  We study a quantum Bose-Fermi mixture near a broad Feshbach resonance at zero
temperature. Within a quantum field theoretical model a two-step Gaussian
approximation allows to capture the main features of the quantum phase diagram.
We show that a repulsive boson-boson interaction is necessary for thermodynamic
stability. The quantum phase diagram is mapped in chemical potential and
density space, and both first and second order quantum phase transitions are
found. We discuss typical characteristics of the first order transition, such
as hysteresis or a droplet formation of the condensate which may be searched
for experimentally.
