We map spin ladders with
nl legs and couplings ${J}^{\ensuremath{'}}$ across all rungs and $J(1\ifmmode\pm\else\textpm\fi{}\ensuremath{\gamma})$ along the legs, staggered in both directions, to a sigma model. Setting its $\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(2m+1)\ensuremath{\pi}$ (where it is known to be gapless), we locate the critical curves in the $\ensuremath{\gamma}$ versus ${J}^{\ensuremath{'}}/J$ plane at each
nl, and spin
S. The phase diagram is rich and has some surprises: When two gapped chains are suitably coupled, the combination becomes gapless. With
nl, $\ensuremath{\gamma}$, and ${J}^{\ensuremath{'}}/J$ to control, the prospects for experimentally observing any one of these equivalent transitions are enhanced. We interpret our results in the framework of the resonating valence bond description of ladders.
The well known Haldane map from spin chains into the O(3) nonlinear sigma model is generalized to the case of spin ladders. This map allows us to explain the different qualitative behaviour between even and odd ladders, in exactly the same way as it explains the difference between integer and half-integer spin chains. Namely, for even ladders the topological term in the sigma model action is absent, while for odd ladders the parameter, which multiplies the topological term, is equal to , where S is the spin of the ladder. Hence even ladders should have a dynamically generated spin gap, while odd ladders with half-integer spin should stay gapless and physically equivalent to a perturbed Wess - Zumino - Witten model in the infrared regime. We also derive some consequences from the dependence of the sigma model coupling constant on the ladder Heisenberg couplings constants.
We study the opening of the Haldane gap in two-leg and four-leg anisotropic spin ladders using bosonization and renormalization group methods, and we determine the phase diagram as a function of the interchain coupling and relative anisotropy
Jz/Jxy. It is found that the opening of the Haldane gap is qualitatively different for the two cases considered. For the two-leg ladder the Haldane gap opens for arbitrarily small interchain coupling, independent of
Jz/Jxy, and the Haldane phase exists in a large region of parameter space. For the four-leg ladder the opening of the Haldane gap is strongly dependent on both the interchain coupling as well as
Jz/Jxy, and the Haldane phase exists only in a narrow region about the isotropic antiferromagnet.
We analyze an anisotropic spin-1/2 two-leg ladder in the presence of various types 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 occurring 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 nondisordered
XY1 phase occurring between the decoupled chains and the singlet phase. Comparison with other disordered gapped systems such as spin one chains is discussed.