High Density Limit of the Fermi Polaron with Infinite Mass

TL;DR

This paper rigorously proves that the ground state energy of a two-dimensional Fermi gas with an impurity in the high-density limit converges to the polaron energy, confirming a key approximation used in physics.

Contribution

It provides a rigorous mathematical proof that the high-density limit of the Fermi polaron energy matches the polaron energy estimate from physics literature.

Findings

Ground state energy converges to polaron energy at high density

No fixed spectral gap required for the Laplacian

Uses Birman-Schwinger type argument for proof

Abstract

We analyze the ground state energy for $N$ identical fermions in a two-dimensional box of volume $L^2$ interacting with an external point scatterer. Since the point scatterer can be considered as an impurity particle of infinite mass, this system is a limit case of the Fermi polaron. We prove that its ground state energy in the limit of high density $N/L^2 \gg 1$ 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 F. Chevy in the physics literature. The relative error in our result is shown to be small uniformly in $L$. Hence, we do not require a gap of fixed size in the spectrum of the Laplacian on the box. The strategy of our proof relies on a twofold Birman-Schwinger type argument applied to the many-particle Hamiltonian of the system.