  Antiferromagnetism (AF) such as Neel ordering is often closely related to
Coulomb interactions such as Hubbard repulsion in two-dimensional (2D) systems.
Whether Neel AF ordering in 2D can be dominantly induced by electron-phonon
couplings (EPC) has not been completely understood. Here, by employing
numerically-exact sign-problem-free quantum Monte Carlo (QMC) simulations, we
show that optical Su-Schrieffer-Heeger (SSH) phonons with frequency $\omega$
and EPC constant $\lambda$ can induce AF ordering for a wide range of phonon
frequency $\omega>\omega_c$. For $\omega<\omega_c$, a valence-bond-solid (VBS)
order appears and there is a direct quantum phase transition between VBS and AF
phases at $\omega_c$. The phonon mechanism of the AF ordering is related to the
fact that SSH phonons directly couple to electron hopping whose second-order
process can induce an effective AF spin exchange. Our results shall shed new
lights to understanding AF ordering in correlated quantum materials.
