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 the Landau-Pekar functional known from the Bose polaron problem, and in its many-image extension in other dimensions. We discuss applications of our theory for the two-dimensional exciton-polaron and the ionic polaron problem and compute the effective mass for these cases, finding a self-trapping transition with order dependent on the dimensionality.