Primary and secondary order parameters in the fully frustrated transverse-field Ising model on the square lattice

Using quantum Monte Carlo simulations and field-theory arguments, we study the fully frustrated transverse-field Ising model on the square lattice for the purpose of quantitatively relating two different order parameters to each other. We consider a ``primary'' spin order parameter and a ``secondary'' dimer order parameter, which both lead to the same phase diagram but detect Z₈ and Z₄ symmetry breaking, respectively. While at $T>0$ their scaling exponents are simply related to each other, as explained by a mapping to a height model, we show that at $T=0$ they correspond to different charge sectors of the O(2) model in $2+1$ dimensions with nontrivial exponents that are not simply related to each other. Our insights are transferrable to a broad class of Ising models whose low-energy physics involves dimer degrees of freedom, and also serve as a guide to treating primary and secondary order parameters more generally.