  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.
