Magnetization jump in one dimensional $J-Q_{2}$ model with anisotropic exchange

We investigate the adiabatic magnetization process of the one-dimensional $J-Q_{2}$ model with XXZ anisotropy $g$ in an external magnetic field $h$ by using density matrix renormalization group (DMRG) method. According to the characteristic of the magnetization curves, we draw a magnetization phase diagram consisting of four phases. For a fixed nonzero pair coupling $Q$, i) when $g<-1$, the ground state is always ferromagnetic in spite of $h$; ii) when $g>-1$ but still small, the whole magnetization curve is continuous and smooth; iii) if further increasing $g$, there is a macroscopic magnetization jump from partially- to fully-polarized state; iv) for a sufficiently large $g$, the magnetization jump is from non- to fully-polarized state. By examining the energy per magnon and the correlation function, we find that the origin of the magnetization jump is the condensation of magnons and the formation of magnetic domains. We also demonstrate that while the experienced states are Heisenberg-like without long-range order, all the \textit{jumped-over} states have antiferromagnetic or N\'eel long-range orders, or their mixing.