Singlet quantum phases and magnetization of the frustrated spin-1/2 ladder with ferromagnetic (F) exchange in legs and alternating F-AF exchange in rungs

The magnetization $M(h)$ is used to identify three singlet quantum phases of the ladder with isotropic exchange interactions. The Dimer phase with frustrated F exchanges in rungs and legs has a first-order $M(h)$ transition at $0$ K from singlet to ferromagnetic at the saturation field $h_s$. The Haldane-DAF phase with strong F exchange in rungs and net AF exchange between rungs has continuous $M(h)$ and is adiabatically connected to the $S=1$ Heisenberg AF chain. The AF phase with strong F exchange in legs and net AF exchange between legs has continuous $M(h)$ and is adiabatically connected to the spin-1/2 $J_1-J_2$ model with $J_1>0$ and $J_2<0$. All three singlet phases have finite gaps to the lowest triplet state.