Quantum phase transitions in trimerized zig-zag spin ladders

We analyze the effects of a trimerized modulation in a quantum spin $S=\frac12$ zig-zag ladder at the magnetization plateau $M=1/3$. Such periodicity is argued to be stemmed from lattice deformations by phonons. The interplay between frustration and exchange modulation is well described by an effective triple sine-Gordon field theory close to the homogeneous ladder and by block-spin perturbation theory in the weakly coupled trimers regime. The characteristic triple degeneracy of the ground state for homogeneous ladders gives place to modulation driven quantum phase transitions, leading to a rich phase diagram including up-up-down, quantum plateau and gapless plateau states.