Interplay between Kondo screening and local singlets in $SU(N)$-symmetric cold atoms

We study collective phenomena in strongly interacting fermionic alkaline-earth atoms (AEAs) loaded in an optical lattice. Owing to the strong decoupling between electronic orbital and nuclear-spin degrees of freedom, AEAs prepared in the two lowest electronic states are predicted to obey an accurate $SU(N > 2 I + 1)$ symmetry in their two-body collisions ($I$ is the nuclear spin). The $SU(N)$ symmetric models offer a great opportunity to generate exotic many-body behavior emerging from the increased degeneracy and strict conservation laws. We focus on a parameter regime that realizes an $SU(N > 2)$ (Coqblin-Schrieffer) generalization of the usual Kondo lattice model, and show that for band fillings above one atom per site, the system exhibits a peculiar interplay between Kondo screening and formation of singlets between localized atoms. In the limit of large Kondo coupling, we derive an effective Hamiltonian and determine its phase diagram. Our results can be tested in experiments with ultracold ${}^{173}{\rm Yb}$ or ${}^{87}{\rm Sr}$ atoms and are relevant for the physics of heavy-fermion materials with magnetic frustration.