  We describe the properties of a mixture of fermionic and bosonic atoms, as
they are tuned across a Feshbach resonance associated with a fermionic
molecular state. Provided the number of fermionic atoms exceeds the number of
bosonic atoms, we argue that there is a critical detuning at which the
Bose-Einstein condensate (BEC) is completely depleted. The phases on either
side of this quantum phase transition can also be distinguished by the distinct
Luttinger constraints on their Fermi surfaces. In both phases, the total volume
enclosed by all Fermi surfaces is constrained by the total number of fermions.
However, in the phase without the BEC, which has two Fermi surfaces, there is a
_second_ Luttinger constraint: the volume enclosed by one of the Fermi surfaces
is constrained by the total number of_bosons_, so that the volumes enclosed by
the two Fermi surfaces are separately conserved. The phase with the BEC may
have one or two Fermi surfaces, but only their total volume is conserved. We
obtain the phase diagram as a function of atomic parameters and temperature,
and describe critical fluctuations in the vicinity of all transitions. We make
quantitative predictions valid for the case of a narrow Feshbach resonance, but
we expect the qualitative features we describe to be more generally applicable.
As an aside, we point out intriguing connections between the BEC depletion
transition and the transition to the fractionalized Fermi liquid in Kondo
lattice models.
