The long-range interacting Fermi polaron

TL;DR

This paper develops a density functional approach to the Fermi polaron problem with long-range interactions, revealing bosonization in 2D and analyzing self-trapping transitions across dimensions.

Contribution

It introduces a simple density functional model for long-range Fermi polarons, showing bosonization in 2D and exploring dimensional effects on self-trapping.

Findings

Fermi polaron is fully bosonized in 2D, described by a Landau-Pekar functional.

In other dimensions, the impurity interacts with infinite images, preventing bosonization.

Identifies a self-trapping transition with dimensionality-dependent order.

Abstract

We construct the simplest density functional for the problem of a single impurity interacting with a Fermi gas via a long--ranged potential using the Thomas--Fermi approach. We find that the Fermi polaron is fully bosonized in two dimensions, as the model results in a suitable Landau--Pekar functional known from the Bose polaron problem which describes a self--interacting impurity. In other dimensions, the impurity self--interacts with an infinite number of its own images, and no bosonization occurs. We discuss applications of our theory for the $2d$ exciton--polaron and the ionic polaron problem and compute the effective mass for these cases, finding a self--trapping transition with order depending on the dimensionality.