  The two-dimensional Su-Schrieffer-Heeger model of electrons coupled to
quantum phonons is investigated using Langevin dynamics within the framework of
auxiliary-field quantum Monte Carlo. Based on an explicit determination of the
density of zeros of the fermion determinant, it is argued that the method is
efficient in the challenging adiabatic limit. Large-scale simulations at the
O(4)-symmetric point establish that the ground state of the 2D SSH model
undergoes a transition from a $(\pi,\pi)$ valence bond solid to an
antiferromagnet with increasing phonon frequency, yet still in the adiabatic
regime. The single-particle spectrum illustrates the renormalization of the
electronic band and suggests the existence of a gapped polaronic band, whereas
the particle-hole channels show gapless modes associated with long-range bond
and magnetic order, respectively. The simulations are supplemented with a
mean-field analysis and a self-consistent Born proximation.
