Heavy-light $N+1$ clusters of two-dimensional fermions

We study binding of N N identical heavy fermions by a light atom in two dimensions assuming zero-range attractive heavy-light interactions. By using the mean-field theory valid for large N N we show that the N+1 N+1 cluster is bound when the mass ratio exceeds 1.074N^2 1.074N2 . The mean-field theory, being scale invariant in two dimensions, predicts only the shapes of the clusters leaving their sizes and energies undefined. By taking into account beyond-mean-field effects we find closed-form expressions for these quantities. We also discuss differences between the Thomas-Fermi and Hartree-Fock approaches for treating the heavy fermions.