  Using quantum Monte Carlo simulations and field-theory arguments, we study
the fully frustrated (Villain) transverse-field Ising model on the square
lattice. We consider a "primary" spin order parameter and a "secondary" dimer
order parameter, which both lead to the same phase diagram but detect $Z_8$ and
$Z_4$ symmetry, respectively. The spin order scales with conventional
exponents, both in the finite temperature critical phase and at the $T = 0$
quantum critical point. The scaling of the dimer order requires more detailed
investigations of the applicable low-energy theories; the height model at $T >
0$ and the $O(2)$ model in 2+1 dimensions at $T = 0$. Relating the order
parameters to operators in these effective models, we predict the secondary
critical exponents and confirm them numerically. The relationships between the
primary and secondary order parameters have not been previously discussed in
this context and provide insight more broadly for Ising models whose low-energy
physics involves dimer degrees of freedom.
