  We study a Su-Schrieffer-Heeger electron-phonon model on a square lattice by
means of auxiliary-field quantum Monte Carlo simulations. The addition of a
symmetry-allowed interaction permits analytical integration over the phonons at
the expense of discrete Hubbard-Stratonovich fields with imaginary-time
correlations. Using single-spin-flip and global updates, we investigate the
phase diagram at the O(4)-symmetric point as a function of hopping $t$ and
phonon frequency $\omega_0$. For $t=0$, where electron hopping is boson
assisted, the model maps onto an unconstrained $\mathbb{Z}_2$ gauge theory. A
key quantity is the emergent effective flux per plaquette, which equals $\pi$
in the assisted-hopping regime and vanishes for large $t$. Phases in the former
regime can be understood in terms of instabilities of emergent Dirac fermions.
Our results support a direct and continuous transition between a $(\pi,0)$
valence bond solid (VBS) and an antiferromagnetic (AFM) phase with increasing
$\omega_0$. For large $t$ and small $\omega_0$, we find finite-temperature
signatures, a disordered pseudogap phase, of a previously reported $(\pi,\pi)$
VBS ground state related to a nesting instability. With increasing $\omega_0$,
AFM order again emerges.
