Exact polaron-polaron interactions between heavy impurities in a quantum Hall fluid

We present an exact solution for effective polaron-polaron interactions between heavy impurities, mediated by a sea of noninteracting light fermions in the quantum Hall regime with highly degenerate Landau levels. For weak attraction between impurities and fermions, where only the manifold of lowest Landau levels is relevant, we obtain an analytical expression of mediated polaron-polaron interactions. Remarkably, polaron interactions are exactly zero when fermions in the lowest Landau levels outnumber heavy impurities. For strong attraction, different manifolds of higher Landau levels come into play and we derive a set of equations that can be used to numerically solve the mediated polaron interaction potential. We find that the potential vanishes when the distance <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>R</a:mi></a:math> between impurities is larger than the magnetic length, but strongly diverges at short range following a Coulomb form <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mo>−</b:mo><b:mn>1</b:mn><b:mo>/</b:mo><b:mi>R</b:mi></b:mrow></b:math>. Our exact results of polaron-polaron interactions might be examined in cold-atom setups, where a system of Fermi polarons in the quantum Hall regime is realized with a synthetic gauge field or under fast rotation. Our predictions could also be useful to understand the effective interaction between exciton-polarons in electron-doped semiconductors under strong magnetic field.