Dualities in spin ladders

Abstract

We introduce a set of discrete modular transformations Tℓ, Uℓ and Sℓ in order to study the relationships between the different phases of the Heisenberg ladders obtained with all possible exchange coupling constants. For the two-legged ladder we show that the resonating valence bond (RVB) phase is invariant under the Sℓ transformation, while the Haldane phase is invariant under Uℓ. These two phases are related by Tℓ. Moreover, there is a 'mixed' phase, that is invariant under Tℓ, and which under Uℓ becomes the RVB phase, while under Sℓ becomes the Haldane phase. For odd ladders there exists only the Tℓ transformation which, for strong coupling, maps the effective antiferromagnetic spin 1/2 chain onto the spin 3/2 chain. Our work is based on a combination of approximate methods such as bosonization, perturbation theory and the nonlinear sigma model, adapted to the different regimes of coupling constants. 1998 IOP Publishing Ltd