Absence of spin order in a two-dimensional orbital optical lattice

Mott insulators with both spin and orbital degeneracy are pertinent to a family of transition metal oxides. The intertwined spin and orbital fluctuations can lead to exotic phases such as quantum spin-orbital liquids. Here we consider two-component spin 1/2 fermionic atoms with strong repulsive interaction on the $p$-band of the square optical lattice. We derive the spin-orbital exchange for quarter filling of the $p$-band in the Mott limit, and show it frustrates the development of long range spin order. Exact diagonalization indicates a spin-disordered ground state with ferro-orbital order. The system dynamically decouples into individual Heisenberg spin chains, each realizing a Luttinger liquid accessible at higher temperatures compared to atoms confined to the $s$-band. Our model serves as an example of how orbital order enhances quantum fluctuations to prevent spin order and leads to dimension reduction in a quantum gas system.