Phase Transitions in Staggered Spin Ladders

We map spin ladders with $n_l$ legs and couplings $J^{\prime }$ 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 $J^{\prime }/J$ 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 $J^{\prime }/J$ 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.