 0  0.2  0.4  0.6  0.8  1  0.5  1  1.5  2  2.5  3  3.5  4 n s (k) k/k$_{F}$ 0.9 0.95  1 1.6 1.9 2.2 2.5 n M /n B (k$_{F}$ a) -1 -6 -4 -2  0 -1 -0.5  0  0.5  1  1.5 B /E F (k$_{F}$a) -1 0 0.25 0.5 0.75  0.5  1  1.5 -1  0 1  2 FIG. 4: Boson (full line) and fermion (dashed-dotted) mo- menta distribution curves for n B = n F , m B = m F , ( k F a ) $_{−}$1 = 1 . 63 and T = T c ≃ 0 . 015 E F . See text for the meaning of the dashed curve. The inset reports the fraction of molecules vs ( k F a ) $_{−}$1 for the same mixture at T = 0 . 1 E F . FIG. 3: Boson chemical potential at the critical temperature T c (in units of E F ) vs the boson-fermion coupling ( k F a ) $_{−}$1 for different values of the density imbalance ( n F − n B ) /n in a mixture with m B = m F . The corresponding fermion chemical potential is reported in the inset. 3 Fig. 2 for a mixture with m$_{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. 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. 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$_{F}$ . 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 The chemical potentials µ$_{B}$ and µ$_{F}$ at T$_{c}$ vs. ( k$_{F}$ a ) − 1