Quantum groups as hidden symmetries of quantum impurities

TL;DR

This paper introduces a novel approach using quantum groups to describe and analyze quantum impurities, revealing hidden symmetries and simplifying complex many-body interactions through a single deformation parameter.

Contribution

The paper demonstrates how quantum groups can encode many-particle interactions via a deformation parameter, providing a new framework for understanding impurity systems.

Findings

Deformation parameter $q$ captures all interaction information.

Predicts system behavior across coupling regimes.

Identifies conditions for stable quasiparticle formation.

Abstract

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 non-perturbative calculations. Furthermore, the value of the deformation parameter allows 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.