  The density matrix renormalization group method is used to investigate the
Peierls transition for the extended Hubbard model coupled to quantized phonons.
Following our earlier work on spin-Peierls systems, we use a phonon spectrum
that interpolates between a gapped, dispersionless (Einstein) limit to a
gapless, dispersive (Debye) limit to investigate the entire frequency range. A
variety of theoretical probes are used to determine the quantum phase
transition, including energy gap crossing, a finite size scaling analysis, and
bipartite quantum entanglement. All these probes indicate that a transition of
Berezinskii-Kosterlitz-Thouless-type is observed at a non-zero electron-phonon
coupling, $g_{\text c}$, for a non-vanishing electron-electron interaction. An
extrapolation from the Einstein limit to the Debye limit is accompanied by an
increase in $g_{\text c}$ for a fixed optical ($q=\pi $) phonon gap. We
therefore conclude that the dimerized ground state is more unstable with
respect to Debye phonons, with the introduction of phonon dispersion
renormalizing the effective electron-lattice coupling for the Peierls-active
mode. By varying the Coulomb interaction, $U$, we observe a generalized Peierls
transition, intermediate to the uncorrelated ($U=0$) and spin-Peierls
($U\to\infty$) limits, where $U$ is the Hubbard Coulomb parameter. Using the
extended Hubbard model with Debye phonons, we investigate the Peierls
transition in \textit{trans}-polyacetylene and show that the transition is
close to the critical regime.
