Bosonization Solution to Spin-Valley Kondo Problem: Finite-Size Spectrum and Renormalization Group Analysis

Spin-valley Anderson impurities (SVAIM) with (anti-)Hund's splitting provide a natural explanation to the origin of pairing potential and pseudogap in the magic-angle graphene. In this work, we derive and analytically solve the low-energy Kondo theories for SVAIM at half-filling, with especial focus on the two anti-Hund's regimes: the impurity is either dominated by a valley doublet, or a trivial singlet. In the doublet regime, we reveal that a novel pair Kondo scattering $\lambda_x$ is required to flip the valley doublet, which involves a quartic operator of bath electrons. Our renormalization group (RG) calculation based on the Coulomb gas analog shows $\lambda_x$ drives a phase transition of the Berezinskii-Kosterlitz-Thouless type. One side of the transition is an anisotropic doublet phase, characterized by non-universal phase shifts of bath electrons and non-analytic impurity susceptibilities, while the other is a Fermi liquid formed by pair-Kondo resonance. The finite-size many-body spectrum, thermodynamic quantities, and correlation functions for both phases are analytically solved. Remarkably, the solution in the pair-Kondo Fermi liquid is achieved via the constructive approach of bosonization-refermionization along a solvable fixed line, where the many-body interaction $\lambda_x$ is mapped into a pseudo-fermion bilinear in a rigorous manner. Finally, we also apply the RG analysis to the singlet regime, and identify a second-order phase transition between the Kondo Fermi liquid and a local singlet phase.