We investigate the general structure of orbital exchange physics in Mott-insulating states of p-orbital systems in optical lattices. Orbital orders occur in both the triangular and kagome lattices. In contrast, orbital exchange in the honeycomb lattice is frustrated as described by a novel quantum 120 degrees model. Its classical ground states are mapped into configurations of the fully packed loop model with an extra U(1) rotation degree of freedom. Quantum orbital fluctuations select a six-site plaquette ground state ordering pattern in the semiclassical limit from the "order from disorder" mechanism. This effect arises from the appearance of a zero energy flat band of orbital excitations.
A gas of strongly interacting single-species (spinless) p-orbital fermionic atoms in 2D optical lattices is proposed and studied. Several interesting new features are found. In the Mott limit on a square lattice, the gas is found to be described effectively by an orbital exchange Hamiltonian equivalent to a pseudospin-1/2 XXZ model. For a triangular, honeycomb, or kagome lattice, the orbital exchange is geometrically frustrated and described by a new quantum 120 degrees model. We determine the orbital ordering on the kagome lattice, and show how orbital wave fluctuations select ground states via the order by disorder mechanism for the honeycomb lattice. We discuss experimental signatures of various orbital ordering.
We propose to go beyond the usual Hubbard model description of atoms in optical lattices and show how few-body physics can be used to simulate many-body phenomena, e.g., an electron-phonon system. We take one atomic species to be trapped in a deep optical lattice at full-filling and another to be untrapped spin-polarized fermions (which do not see the optical lattice) but to have an
s-wave contact interaction with the first species. For large positive scattering length on the order of lattice spacing, the usual two-body bound (dimer) states overlap forming giant orbitals extending over the entire lattice, which can be viewed as an ``electronic'' band for the untrapped species while the trapped atoms become the ``ions'' with their own on-site dynamics, thereby simulating an electron-phonon system with renormalization of the phonon frequencies and Peierls transitions. This setup requires large scattering lengths but minimizes losses, does not need higher bands, and adds degrees of freedom which cannot easily be described in terms of lattice variables, thus opening up intriguing possibilities to explore interesting physics at the interface between few-body and many-body systems.