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We consider various two-leg ladder models exhibiting gapped phases. All ofthese phases have short-ranged valence bond ground states, and they all exhibitstring order. However, we show that short-ranged valence bond ground statesdivide into two topologically distinct classes, and as a consequence, thereexist two topologically distinct types of string order. Therefore, not allgapped phases belong to the same universality class. We show that phasetransitions occur when we interpolate between models belonging to differenttopological classes, and we study the nature of these transitions.Comment: 11 pages, 16 postscript figure

Phase transitions between topologically distinct gapped phases in isotropic spin ladders