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A Lower Bound on the Critical Momentum of an Impurity in a Bose-Einstein Condensate

Benjamin Hinrichs, Jonas Lampart

TL;DR

The paper studies a translation-invariant Bogoliubov--Fröhlich polaron in $\mathbb{R}^3$ and proves a universal lower bound $P_* \ge c$ for the critical momentum, ensuring a stable ground-state eigenstate for all $|P|<c$ despite massless phonons. It develops a rigorous UV renormalization that subtracts divergences without an ultraviolet cutoff and introduces infrared regularization to enable a compactness-based existence proof. Ground states are shown to exist for all $|P|<c$ by passing to the infrared limit $\kappa\to0$ and removing the UV regulator, via a pull-through formula and an HVZ-type analysis. The result implies an effective polaron mass $m_*$ with $P_* \approx m_* c$ and $m_* \ge 1$, consistent with phonon dressing and superfluid behavior up to the critical momentum.

Abstract

In the Bogoliubov-Fröhlich model, we prove that an impurity immersed in a Bose-Einstein condensate forms a stable quasi-particle when the total momentum is less than its mass times the speed of sound. The system thus exhibits superfluid behavior, as this quasi-particle does not experience friction. We do not assume any infrared or ultraviolet regularization of the model, which contains massless excitations and point-like interactions.

A Lower Bound on the Critical Momentum of an Impurity in a Bose-Einstein Condensate

TL;DR

The paper studies a translation-invariant Bogoliubov--Fröhlich polaron in and proves a universal lower bound for the critical momentum, ensuring a stable ground-state eigenstate for all despite massless phonons. It develops a rigorous UV renormalization that subtracts divergences without an ultraviolet cutoff and introduces infrared regularization to enable a compactness-based existence proof. Ground states are shown to exist for all by passing to the infrared limit and removing the UV regulator, via a pull-through formula and an HVZ-type analysis. The result implies an effective polaron mass with and , consistent with phonon dressing and superfluid behavior up to the critical momentum.

Abstract

In the Bogoliubov-Fröhlich model, we prove that an impurity immersed in a Bose-Einstein condensate forms a stable quasi-particle when the total momentum is less than its mass times the speed of sound. The system thus exhibits superfluid behavior, as this quasi-particle does not experience friction. We do not assume any infrared or ultraviolet regularization of the model, which contains massless excitations and point-like interactions.
Paper Structure (5 sections, 11 theorems, 53 equations)

This paper contains 5 sections, 11 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. The Bogoliubov--Fröhlich Polaron
  3. Critical Momentum
  4. Renormalization and Properties of the Bogoliubov--Fröhlich Polaron
  5. Existence of Ground States

Key Result

Proposition 1.1

There exist $(\Sigma_\Lambda)_{\Lambda\ge 0}\subset {\mathbb R}$ and, for all $P\in{\mathbb R}^3$, a selfadjoint lower-semibounded operator $H(P)$ (given in thm:ren) such that $H_\Lambda(P)-\Sigma_\Lambda$ converges to $H(P)$ as $\Lambda\to\infty$ in the norm resolvent sense.

Theorems & Definitions (24)

  • Proposition 1.1
  • proof
  • Theorem 1.2
  • proof
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1: Lampart.2020
  • proof : Sketch of the proof
  • Lemma 2.2
  • proof
  • ...and 14 more