We investigate the competition between different orders in the two-leg spin ladder with a ring-exchange interaction by means of a bosonic approach. The latter is defined in terms of spin-1 hardcore bosons which treat the Neel and vector chirality order parameters on an equal footing. A semiclassical approach of the resulting model describes the phases of the two-leg spin ladder with a ring-exchange. In particular, we derive the low-energy effective actions which govern the physical properties of the rung-singlet and dominant vector chirality phases. As a by-product of our approach, we reveal the mutual induction phenomenon between spin and chirality with, for instance, the emergence of a vector-chirality phase from the application of a magnetic field in bilayer systems coupled by four-spin exchange interactions.
We study the ground state and the magnetization process of a spin-1/2 $J_1$-$J_2$ model with a plaquette structure by using various methods. For small inter-plaquette interaction, this model is expected to have a spin-gap and we computed the first- and the second excitation energies. If the gap of the lowest excitation closes, the corresponding particle condenses to form magnetic orders. By analyzing the quintet gap and magnetic interactions among the quintet excitations, we find a spin-nematic phase around $J_1/J_2sim -2$ due to the strong frustration and the quantum effect. When high magnetic moment is applied, not the spin-1 excitations but the spin-2 ones soften and dictate the magnetization process. We apply a mean-field approximation to the effective Hamiltonian to find three different types of phases (a conventional BEC phase, ``striped supersolid phases and a 1/2-plateau). Unlike the BEC in spin-dimer systems, this BEC phase is not accompanied by transverse magnetization. Possible connection to the recently discovered spin-gap compound (CuCl)LaNb2O7 is discussed.
