Magnetic Order through Kondo coupling to a quantum spin liquid.

We explore magnetic ordering in Kondo superlattices induced by quantum spin liquids, focusing on effective interactions between local moments mediated by Majorana fermions. Using the Kitaev model and its two gamma matrix generalizations, we compute the effective superexchange interactions through perturbation theory in cases involving vison excitations, and in other scenarios, we employ exact diagonalization to study long-range Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions. In the Kitaev spin liquid, short-range interactions among local moments result in an exactly solvable model without opening a gap in the spin liquid's spectrum. One gamma matrix generalization, defined on the honeycomb lattice, generates long-range RKKY interactions akin to those found in graphene. The other generalization, on the square lattice, exhibits long-range RKKY interactions only when the local moments align in the z-direction, while x or y alignments produce short-range superexchange interactions. Notably, the magnetic ordering of local moments in these generalized models induces gaps in the Majorana fermion spectra, revealing a rich interplay between spin liquids and magnetic ordering in Kondo superlattices.