  Over the last several years, a new generation of quantum simulations has
greatly expanded our understanding of charge density wave phase transitions in
Hamiltonians with coupling between local phonon modes and the on-site charge
density. A quite different, and interesting, case is one in which the phonons
live on the bonds, and hence modulate the electron hopping. This situation,
described by the Su-Schrieffer-Heeger (SSH) Hamiltonian, has so far only been
studied with quantum Monte Carlo in one dimension. Here we present results for
the 2D SSH model, and show that a bond ordered wave (BOW) insulator is present
in the ground state at half-filling, and argue that a critical value of the
electron-phonon coupling is required for its onset, in contradistinction with
the 1D case where BOW exists for any nonzero coupling. We determine the precise
nature of the bond ordering pattern, which has hitherto been controversial, and
the critical transition temperature, which is associated with a spontaneous
breaking of ${\cal Z}_4$ symmetry.
