The long-range interacting Fermi polaron

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.