  We study theoretically the phase diagram of strongly-coupled two-dimensional
Bose-Fermi mixtures interacting with attractive short-range potentials as a
function of the particle densities. We focus on the limit where the size of the
bound state between a boson and a fermion is small compared to the average
inter-boson separation and develop a functional renormalization group approach
that accounts for the bound-state physics arising from the extended Fr\"ohlich
Hamiltonian. 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 predict frequency- and momentum-resolved
spectral functions and study the impact of three-body correlations on
quasiparticle properties. At finite boson density, we find that when the bound
state energy exceeds the Fermi energy by a critical value, the fermions and
bosons can form a fermionic composite with a well-defined Fermi surface. These
composites constitute 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 increased further, 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. We highlight the universal connection between two-dimensional
semiconductors and ultracold atoms and we discuss perspectives for further
exploring the rich structure of strongly-coupled Bose-Fermi mixtures in these
complementary systems.
