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We investigate the competition between different orders in the two-leg spinladder with a ring-exchange interaction by means of a bosonic approach. Thelatter is defined in terms of spin-1 hardcore bosons which treat the N\'eel andvector chirality order parameters on an equal footing. A semiclassical approachof the resulting model describes the phases of the two-leg spin ladder with aring-exchange. In particular, we derive the low-energy effective actions whichgovern the physical properties of the rung-singlet and dominant vectorchirality phases. As a by-product of our approach, we reveal the mutualinduction phenomenon between spin and chirality with, for instance, theemergence of a vector-chirality phase from the application of a magnetic fieldin bilayer systems coupled by four-spin exchange interactions.

Semiclassical Approach to Competing Orders in Two-leg Spin Ladder with Ring-Exchange