Frustration and Entanglement in Compass and Spin-Orbital Models

We review the consequences of intrinsic frustration of the orbital superexchange and of spin-orbital entanglement. While the Heisenberg perturbing interactions remove frustration in the compass model, the lowest columnar excitations are robust in the nanoscopic compass clusters and might be used for quantum computations. Entangled spin-orbital states determine the ground states in some cases, while in others concern excited states and lead to measurable consequences, as in the RVO3 perovskites. On-site entanglement for strong spinorbit coupling generates the frustrated KitaevHeisenberg model with a rich magnetic phase diagram on the honeycomb lattice.