LAUSR

We theoretically study the influence of the $k$-cubic Rashba spin-orbit coupling (SOC) on the correlation effects of magnetic impurities by combining the variational method and the Hirsch-Fye quantum Monte Carlo (HFQMC) simulations. Markedly different from the normal $k$-linear Rashba SOC, even a small cubic Rashba term can greatly alter the band structure and induce a Van Hove singularity in a wide range of energy, thus the single impurity local moment becomes largely tunable. The cubic Rashba SOC adopted in this work breaks the rotational symmetry, but the host material is still invariant under the operations $\mathcal{R}^z(\pi)$, $\mathcal{IR}^z(\pi/2)$, $\mathcal{M}_{xz}$, $\mathcal{M}_{yz}$, where $\mathcal{R}^z(\theta)$ is the rotation of angle $\theta$ about the $z$-axis, $\mathcal{I}$ is the inversion operator and $\mathcal{M}_{xz}$ ($\mathcal{M}_{yz}$) is the mirror reflection about the $x$-$z$ ($y$-$z$) principal plane. Saliently, various components of spin-spin correlation between the single magnetic impurity and the conduction electrons show three- or six-fold rotational symmetry. This unique feature is due to the triple winding of the spins with a $2\pi$ rotation of $\mathbf{k}$, which is a hallmark of the cubic Rashba effect and can possibly be an identifier to distinguish the cubic Rashba SOC from the normal $k$-linear Rashba term in experiments. Although the cubic Rashba term drastically alters the electronic properties of the host, we find that the spatial decay rate of the spin-spin correlation function remains essentially unchanged. Moreover, the carrier-mediated Ruderman-Kittel-Kasuya-Yosida interactions between two magnetic impurities show twisted features, the ferromagnetic diagonal terms dominate when two magnetic impurities are very close, but the off-diagonal terms become important at long distances.