  We consider various two-leg ladder models exhibiting gapped phases. All of
these phases have short-ranged valence bond ground states, and they all exhibit
string order. However, we show that short-ranged valence bond ground states
divide into two topologically distinct classes, and as a consequence, there
exist two topologically distinct types of string order. Therefore, not all
gapped phases belong to the same universality class. We show that phase
transitions occur when we interpolate between models belonging to different
topological classes, and we study the nature of these transitions.

Strongly Correlated Electrons