Phase separation and collapse in Bose-Fermi mixtures with a Feshbach resonance

We consider a mixture of single-component bosonic and fermionic atoms with an interspecies interaction that is varied using a Feshbach resonance. By performing a mean-field analysis of a two-channel model, which describes both narrow and broad Feshbach resonances, we find an unexpectedly rich phase diagram at zero temperature: Bose-condensed and non-Bose-condensed phases form a variety of phase-separated states that are accompanied by both critical and tricritical points. We discuss the implications of our results for the experimentally observed collapse of Bose-Fermi mixtures on the attractive side of the Feshbach resonance, and we make predictions for future experiments on Bose-Fermi mixtures close to a Feshbach resonance.

Figure 1

(Color online) Surface of first-order phase transitions in the three-dimensional (3D) space $\left\{{\mu }_f^{\left(r\right)},{\nu }^{\left(r\right)},{\mu }_b^{\left(r\right)}\right\}\equiv c^{1/3}\left\{c^{1/3}{\mu }_f/{\gamma }^2,c^{1/3}\left(\nu -{\mu }_b\right)/{\gamma }^2,{\mu }_b/{\gamma }^2\right\}$, where $c=\lambda m_b^{2/3}\gamma$, that has been projected onto the $\left({\nu }^{\left(r\right)},{\mu }_b^{\left(r\right)}\right)$ plane. The gray shaded area corresponds to the first-order region. For $\left|{\nu }^{\left(r\right)}\right|$ greater than the tetracritical points $O\left({\Phi }^2\right)=O\left({\Phi }^4\right)=O\left({\Phi }^6\right)=0$ (open circles), we have lines of tricritical points represented by the (blue) filled circles. Otherwise, they are replaced by critical points [(red) filled squares]. The straight lines correspond to the cuts at fixed $\nu /{\gamma }^2$ and $c$ in Fig. 2, where we have $c\simeq 0.044$ and $\nu /{\gamma }^2=-80$ (dotted-dashed), 0 (solid), and 70 (dashed) going from left to right.

Figure 2

(Color online) Zero-temperature mean-field phase diagrams for ${}^{23}\text{N}\text{a-}{}^6\text{L}\text{i}$ mixtures, where $c=\lambda m_b^{3/2}\gamma \simeq 0.044$. The top and bottom rows correspond to chemical potential and density space, respectively, while the columns represent different detunings: $\nu /{\gamma }^2=-80$ $\left(a_{\text{BF}}\simeq -475a_0\right)$, $\nu /{\gamma }^2=0$, and $\nu /{\gamma }^2=70$ $\left(a_{\text{BF}}\simeq 543a_0\right)$, going from left to right. The various phases can be distinguished based on the number of Fermi surfaces and whether or not there is a BEC (light gray shaded region). The solid thick (red) lines represent first-order phase transitions, which are accompanied by regions of phase separation (dark gray shaded region) in density space, while the thin solid (black) lines of the phase boundaries are continuous. The dotted-dashed lines separate regions with a different number of Fermi surfaces. The dotted lines that connect points on the first-order boundary depict which phases constitute the phase-separated state at a given density. A filled (blue) circle denotes a tricritical point, a filled (red) square denotes a critical point, and a dashed (blue) line denotes a spinodal line.

Figure 3

(Color online) Trap density profiles at $\nu =0$ for boson number $n_b$, fermion number $n_f$, and condensed boson number $n_{\text{cond}}\equiv {\Phi }^2/g^2$ in units of $m^{3/2}{\gamma }^3$, where $\lambda m_b^{3/2}\gamma \simeq 0.044$. The profiles correspond to three different cuts (a), (b), and (c) across the chemical-potential phase diagram as shown in the top panel, where the gray arrows represent trajectories from the center to the edge of the trap. In all three cases $r=r_0$ has been fixed as the point at which the first-order transition occurs.