Phase transition of spin-orbital models with a four-spin ring exchange

We extend the bond-operator mean-field theory to study the rung singlet phase and its phase boundary, the triplet excitation, and the spin gap of the spin-orbital models with four-spin exchanges. The theory gives a well description of the rung singlet phase and phase boundaries in two-dimensional (2D) and three-dimensional (3D) cases are predicted. It is shown that consideration of the ring exchange suppresses the excitation spectrum and decreases the spin gap. For 2D and 3D spin-orbital models, positive ring and leg coupling tend to collaborate with each other to break the rung singlet phase. On the boundary line Jleg=Jring