Quantized Lattice Dynamic Effects on the Peierls transition of the Extended Hubbard Model

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.