Antiferromagnetic spin ladders effectively coupled by one-dimensional electron liquids

We study a model of the stripe state in strongly correlated systems consist ing of an array of antiferromagnetic spin ladders, each with n(leg) legs, c oupled to each other through the spin-exchange interaction to charged strip es in between each pair of ladders. The charged stripes are assumed to be L uttinger liquids in a spin-gap regime. An effective interaction for a pair of neighboring ladders is calculated by integrating out the gapped spin deg rees of freedom in the charged stripes. The low energy effective theory of each ladder is a nonlinear sigma model with additional cross couplings of n eighboring ladders, which favor either in-phase or antiphase short-range sp in orderings depending on the physical parameters of the charged stripe.