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Fig. 2 for a mixture with m$_{B}$ = m$_{F}$ at several values of density imbalance. [26] All curves start in weak-coupling from the corresponding noninteracting values, and de- crease monotonically for increasing coupling due to the growing pairing correlations, which tend to deplete the zero momentum mode and distribute the bosons over a vast momentum region, as required to build the internal molecular wave-function. The critical temperature van- ishes eventually at a critical coupling, corresponding to a quantum critical point which separates a phase with a condensate from a phase where molecular correlations are so strong to deplete the condensate completely. 

Remarkably, the critical coupling value depends very weakly on the degree of density imbalance: all curves terminate in the narrow region 1 . 6 < ( k$_{F}$ a ) − 1 < 1 . 8. In this respect, it is interesting to consider the limit n$_{B}$ → 0, where the critical coupling can be calculated independently by solving the problem of a single boson immersed in a Fermi sea. This is actually the same as a spin-down fermion immersed in a Fermi sea of spin-up fermions, since for a single particle the type of statistics is immaterial. The critical coupling reduces then to that for the polaron-to-molecule transition, recently studied in the context of strongly imbalanced two-component Fermi gases [27–33]. In particular, the solution of our equations in this limit yields the critical coupling ( k$_{F}$ a ) − 1 = 1 . 60, in full agreement with the value ( k ↑ $_{F}$a ) − 1 = 1 . 27 (where k ↑ F = 2 1 / $^{3}$k$_{F}$ ) reported in [32, 33] for the polaron-to- molecule transition, when treated at the same level of ap- proximation [34, 36]. The meeting of the properties of a Bose-Fermi mixture with those of a two-component Fermi mixture at this point of the phase diagram is quite re- markable, especially because, according to our results of Fig. 1, this “universal” point sets the scale for the quan- tum phase transition for all boson densities n$_{B}$ ≤ n$_{F}$ . 1 

The chemical potentials µ$_{B}$ and µ$_{F}$ at T$_{c}$ vs. ( k$_{F}$ a ) − 

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are reported in Fig. 3, for different values of the density imbalance. The two chemical potentials behave quite dif- ferently. The fermion chemical potential remains almost constant in the whole range of coupling considered; the boson chemical potential, on the other hand, diminishes quite rapidly with increasing coupling while depending little on the density imbalance. This different behav- ior results from the concurrence of several factors. For weak-coupling, the increasing (negative) mean-field shift of the fermion chemical potential for increasing coupling is partially compensated by the decrease of the tempera- ture when moving along the critical line. On the molecu- lar side, the fermion chemical potential is instead deter- mined by the Fermi energy of the unpaired fermions plus a mean-field shift caused by interaction with molecules. Pauli repulsion makes this interaction repulsive [8, 25] thus keeping the fermion chemical potential positive. The boson chemical potential, on the other hand, interpolates between the mean-field value 2 πn$_{F}$ a/m$_{r}$ in weak cou- pling and µ$_{B}$ ≈ − ϵ 0 in strong coupling, as required by molecule formation. 

Figure 4 reports the momentum distributions n$_{B}$ ( k ) and n$_{F}$ ( k ) (as obtained before momentum integration in Eqs. (5)-(6)) at T$_{c}$ , for a mixture with n$_{B}$ = n$_{F}$ , at the coupling value ( k$_{F}$ a ) − 1 = 1 . 63 (approximately at the quantum critical point). The two distributions are markedly different at low momenta, consistently with their different statistics, but coalesce into the same be- havior just after the step in the fermion momentum distribution. This common behavior corresponds to the function n$_{M}$ | φ ( k ) | 2 (dashed line in Fig. 4), where φ ( k ) = (8 πa $^{3}$)1 / $^{2}$/ ( k $^{2}$a2 + 1) is the normalized two-body internal wave function of the molecules, while the coeffi- cient n$_{M}$ represents their density. In the present case n$_{M}$ = 0 . 89 n$_{B}$ , showing that a fraction of bosons re- mains unpaired but still does not condense even at such 

