p-band Mott insulators on an optical diamond lattice

The Mott-insulating states of p-orbital spinless fermions in a three-dimensional optical
lattice exhibit an unusual degeneracy due to the anisotropic orbital interaction and
geometrical frustration. Here we demonstrate the existence of orbital Coulomb phase
as the exact ground state of the $p$-orbital exchange Hamiltonian on the diamond lattice.
We show that the extensively degenerate orbital configurations of this Coulomb phase
can be mapped to Ising spins obeying the ice rule on the pyrochlore lattice.
An experimental signature is the pinch-point singularities in diffusion-scattering
measurement which originate from a dipolar-like correlation function in the orbital ice state.