Antiferromagnetic spin ladders effectively coupled by one-dimensional electron liquids

We study a model of the stripe state in strongly correlated systems consisting of an array of antiferromagnetic spin ladders, each with $n_{leg}$ legs, coupled to each other through the spin-exchange interaction to charged stripes in between each pair of ladders. The charged stripes are assumed to be Luttinger liquids in a spin-gap regime (Luther-Emery). An effective interaction for a pair of neighboring ladders is calculated by integrating out the gapped spin degree of freedom in the charged stripe. The low energy effective theory of each ladder is the usual nonlinear $\sigma$-model with additional cross couplings of neighboring ladders. These interactions are found to favor either in-phase or anti-phase short range spin orderings depending on whether the charge stripe is site-centered or bond-centered as well as on its filling factor and other physical parameters of the charged stripe.