Analytic Bounds on Conformal Defect Spectra
Conformal defects describe impurities and boundaries at criticality. Their stability depends on the scaling dimensions of operators localized on the defects, but these dimensions are often difficult to determine. We derive analytic upper bounds on the lowest nonvacuum scaling dimension using only two defect entropies and the dimension of a defect-changing operator. The bounds identify when relevant operators must appear on the defect with larger defect entropy. They follow from the conformal bootstrap of four-point functions of defect-changing operators and apply to unitary conformal line defects and to conformal boundaries in two-dimensional CFTs. The Ising and Potts models provide exact checks. In three-dimensional $O(N)$ models, we apply the bounds to direct sums of pinning field defects with symmetry-related orientations. Data from a single such defect exclude stable impurity order with three or more orientations in $O(2)$ and $O(3)$, or four or more in $O(4)$, for any geometry of the orientations, and require a relevant domain-wall operator between individually stable pinning field defects at every nonzero relative angle up to $120^\circ$ in $O(3)$ and up to $109.47^\circ$ in $O(4)$. A direct fuzzy-sphere computation of the lowest domain-wall dimension at different relative angles for $N=2,3,4$ is consistent with these bounds and indicates that, within this class of impurities, only two-state order remains a candidate for stable order in $O(3)$ and $O(4)$, while none survives in $O(2)$. A separate application to AdS/BCFT shows that, at fixed unequal brane tensions, a light boundary-condition-changing operator requires additional light primaries on the boundary with larger boundary entropy.