  We study one-dimensional Su-Schrieffer-Heeger (SSH) models with quantum
phonons using a continuous-time quantum Monte Carlo method. Within statistical
errors, we obtain identical results for the SSH model with acoustic phonons,
and a related model with a coupling to an optical bond phonon mode. Based on
this agreement, we first study the Peierls metal-insulator transition of the
spinless SSH model, and relate it to the Kosterlitz-Thouless transition of a
spinless Luttinger liquid. In the Peierls phase, the spectral functions reveal
the single-particle and charge gap, and a central peak related to long-range
order. For the spinful SSH model, which has a dimerized ground state for any
nonzero coupling, we reveal a symmetry-related degeneracy of spin and charge
excitations, and the expected spin and charge gaps as well as a central peak.
Finally, we study the SSH-$UV$ model with electron-phonon and electron-electron
interaction. We observe a Mott phase with critical spin and bond correlations
at weak electron-phonon coupling, and a Peierls phase with gapped spin
excitations at strong coupling. We relate our findings to the extended Hubbard
model, and discuss the physical origin of the agreement between optical and
acoustic phonons.
