  The quantum phase transition between a repulsive Luttinger liquid and an
insulating Peierls state is studied in the framework of the one-dimensional
spinless Holstein model. We focus on the adiabatic regime but include the full
quantum dynamics of the phonons. Using continuous-time quantum Monte Carlo
simulations, we track in particular the dynamic charge structure factor and the
single-particle spectrum across the transition. With increasing electron-phonon
coupling, the dynamic charge structure factor reveals the emergence of a charge
gap, and a clear signature of phonon softening at the zone boundary. The
single-particle spectral function evolves continuously across the transition.
Hybridization of the charge and phonon modes of the Luttinger liquid
description leads to two modes, one of which corresponds to the coherent
polaron band. This band acquires a gap upon entering the Peierls phase, whereas
the other mode constitutes the incoherent, high-energy spectrum with backfolded
shadow bands. Coherent polaronic motion is a direct consequence of quantum
lattice fluctuations. In the strong-coupling regime, the spectrum is described
by the static, mean-field limit. Importantly, whereas finite electron density
in general leads to screening of polaron effects, the latter reappear at half
filling due to charge ordering and lattice dimerization.
