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 ${\mathbb{Z}}_8$ and ${\mathbb{Z}}_4$ 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.

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• BKT transition
• Effective field theory
• Frustrated magnetism
• Ising model
• O(N) symmetry
• Quantum Monte Carlo