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We present a universal theory for the critical behavior of an impurity at thetwo-dimensional superfluid-Mott insulator transition. Our analysis is motivatedby a numerical study of the Bose-Hubbard model with an impurity site by Huanget al. (Phys. Rev. B 94, 220502 (2016)), who found an impurity phase transitionas a function of the trapping potential. The bulk theory is described by the$O(2)$ symmetric Wilson-Fisher conformal field theory, and we model theimpurity by a localized spin-1/2 degree of freedom. We also consider ageneralized model by considering an $O(N)$ symmetric bulk theory coupled to aspin-$S$ degree of freedom. We study this field theory using the $\epsilon = 3- d$ expansion, where the impurity-bulk interaction flows to an infrared stablefixed point at the critical trapping potential. We determine the scalingdimensions of the impurity degree of freedom and the associated criticalexponents near the critical point. We also determine the universal contributionof the impurity to the finite temperature compressibility of the system atcriticality. Our results are compared with recent numerical simulations.

Critical behavior of an impurity at the boson superfluid-Mott insulator transition