Rung-singlet phase of the $S=\frac{1}{2}$ two-leg spin-ladder with four-spin cyclic exchange

Using continuous unitary transformations (CUTs) we calculate the one-triplet gap for the antiferromagnetic $S=\frac{1}{2}$ two-leg spin ladder with additional four-spin exchange interactions in a high order series expansion about the limit of isolated rungs. By applying an efficient extrapolation technique we calculate the transition line between the rung-singlet phase and a spontaneously dimerized phase with dimers on the legs. Using this extrapolation technique we are able to analyze the crossover from strong rung coupling to weakly coupled chains.