  The interplay between electron-electron and electron-phonon interactions is
studied in a one-dimensional lattice model, by means of a variational Monte
Carlo method based on generalized Jastrow-Slater wave functions. Here, the
fermionic part is constructed by a pair-product state, which explicitly depends
on the phonon configuration, thus including the electron-phonon coupling in a
backflow-inspired way. We report the results for the Hubbard model in presence
of the Su-Schrieffer-Heeger coupling to optical phonons, both at half-filling
and upon hole doping. At half-filling, the ground state is either a
translationally invariant Mott insulator, with gapless spin excitations, or a
Peierls insulator, which breaks translations and has fully gapped excitations.
Away from half-filling, the charge gap closes in both Mott and Peierls
insulators, turning the former into a conventional Luttinger liquid (gapless in
all excitation channels). The latter, instead, retains a finite spin gap that
closes only above a threshold value of the doping. Even though consistent with
the general theory of interacting electrons in one dimension, the existence of
such a phase (with gapless charge but gapped spin excitations) has never been
demonstrated in a model with repulsive interaction and with only two Fermi
points. Since the spin-gapped metal represents the one-dimensional counterpart
of a superconductor, our results furnish evidence that a true off-diagonal
long-range order may exist in the two-dimensional case.
