Magnetic impurities at quantum critical points: large-$N$ expansion and SPT connections

In symmetry protected topological (SPT) phases, the combination of symmetries and a bulk gap stabilizes protected modes at surfaces or at topological defects. Understanding the fate of these modes at a quantum critical point, when the protecting symmetries are on the verge of being broken, is an outstanding problem. This interplay of topology and criticality must incorporate both the bulk dynamics of critical points, often described by nontrivial conformal field theories, and SPT physics. Here, we study the simplest nontrivial setting - that of a 0+1 dimensional topological mode - a quantum spin - coupled to a 2+1D critical bulk. Using the large-$N$ technique we solve a series of models which, as a consequence of topology, demonstrate intermediate coupling fixed points. We compare our results to previous numerical simulations and find good agreement. We also point out intriguing connections to generalized Kondo problems and Sachdev-Ye-Kitaev (SYK) models. In particular, we show that a Luttinger theorem derived for the complex SYK models, that relates the charge density to particle-hole asymmetry, also holds in our setting. These results should help stimulate further analytical study of the interplay between SPT physics and quantum criticality.