Simulating the Palmer-Chalker state in an orbital superfluid.

We consider a bosonic $s$ and $p$ orbital system in a face-centered cubic (FCC) optical lattice, and predict a fluctuation-induced instability towards the orbital analogue of Palmer-Chalker state, which is originally proposed in an electronic spin system. For bosons loaded in the FCC optical lattice, the single-particle spectrum has four degenerate band minima with their crystal momenta forming a tetrahedron in Brillouin zone. In the weakly interacting regime, the ensuing many-particle ground state, at the classical level, underlies a four-sublattice tetrahedral supercell of spontaneously generated $p$-orbital angular momenta through the Bravias-Bloch duality between real and momentum space, and is macroscopically degenerate originating from the geometric frustration. The fluctuations on top of the classical ground state lift its degeneracy and select the Palmer-Chalker ordering of $p$-orbital angular momenta as the quantum ground state through order-by-disorder mechanism. These findings raise the exciting possibility of simulating the Palmer-Chalker state with its orbital counterpart in ultracold atomic gases.