  We studied quantum phase transitions in the antiferromagnetic dimerized
spin-1/2 XY chain andvtwo-leg ladders. From analysis of several spin models we
present our main result: the framework to deal with topological orders and
hidden symmetries within the Landau paradigm. After mapping of the spin
Hamiltonians onto the tight-binding models with Dirac or Majorana fermions and,
when necessary, the mean-field approximation, the analysis can be done
analytically. By utilizing duality transformations the calculation of nonlocal
string order parameters is mapped onto the local order problem in some dual
representation and done without further approximations. Calculated phase
diagrams, phase boundaries, order parameters and their symmetries for each of
the phases provide a comprehensive quantitative Landau description of the
quantum critical properties of the models considered. Complementarily, the
phases with hidden orders can also be distinguished by the Pontryagin (winding)
numbers which we have calculated as well. This unified framework can be
straightforwardly applied for various spin chains and ladders, topological
insulators and superconductors. Applications to other systems are under way.
