Snakes and Ladders

We map spin ladders with $n_l$ legs and couplings $J'$ across all rungs and $J(1 \pm \gamma)$ along the legs, staggered in both directions, to a sigma model. Setting its $\theta = (2m+1)\pi$ (where it is known to be gapless), we locate the critical curves in the $\gamma$ versus ${J'\over J}$ plane at each $n_l$, and spin $S$. The phase diagram is rich and has some surprises: when two gapped chains are suitably coupled, the combination becomes gapless. With $n_l, \gamma$ and $J'/J$ to control, the prospects for experimentally observing any one of these equivalent transitions seem bright. We discuss the order parameters and the behavior of holes in the RVB description.