The matrix product approach to quantum spin ladders

We present a manifestly rotational invariant formulation of the matrix product method valid for spin chains and ladders. We apply it to two-legged spin ladders with spins , 1 and and different magnetic structures labelled by the exchange coupling constants, which can be ferromagnetic or antiferromagnetic along the legs and the rungs of the ladder. We compute ground-state energy densities, correlation lengths and string order parameters. We present numerical evidence of the duality properties of the three different nonferromagnetic spin ladders. We show that the long-range topological order characteristic of isolated spin 1 chains is broken by the interchain coupling. The string order correlation function decays exponentially with a finite correlation length that we compute. A physical picture of the spin 1 ladder is given in terms of a collection of resonating spin 1 chains. Finally, for ladders with spin equal to or greater than we define a class of AKLT states whose matrix product coefficients are given by 9- j symbols.