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arXiv:1709.04919 · Physical Review A

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

Seth Whitsitt, Subir Sachdev A

Published Nov 15, 2017• 6 citations
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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)]; 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 $\text{O}(2)$ 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 $\text{O}(N)$ symmetric bulk theory coupled to a spin-$S$ degree of freedom. We study this field theory using the $\ensuremath{\epsilon}=3\ensuremath{-}d$ 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.

DOI
10.1103/physreva.96.053620

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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 O(2)\text{O}(2)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)\text{O}(N)O(N) symmetric bulk theory coupled to a spin-SSS degree of freedom. We study this field theory using the \ensuremathϵ=3\ensuremath−d\ensuremath{\epsilon}=3\ensuremath{-}d\ensuremathϵ=3\ensuremath−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.
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I argue that the system of interacting bosons at zero temperature and in random external potential possesses a simple critical point that describes the proliferation of disorder-induced defects in the superfluid ground state, and that is located at weak disorder close to and above one dimension. This makes it possible to address the critical behavior at the superfluid-Bose glass transition in dirty boson systems by expanding around the lower critical dimension of d=1.d=1.d=1. Within the formulated renormalization procedure near d=1,d=1,d=1, the dynamical critical exponent is obtained exactly and the correlation length critical exponent is calculated as a Laurent series in the parameter $\sqrt{\ensuremath{\epsilon}},$ with $\ensuremath{\epsilon}=d\ensuremath{-}1:$ z=d,z=d,z=d, $\ensuremath{\nu}=1/\sqrt{3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}})$ for the short-range, and z=1,z=1,z=1, $\ensuremath{\nu}=\sqrt{2/3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}}),$ for the long-range Coulomb interaction between bosons. The identified critical point should be stable against the residual perturbations in the effective action for the superfluid, at least in dimensions 1< d< 2,1<~d<~2,1< d< 2, for both short-range and Coulomb interactions. For the superfluid-Mott insulator transition in the system in a periodic potential and at a commensurate density of bosons, I find $\ensuremath{\nu}=(\frac{1}{2}\sqrt{\ensuremath{\epsilon}})+\frac{1}{4}+O(\sqrt{\ensuremath{\epsilon}}),$ which yields a result in reasonable agreement with the known XY\mathrm{XY}XY critical exponent in d=2.d=2.d=2. The critical behavior of the superfluid density, phonon velocity, and the compressibility in the system with short-range interactions is discussed.
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