Functional renormalization group approach to strongly-coupled Bose-Fermi mixtures in two dimensions

We study theoretically the phase diagram of strongly-coupled two-dimensional Bose-Fermi mixtures that interact with attractive short-range potentials as a function of the boson and fermion densities. We focus on the limit where the bound state size is small compared to the average distance between bosons. To approach the problem we develop a functional renormalization group approach that accounts for the bound-state physics arising from the extended Fröhlich Hamiltonian and by including three-body correlations we are able to reproduce the polaron-to-molecule transition in two-dimensional Fermi gases in the extreme limit of vanishing boson density. We then extend our framework to describe Bose-Fermi mixtures at finite boson density. We find that when the bound state energy exceeds the Fermi energy, the fermions and bosons can form a fermionic composite with a well-defined Fermi surface constituting a Fermi sea of dressed Feshbach molecules in the case of ultracold atoms while in the case of atomically thin semiconductors a trion liquid emerges. As the boson density is further increased, the effective energy gap of the composites decreases, leading to a transition into a strongly-correlated phase where polarons are hybridized with molecular degrees of freedom.