Quantum Groups as Hidden Symmetries of Quantum Impurities

We present an approach to interacting quantum many-body systems based on the notion of quantum groups, also known as $q$-deformed Lie algebras. In particular, we show that, if the symmetry of a free quantum particle corresponds to a Lie group $G$, in the presence of a many-body environment this particle can be described by a deformed group, $G_q$. Crucially, the single deformation parameter, $q$, contains all the information about the many-particle interactions in the system. We exemplify our approach by considering a quantum rotor interacting with a bath of bosons, and demonstrate that extracting the value of $q$ from closed-form solutions in the perturbative regime allows one to predict the behavior of the system for arbitrary values of the impurity-bath coupling strength, in good agreement with nonperturbative calculations. Furthermore, the value of the deformation parameter allows one to predict at which coupling strengths rotor-bath interactions result in a formation of a stable quasiparticle. The approach based on quantum groups does not only allow for a drastic simplification of impurity problems, but also provides valuable insights into hidden symmetries of interacting many-particle systems.

Figure 1

Comparison of the renormalized rotational constant, $B^{*}/B$, obtained within the quantum group (QG) approach and using the variational method of Ref. [19] (a) as a function of dimensionless bath density, $\tilde{n}$, for the parameters of Ref. [18], and (b) as a function of dimensionless rotational constant $\tilde{B}$ for the parameters of Ref. [19]. (c) The comparison with the diagrammatic Monte Carlo (DiagMC) approach as a function of $\tilde{n}$ for the parameters used in Ref. [26]. See the text.

Figure 2

(a) The deformation parameter, $\tau$, as a function of the dimensionless bath density, $\tilde{n}$. (b) The spectral function of the angulon quasiparticle as a function of the dimensionless energy, $\tilde{E}=E/B$, and $\tilde{n}$, obtained using the variational approach of Ref. [19]. The blue sharp peaks correspond to quasiparticle states, whereas yellowish blurred peaks show angulon instabilities, which match with the domain where $\tau$ is imaginary. The model parameters are the same as in Ref. [26]. See the text.