Formation of energy gap in higher dimensional spin-orbital liquids

A Schwinger boson mean field theory is developed for spin liquids in a symmetric spin-orbital model in higher dimensions. Spin, orbital and coupled spin-orbital operators are treated equally. We evaluate the dynamic correlation functions and collective excitations spectra. As the collective excitations have a finite energy gap, we conclude that the ground state is a spin-orbital liquid with a two-fold degeneracy, which breaks the discrete spin-orbital symmetry. Possible relevence of this spin liquid state to several realistic systems, such as CaV${}_4$V${}_9$ and Na${}_2$Sb${}_2$Ti${}_2$O, are discussed.