Magnetic impurities at quantum critical points: Large- N expansion and connections to symmetry-protected topological states

Abstract

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