Using a controlled analytic non-perturbative treatment, that accounts for the quantum nature of the phonons, we derive a model that generically describes cooperative breathing-mode at strong electron-phonon interaction in one-band one-dimensional systems. The effective model involves a {\em next-nearest-neighbor} hopping (that dominates over the nearest-neighbor hopping at strong coupling) and a nearest-neighbor repulsion that is significantly enhanced due to incompatibility of neighboring dilations/compressions. At non-half filling, upon tuning the electron-phonon coupling, the system undergoes a period-doubling second-order quantum phase transition from a Luttinger liquid to a {\em conducting commensurate} charge-density-wave state: a phenomenon absent in both the Holstein model and the t-V model. Using fidelity to study the nature of the quantum phase transition, we find that the fidelity susceptibility shows a superextensive power law divergence as well as a remarkable scaling behavior: both together establish a second-order transition.

Added new Section IV and revised Appendix B