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.
Highlights
The quantum phase transition between a superfluid and a Mott insulator in two dimensions represents one of the best studied examples of quantum critical matter, both theoretically and experimentally
Huang et al [6] recently found a novel impurity quantum critical point in their study of the Bose-Hubbard model on the square lattice. They held the bulk square lattice at the superfluid-insulator quantum critical point, and varied the strength of the trapping potential at a single site. They found a quantum phase transition, with a diverging length scale, at a critical value of the trapping potential where the impurity site occupation number jumped by unity
In an earlier study of quantum antiferromagnets with SU(2) spin rotation symmetry, Ref. 9 examined impurities in dimerized, two-dimensional antiferromagnets at the bulk critical point point between a spin-gap state and Neel order described by the O(3) Wilson-Fisher conformal field theory. They found that impurities were universally characterized by a single spin quantum number, S, which specified a renormalization group fixed point with no relevant directions in the impurity field theory
Summary
The quantum phase transition between a superfluid and a Mott insulator in two dimensions represents one of the best studied examples of quantum critical matter, both theoretically and experimentally. Despite the jump in the boson number, the transition is second order because it is associated with divergence in the size of the screening cloud They find the emergence of scale-invariant behavior for a critical value of the trapping potential, suggesting the emergence of a new universality class associated with the impurity degree of freedom. The impurities are represented by a localized spin degree of freedom which coupled to the bulk quantum field theory, and a stable interacting fixed point was found perturbatively in the = 3 − d expansion. This novel impurity-driven critical behavior led to new observables associated with the impurity degree of freedom.
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