  We present a density-matrix embedding theory (DMET) study of the
one-dimensional Hubbard-Holstein model, which is paradigmatic for the interplay
of electron-electron and electron-phonon interactions. Analyzing the
single-particle excitation gap, we find a direct Peierls insulator to Mott
insulator phase transition in the adiabatic regime of slow phonons in contrast
to a rather large intervening metallic phase in the anti-adiabatic regime of
fast phonons. We benchmark the DMET results for both on-site energies and
excitation gaps against density-matrix renormalization group (DMRG) results and
find excellent agreement of the resulting phase boundaries. We also compare the
fully quantum treatment of phonons against the standard Born-Oppenheimer (BO)
approximation. The BO approximation gives qualitatively similar results to DMET
in the adiabatic regime, but fails entirely in the anti-adiabatic regime, where
BO predicts a sharp direct transition from Mott to Peierls insulator, whereas
DMET correctly shows a large intervening metallic phase. This highlights the
importance of quantum fluctuations in the phononic degrees of freedom for
metallicity in the one-dimensional Hubbard-Holstein model.
