Analytical approach to the Bose polaron \\ via $q$-deformed Lie algebra

Abstract

We present a novel approach to the Bose polaron based on the notion of quantum groups, also known as $q$-deformed Lie algebras. In this approach, a mobile impurity can be depicted as a deformation of the Lie algebra of the bosonic creation and annihilation operators of the bath, in which the impurity is immersed. Accordingly, the Bose polaron can be described as a bath of noninteracting $q$-deformed bosons, which allows us to provide an analytical formulation of the Bose polaron at arbitrary couplings. Particularly, we derive its ground state energy in the phonon branch of the Bogoliubov dispersion and demonstrate that the previously observed transition from a repulsive to an attractive polaron occurs at the vicinity where the quantum group symmetry is broken. Furthermore, our approach has the potential to open up new avenues in polaron physics by connecting it with seemingly unrelated research topics where quantum groups play an essential role, such as anyons.