$p$ orbitals in 3D lattices; fermions, bosons and (exotic) models of magnetism

We demonstrate how different types of $SU(3)$ Heisenberg models can be implemented with the use of the $p$ orbitals of three dimensional optical lattices. By considering a Mott insulator with unit filling, the dynamics is well described by an effective model derived from the perturbative treatment of the tunneling elements relative to the onsite interaction terms. This yields systems with degrees of freedom that are generators of the $SU(3)$ group, which extends the Heisenberg models frequently used to analyze quantum magnetism. Due to the different character of interactions in the bosonic and fermionic cases, the choice of atom determines what type of anisotropies will appear in the couplings of the corresponding effective Hamiltonians. Experimental schemes for detection and manipulation of these systems are presented, and properties of the ground states of selected examples are discussed.