We study the Kondo screening of a spin-1/2 magnetic impurity in a $d$-wave altermagnetic metal in the presence of cubic Rashba spin-orbit coupling (SOC) by combining the variational method and the Hirsch-Fye quantum Monte Carlo simulations. Interestingly, we find that the cubic Rashba SOC plays a dominant role in the Kondo effect when the chemical potential deviates from the band center, while the spatial symmetry of the Kondo screening cloud is solely determined by the altermagnetism. Altermagnetism has a minor influence on the density of states, which is crucial for the Kondo effect, especially away from the band center. Nevertheless, from the perspective of the Kondo screening cloud, its spatial symmetry is predominantly determined by altermagnetism. The addition of a cubic Rashba term modifies the ${\mathcal{C}}_{4z}\mathcal{T}$ symmetry of a $d$-wave altermagnet to $\mathcal{I}{\mathcal{C}}_{4z}\mathcal{T}$. Notably, this change in symmetry does not manifest in the diagonal components of the spatial Kondo screening cloud. The major contribution of cubic Rashba SOC to the spatial Kondo screening cloud is that it can induce off-diagonal correlations between local spin and conduction electron spins. In addition, we find that the spatial Kondo screening cloud decays with $1/r^3$ in space, while the indirect Ruderman-Kittel-Kasuya-Yosida interaction decays with $1/r^2$ with the coexistence of cubic Rashba SOC and altermagnetism.

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• Kondo effect
• RKKY interaction
• Altermagnets
• Quantum Monte Carlo
• Variational wave functional methods