  We demonstrate the existence of orbital Coulomb phase as the exact ground
state of p-orbital exchange Hamiltonian on the diamond lattice. The Coulomb
phase is an emergent state characterized by algebraic dipolar correlations and
a gauge structure resulting from local constraints (ice rules) of the
underlying lattice models. For most ice models on the pyrochlore lattice, these
local constraints are a direct consequence of minimizing the energy of each
individual tetrahedron. On the contrary, the orbital ice rules are emergent
phenomena resulting from the quantum orbital dynamics. We show that the orbital
ice model exhibits an emergent geometrical frustration by mapping the
degenerate quantum orbital ground states to the spin-ice states obeying the
2-in-2-out constraints on the pyrochlore lattice. We also discuss possible
realization of the orbital ice model in optical lattices with p-band fermionic
cold atoms.
