DUALITIES IN SPIN LADDERS

We introduce a set of discrete modular transformations T-l, U-l and S- l in order to study the relationships between the different phases of the Heisenberg ladders obtained with all possible exchange coupling co nstants. For the two-legged ladder we show that the resonating valence bond (RVB) phase is invariant under the S-l transformation, while the Haldane phase is invariant under U-e. These two phases are related by T-l. Moreover, there is a 'mixed' phase, that is invariant under T-l, and which under lie becomes the RVB phase, while under S-l becomes th e Haldane phase. For odd ladders there exists only the T-l transformat ion which, for strong coupling, maps the effective antiferromagnetic s pin 1/2 chain onto the spin 3/2 chain. Our work is based on a combinat ion of approximate methods such as bosonization, perturbation theory a nd the nonlinear sigma model, adapted to the different regimes of coup ling constants.