  The Hubbard and Su-Schrieffer-Heeger Hamiltonians (SSH) are iconic models for
understanding the qualitative effects of electron-electron and electron-phonon
interactions respectively. In the two-dimensional square lattice Hubbard model
at half filling, the on-site Coulomb repulsion, $U$, between up and down
electrons induces antiferromagnetic (AF) order and a Mott insulating phase. On
the other hand, for the SSH model, there is an AF phase when the
electron-phonon coupling $\lambda$ is less than a critical value $\lambda_c$
and a bond order wave when $\lambda > \lambda_c$. In this work, we perform
numerical studies on the square lattice optical Su-Schrieffer-Heeger-Hubbard
Hamiltonian (SSHH), which combines both interactions. We use the determinant
quantum Monte Carlo (DQMC) method which does not suffer from the fermionic sign
problem at half filling. We map out the phase diagram and find that it exhibits
a direct first-order transition between an antiferromagnetic phase and a
bond-ordered wave as $\lambda$ increases. The AF phase is characterized by two
different regions. At smaller $\lambda$ the behavior is similar to that of the
pure Hubbard model; the other region, while maintaining long range AF order,
exhibits larger kinetic energies and double occupancy, i.e. larger quantum
fluctuations, similar to the AF phase found in the pure SSH model.
