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 J l e g = J r i n g < J r u n g / 4 , the rung singlet density is one and a second-order phase transition occurs.