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

TL;DR

This paper develops a universal field theory describing the critical behavior of an impurity at the superfluid-Mott insulator transition in two dimensions, providing analytical predictions for impurity scaling and thermodynamic contributions.

Contribution

It introduces a field-theoretic model for impurity criticality at the transition, extending to O(N) symmetry and calculating impurity scaling dimensions and universal thermodynamic effects.

Findings

Impurity scaling dimensions are computed near the critical point.

Universal impurity contribution to finite temperature compressibility is determined.

Results agree with recent numerical simulations.

Abstract

We present a universal theory for the critical behavior of an impurity at the two-dimensional superfluid-Mott insulator transition. Our analysis is motivated by a numerical study of the Bose-Hubbard model with an impurity site by Huang et al. (Phys. Rev. B 94, 220502 (2016)), who found an impurity phase transition as a function of the trapping potential. The bulk theory is described by the $O(2)$ symmetric Wilson-Fisher conformal field theory, and we model the impurity by a localized spin-1/2 degree of freedom. We also consider a generalized model by considering an $O(N)$ symmetric bulk theory coupled to a spin-$S$ degree of freedom. We study this field theory using the $\epsilon = 3 - d$ expansion, where the impurity-bulk interaction flows to an infrared stable fixed point at the critical trapping potential. We determine the scaling dimensions of the impurity degree of freedom and the associated critical exponents near the critical point. We also determine the universal contribution of the impurity to the finite temperature compressibility of the system at criticality. Our results are compared with recent numerical simulations.