Mean-field theory for the spin-ladder system

In the present paper, we propose a mean-field approach for spin ladders based upon the Jordan-Wigner transformation along an elaborately ordered path. We show on the mean-field level that ladders with even-numbered legs open an energy gap in their low-energy excitation with a magnitude close to the corresponding experimental values, whereas the low-energy excitation of the odd-numbered-leg ladders are gapless. It supports the validity of our approach. We then calculate the gap size and the excitation spectra of a two-leg-ladder system. Our result is in good agreement with both the experimental data and the numerical results.