  Using extensive classical and quantum Monte Carlo simulations, we investigate
the ground-state phase diagram of the fully frustrated transverse field Ising
model on the square lattice. We show that pure columnar order develops in the
low-field phase above a surprisingly large length scale, below which an
effective U(1) symmetry is present. The same conclusion applies to the Quantum
Dimer Model with purely kinetic energy, to which the model reduces in the
zero-field limit, as well as to the stacked classical version of the model. By
contrast, the 2D classical version of the model is shown to develop plaquette
order. Semiclassical arguments show that the transition from plaquette to
columnar order is a consequence of quantum fluctuations.
