Criticality on R\'enyi defects at (2+1)$d$ O(3) quantum critical points

At a quantum critical point, the universal scaling behavior of R\'enyi entanglement entropy is controlled by the universality class of the codimension-two R\'enyi (or conical) defects in the infrared theory. In this work we perform a systematic study of critical correlations along R\'enyi defect lines in (2+1)d quantum spin models realizing quantum phase transitions described by the O(3) Wilson-Fisher universality class, using large-scale quantum Monte Carlo simulations. We present numerical evidence that, for a fixed R\'enyi index $n$, there exist multiple R\'enyi defect universality classes, with distinct critical exponents for the O(3) order parameter on the defect. These universality classes are realized by choosing microscopically different entanglement cuts in lattice models, which we classify as ordinary, special and extraordinary according to their relation to surface criticality. For the extraordinary entanglement cut, we further find evidence for a phase transition on the defect as a function of the R\'enyi index. Our results highlight the key role of defect universality classes in determining the universal scaling of R\'enyi entropy, and provide a framework for understanding the previously observed dependence of R\'enyi entropy scaling on microscopic lattice details.