Kitaev spin-orbital bilayers and their moiré superlattices

We study the phase diagram of bilayer, SU(2) invariant Kitaev spin-orbital models on a honeycomb lattice with interlayer Heisenberg exchange couplings, for different stacking orders and their moiré superlattices. In the absence of interlayer exchange, each layer is in a gapless spin-liquid phase with three flavors of Majorana excitations. The interlayer exchange commutes with the intralayer flux operators and preserves the topological order for weak interactions. We show that for AA stacking, the model maps onto a SU(3) Hubbard model with complex fermions where the sign of the Hubbard coupling U is positive or negative for ferromagnetic or antiferromagnetic exchange couplings, respectively. The Hubbard model has a charge density wave instability for finite negative U. We show that this instability is associated with a mirror symmetry breaking and that it corresponds to a gapped abelian spin liquid state in the Kitaev bilayer. For AB stacking, the Kitaev bilayer does not map onto a simple Hubbard model. Here, the exchange coupling induces an effective interlayer hybridization and the Majorana spectrum resembles that of bilayer graphene with quadratic band touching. We also study the bilayer model with twisting and discuss how its instabilities evolve with twist angle. Our study provides insight into the stability of the abelian spin-liquid phases in candidate bilayer spin-orbital Kitaev materials.