Excitations of a low-dimensional dimerized spin ladder under a magnetic field

We study the dimerized spin ladder with nearest-neighbor ($J_1$) and next-nearest-neighbor ($J_2$) anti-ferromagnetic interaction under a magnetic field. We predict the existence of different magnetization plateaus for the presence of spin-Peierls interaction on both $J_1$ and $J_2$. Magnetization plateau at $m=0$ for $J_1$ dimerization is spontaneous due to XY interaction, but it is absent for $J_2$ dimerization, only intrinsic umklapp term leads to plateau (spin gap) state for some specific values of XXZ anisotropy ($\Delta$) and $J_2$. Here we predict a saturation plateau which is the classical phase of the system. There are some numerical support of our analytical approach already existing in the literature. The transition from commensurate gapped phase to incommensurate Luttinger liquid phase is the Mott-$\delta$ type of transition.