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Scaling dimensions of higher-charge monopoles at deconfined critical points

The classical cubic dimer model has a columnar ordering transition that is continuous and described by a critical Anderson--Higgs theory containing an SU(2)-symmetric complex field minimally coupled to a noncompact U(1) gauge theory. Defects in the dimer constraints correspond to monopoles of the gauge theory, with charge determined by the deviation from unity of the dimer occupancy. By introducing such defects into Monte Carlo simulations of the dimer model at its critical point, we determine the scaling dimensions $y_2 = 1.48\pm0.07$ and $y_3 = 0.20 \pm 0.03$ for the operators corresponding to defects of charge $q=2$ and $3$ respectively. These results, which constitute the first direct determination of the scaling dimensions, shed light on the deconfined critical point of spin-1/2 quantum antiferromagnets, thought to belong to the same universality class. In particular, the positive value of $y_3$ implies that the transition in the JQ model on the honeycomb lattice is of first order.