  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.
