Weak coupling limit for the ground state energy of the 2D Fermi polaron

TL;DR

This paper rigorously derives the ground state energy of a 2D Fermi polaron in the weak coupling limit, confirming it matches the polaron energy predicted by physics, using advanced mathematical techniques.

Contribution

It provides a rigorous proof that the ground state energy converges to the polaron energy in the weak coupling limit for 2D Fermi polarons, extending previous physics predictions.

Findings

Ground state energy equals polaron energy in the weak coupling limit.

Applicable to large box sizes with fixed Fermi energy.

Introduces a localization technique for the polaron energy.

Abstract

We analyze the ground state energy for N fermions in a two-dimensional box interacting with an impurity particle via two-body point interactions. We allow for mass ratios M > 1.225 between the impurity mass and the mass of a fermion and consider arbitrarily large box sizes while keeping the Fermi energy fixed. Our main result shows that the ground state energy in the limit of weak coupling is given by the polaron energy. The polaron energy is an energy estimate based on trial states up to first order in particle-hole expansion, which was proposed by Chevy in the physics literature. For the proof we apply a Birman-Schwinger principle that was recently obtained by Griesemer and Linden. One main new ingredient is a suitable localization of the polaron energy.