Bose-Einstein condensation and density collapse in a weakly coupled boson-fermion mixture

We investigate the mechanical instability toward density collapse and the transition temperature of Bose-Einstein condensation (BEC) in the weakly coupled boson-fermion many-body mixture of single-component boson and fermion gases, in the case of the repulsive boson-boson and attractive boson-fermion interactions; no fermion-fermion interaction is assumed to be originated from the Pauli exclusion because of the diluteness of the mixture. The mechanical instability occurs in the bosonic gas where the induced boson-boson attraction mediated by the fermion polarization overcomes the boson-boson repulsive interaction at low temperatures. We calculate the onset temperature of the mechanical instability and the BEC transition temperature for the interaction strength and the boson-fermion number ratio. We also discuss the relation between the mechanical instability and the BEC transition using a diagrammatic method.

Figure 1

Leading diagrams contributing to the low-momentum structure of the boson self-energy at the critical point. the solid (dashed) line implies the boson (fermion) propagator.

Figure 2

Boson bubbles with an effective boson-boson interaction (solid circle) including fermion polarization.

Figure 3

Phase diagrams of the boson-fermion mixture at a fixed boson density $n_b$: the normalized BEC transition temperature (solid line), the instability boundary to density collapse (dashed line), and zeros of the effective boson-boson interaction (dotted line), for the inverse of the dimensionless boson-fermion scattering length. Here $T_0$ is the free boson BEC transition temperature, $1/a_{bb}{n}_b^{1/3}=10$ (top row) and 100 (bottom row), and $m_f/m_b=1$. From left to right $n_f/n_b=2,1,0.5$.