Haldane and dimer gap in general double spin-chain models

We have obtained an analytic expression for the $k$ dependence of excitation energy gap for an arbitrary double $S=1/2$ spin-chain by using the nonlocal unitary transformation and the variational method. It is checked to explain the gap behavior of various systems, which include the Haldane system and the dimer system in both extreme limits, and also the ladder model and the Majumdar-Ghosh model. The string order parameter, the dimer order parameter, and the local spin value are also calculated in the ground state. The ground-state energy exhibits a great stabilization by an antiferromagnetic bond dimerization, which might be realized in various new compounds. We also mention the relation of the convergence to the Haldane state with the spin-exchange symmetry of the model. The excited state has one domain wall of a local triplet type except in the vicinity of the Majumdar-Ghosh point, where a local triplet is decomposed to two $S=1/2$ free spins moving among the singlet dimers.