Scaling theory of intrinsic Kondo and Hund's rule interactions in magic-angle twisted bilayer graphene

Motivated by the recent studies of intrinsic local moments and Kondo-driven phases in magic-angle twisted bilayer graphene, we investigate the renormalization of Kondo coupling (${J}_K$) and the competing Hund's rule interaction ($J$) in the low-energy limit. Specifically, we consider a surrogate single-impurity generalized Kondo model and employ the poor man's scaling approach. The scale dependent ${J}_K$ and $J$ are derived analytically within the one-loop poor man's scaling approach, and the Kondo temperature (${T}_K$) and the characteristic Hund's rule coupling (${J}^*$, defined by the renormalized value of $J$ at some small finite-energy scale) are estimated over a wide range of filling factors. We find that ${T}_K$ depends strongly on the filling factors as well as the value of ${J}_K$. Slightly doping away from integer fillings and/or increasing ${J}_K$ may substantially enhance ${T}_K$ in the parameter regime relevant to experiments. ${J}^*$ is always reduced from the bare value of $J$, but the filling-factor dependence is not as significant as it is for ${T}_K$. Our results suggest that it is essential to incorporate the renormalization of ${J}_K$ and $J$ in the many-body calculations, and Kondo screening should occur for a wide range of fractional fillings in magic-angle twisted bilayer graphene, implying the existence of Kondo-driven correlated metallic phases. We also point out that the observation of distinct phases at integer fillings in different samples may be due to the variation of ${J}_K$ in addition to disorder and strain in the experiments.