I address the long-range behavior of multi-spin MEI on the basis of a multiple-scattering formalism implementable in Green functions based methods. I consider the impact of spin-orbit coupling (SOC) as described in the one- (1D) and two-dimensional (2D) Rashba model, from which the analytical forms of the four- and six-spin interactions are extracted and compared to the bilinear isotropic, anisotropic and Dzyaloshinskii-Moriya interactions (DMI). Similarly to the DMI between two sites $i$ and $j$, there is a four-spin chiral vector perpendicular to the bond connecting the two sites. The oscillatory behavior of the MEI and their decay as function of interatomic distances are analysed and quantified for the Rashba surfaces states characterizing Au surfaces. The interplay of beating effects and strength of SOC gives rise to a wide parameter space where chiral MEI are more prominent than the isotropic ones. The multi-spin interactions for a plaquette of $N$ magnetic moments decay like $\{q_F^{N-d} P^{\frac{1}{2}(d-1)}L\}^{-1}$ simplifying to $\{q_F^{N-d} R^{\left[1+\frac{N}{2}(d-1)\right]}N\}^{-1}$ for equidistant atoms, where $d$ is the dimension of the mediating electrons, $q_F$ the Fermi wave vector, $L$ the perimeter of the plaquette while $P$ is the product of interatomic distances. This recovers the behavior of the bilinear MEI, $\{q_F^{2-d} R^{d}\}^{-1}$, and shows that increasing the perimeter of the plaquette weakens the MEI. More important, the power-law pertaining to the distance-dependent 1D MEI is insensitive to the number of atoms in the plaquette in contrast to the linear dependence associated with the 2D MEI.

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多自旋手性磁相互作用的多重散射方法：在一维和二维 Rashba 电子气中的应用

最近公布了各种手性性质的多自旋磁交换相互作用（MEI）。由于它们对实现扭曲自旋纹理的潜在影响，它们在自旋电子学或量子计算中的意义非常有前途。在这里，我基于可在基于格林函数的方法中实现的多重散射形式来解决多自旋 MEI 的远程行为。我考虑了一（1D）和二维（2D）Rashba模型中描述的自旋轨道耦合（SOC）的影响，从中提取了四和六自旋相互作用的分析形式并与双线性各向同性、各向异性和 Dzyaloshinskii-Moriya 相互作用 (DMI)。与两个位点 $i$ 和 $j$ 之间的 DMI 类似，存在一个垂直于连接两个位点的键的四自旋手性矢量。对于表征 Au 表面的 Rashba 表面状态，对 MEI 的振荡行为及其作为原子间距离函数的衰减进行了分析和量化。跳动效应和 SOC 强度的相互作用产生了一个广泛的参数空间，其中手性 MEI 比各向同性 MEI 更突出。$N$ 磁矩衰减的多自旋相互作用，如 $\{q_F^{Nd} P^{\frac{1}{2}(d-1)}L\}^{-1}$对于等距原子，简化为 $\{q_F^{Nd} R^{\left[1+\frac{N}{2}(d-1)\right]}N\}^{-1}$，其中 $ d$ 是介导电子的尺寸，$q_F$ 是费米波矢量，$L$ 是小块的周长，而 $P$ 是原子间距离的乘积。这恢复了双线性 MEI 的行为，$\{q_F^{2-d} R^{d}\}^{-1}$，并表明增加血小板的周长会削弱 MEI。更重要的是，与与 2D MEI 相关的线性相关性相比，与距离相关的 1D MEI 相关的幂律对斑块中的原子数不敏感。