Validity of the Fröhlich model for a mobile impurity in a Bose-Einstein condensate

TL;DR

The paper addresses the validity of the Bogoliubov–Fröhlich (BF) Hamiltonian for a mobile impurity in a dilute Bose gas, establishing rigorous error bounds in the moderately strong, repulsive regime and deriving a universal logarithmic correction to the ground‑state energy from impurity‑mediated phonon interactions. Using a sequence of exact unitary transformations and careful renormalization, the authors map the full many‑body problem to the BF model and obtain a finite, renormalized energy expansion with $E_0 = \frac{1}{4m} N + \frac{2\pi}{\mu}\sqrt{\alpha} N^{1/2} - \frac{16\pi \alpha^2}{\mu}\left(\mu^{-1}\arcsin \mu - \sqrt{1-\mu^2}\right) \log N + \mathcal{O}(1)$, highlighting a universal $\log N$ term that vanishes in the heavy‑impurity limit. The excitation spectrum is shown to be well described by the BF Hamiltonian, and the analysis provides nonperturbative renormalization of short‑range physics via unitary transformations. Together, these results validate the BF description for Bose polarons in the stated regime and quantify measurable energy shifts arising from impurity‑phonon interactions. The methods combine Bogoliubov transformations, Weyl maps, and careful normal‑ordering to control divergences and connect the microscopic model to an effective, renormalized BF theory.

Abstract

We analyze the many-body Hamiltonian describing a mobile impurity immersed in a Bose-Einstein condensate (BEC). Using exact unitary transformations and rigorous error estimates, we show the validity of the Bogoliubov-Fröhlich Hamiltonian for the Bose polaron in the regime of moderately strong, repulsive interactions with a dilute BEC. Moreover, we calculate analytically the universal logarithmic correction to the ground state energy that arises from an impurity mediated phonon-phonon interaction.