Weakly disordered spin ladders

We analyze an anisotropic spin 1/2 two legs ladder in the presence of various type of random perturbations. The generic phase diagram for the pure system, in a way similar to spin one chains, consists of four phases: an Antiferromagnet, a Haldane gap or a Singlet phase (depending on the sign of interchain coupling) and two XY phases designated by XY1 and XY2. The effects of disorder depend crucially on whether it preserves XY symmetry (random field along z and random exchange) or not (random anisotropy or random XY fields). In all cases we computed the new phase diagram and the correlation length for the disordered system. The ladder exhibits a remarkable stability to disorder with XY symmetry. Not only the singlet phases but also the massless XY1 phase are totally unaffected by disorder in stark contrast with the single chain case. Even in the presence of an external magnetic field breaking the spin gap most of the phase diagram (including the XY point) remains unaffected by the disorder, again in opposition with the single chain case. This stabilization towards disorder is similar, albeit stronger, to the one occuring for fermionic ladders. On the other hand the XY2 phase is very strongly suppressed. Disorder breaking XY symmetry has much stronger effects and most of the phases at the exception of the singlet one are now destroyed. Interestingly, the disordered XY1 phase has a much shorter correlation length than the disordered XY2 phase contrarily with the case of perturbations preserving the rotation symmetry around the z axis. The ladder system is thus in fact much more anisotropic than its single chain counterpart with identical exchanges. Finally we examine the case of strong disorder or weak interchain exchange for XY symmetric disorder. Close to the isotropic point, when the interchain exchange is increased the system undergoes a transition between two decoupled disordered spin 1/2 chains to a singlet ladder unaffected by disorder. For more XY like systems, the transition occurs in two steps with a non disordered XY1 phase occuring between the decoupled chains and the singlet phase. Comparison with other disordered gapped systems such as spin one chains is discussed.