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

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)]; they found an impurity phase transition as a function of the trapping potential, while the bulk was critical. The bulk theory is described by the $\text{O}\left(2\right)$ 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 $\text{O}\left(N\right)$ 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 numerical simulations.

Figure 1

The diagrammatic expansion for the spin-spin correlation function at one loop, using the Feynman rules specified in Sec. 3. The diagrams contributing to the numerator and denominator of the correlation function are pictured in (a) and (b), respectively. As described in the main text, the integrals contributing to the numerator and denominator can be combined, so we only need to keep track of differing spin traces.

Figure 2

The diagrams which renormalize the impurity interaction $\gamma$.

Figure 3

The diagrams contributing to the numerator of the two-point function at two loops.

Figure 4

The diagrams contributing to the denominator of the two-point function at two loops.