Pair Density Waves in coupled doped two-leg Ladders

Motivated by Resonant X-ray scattering experiments in cuprate ladder materials showing charge order modulation of period $\lambda=3$ and 5 at specific hole densities, we investigate models involving the electronic t-J ladders and bosonic chains coupled via screened Coulomb repulsion. Extensive density matrix renormalization group calculations applied to the ladders/chains supplemented by a self-consistent mean-field treatment of the inter-ladder/chain coupling provide quantitative estimates of the charge order for $\lambda=3,4$ and 5. As previously proposed, such patterns correspond to the emergence of pair density waves which stem from the strong electronic correlations. We comment on the existence of a $\lambda=4$ modulation not seen so far in experiment.