Solution to a Quantum Impurity Model for Moire Systems: Fermi Liquid, Pairing, and Pseudogap

Recent theoretical and experimental studies have revealed the co-existence of heavy and light electrons in moir\'e graphene systems, which together form multi-orbital periodic Anderson models. This work demonstrates that nontrivial physics - such as non-Fermi-liquid susceptibilities and continuous quantum phase transitions - already appears at the single-impurity level, if an anti-Hund's rule is included to favor either a singlet ($J_S$) or doublet ($J_D$) impurity configuration. We derive a complete phase diagram and analytically solve the impurity problem at several fixed points. When a positive $J_D$ dominates, the valley degree of freedom couples only via pair-hopping processes to the conduction electrons, in sharp contrast to the conventional Kondo scenario.A strong $J_D$ stabilizes an anisotropic local moment phase exhibiting power-law susceptibilities with non-universal exponents, while a weak $J_D$ leads to a Fermi liquid with an attractive renormalized interaction favoring a doublet pairing. Remarkably, both phases are analytically solvable via bosonization and refermionization, and are separated by a quantum critical point in the BKT universality class. When a positive $J_S$ dominates, we identify a Fermi liquid with a local singlet ground state in the strong $J_S$ limit and a Kondo Fermi liquid in the weak $J_S$ limit, separated by a quantum critical point of second order. Both exhibit attractive renormalized interactions favoring singlet pairing. Negative $J_{S}$ and $J_D$ stabilize Fermi liquids with attractive interactions favoring triplet pairing. Based on the solutions, we construct analytical ansatz for self-energies of Green's functions at several fixed points, which account for the ``pseudogap'' features seen in recent spectroscopic experiments. In particular, we obtain non-analytic V-shaped spectral function with non-universal exponents. All the results are further verified by numerical renormalization group calculations. Implications for generalizing the single-impurity physics to the lattice model are also discussed.