The isotropic $S=1/2\mathrm{}$ antiferromagnetic spin ladder with additional four-spin ring exchange is studied perturbatively in the strong coupling regime with the help of cluster expansion technique, and by means of bosonization in the weak coupling limit. It is found that a sufficiently large strength of ring exchange leads to a second-order phase transition, and the shape of the boundary in the vicinity of the known exact transition point is obtained. The critical exponent for the gap is found to be $\eta \simeq 1,$ in agreement both with exact results available for the dimer line and with the bosonization analysis. The phase we determined after crossing the critical line is gapped and spontaneously dimerized. The results for the transition line from strong coupling and from weak coupling match with each other naturally.

• Received 13 June 2002

DOI:https://doi.org/10.1103/PhysRevB.66.134423