  We use quantum Monte Carlo methods and single-mode approximation to study the
magnon dispersion in the 2D half-filled Hubbard and phonon-coupled Heisenberg
models. We find that in the Hubbard model with $U/t< 8$, high-energy magnon
dispersion is similar to those observed in inelastic neutron scattering
experiments in ${La}_2{CuO}_4$. On the other hand, our studies of a 2D
Heisenberg model coupled to dynamic optical bond phonons, fails to reproduce
the experimental dispersion. These results can be interpreted as evidence for
intermediate $U/t$ and charge fluctuations in the cuprate materials.
