I will discuss the quantum criticality in dimerized two- and three-leg anti-ferromagnetic spin -1/2  ladders with different in-ladder dimerization patterns. The results of both the mean-field theory and the exact diagonalization technique will be presented.

It is shown that the existence or absence of the quantum critical phase transition between gapped phases is dependent on the dimerization pattern. A columnar arrangement of the strong and weak couplings on the legs is never critical, whereas a staggered configuration possesses a quantum critical point. These gapped phases cannot be distinguished by the local Landau long-range order parameter. However, they possess a non-local string order parameter, which is non-zero in one phase and vanishes in the other. We use various techniques to calculate ground state energies, energy gaps, string order parameters, and to yield estimates of the critical exponents nu and beta.

Based on these results, a qualitative analysis of the hidden orders and symmetries of the Hamiltonian will be given. The broader implications of the hidden string order for the exotic quantum and some well-known classical phases will also be discussed.