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Nature of ground states in one-dimensional electron-phonon Hubbard models at half-filling

The renormalization group technique is applied to one-dimensional electron-phonon Hubbard models at half-filling and zero temperature. For the Holstein-Hubbard model, the results of one-loop calculations are congruent with the phase diagram obtained by quantum Monte Carlo simulations in the $(U,g_{\rm ph})$ plane for the phonon-mediated interaction $g_{\rm ph}$ and the Coulomb interaction $U$. The incursion of an intermediate phase between a fully gapped charge-density-wave state and a Mott antiferromagnet is supported along with the growth of its size with the molecular phonon frequency $ω_0$. We find additional phases enfolding the base boundary of the intermediate phase. A Luttinger liquid line is found below some critical $ U^*\approx g^*_{\rm ph}$, followed at larger $U\sim g_{\rm ph}$ by a narrow region of bond-order-wave ordering which is either charge or spin gapped depending on $U$. For the Peierls-Hubbard model, the region of the $(U,g_{\rm ph})$ plane with a fully gapped Peierls-bond-order-wave state shows a growing domination over the Mott gapped antiferromagnet as the Debye frequency $ω_D$ decreases. A power law dependence $g_{\rm ph} \sim U^{2η}$ is found to map out the boundary between the two phases, whose exponent is in good agreement with the existing quantum Monte Carlo simulations performed when a finite nearest-neighbor repulsion term $V$ is added to the Hubbard interaction.