Collapse in Mixture of Two Component Fermi Gases

Trapped Fermi alkali gases are spin polarized and cannot interact in the s-wave channel. A two component system realized by mixtures of two Fermi alkali gases, say $^6$Li and $^{40}$K, interacts in an s-wave channel and can be viewed as a Fermi gas with effective scattering length $a_{LK}$. Our consideration based on the thermodynamic properties gives strong evidence that the compressibility becomes negative as soon as $a_{LK}$ is negative and sufficiently large. As a result, the system collapses when $p_F|a_{LK}|\sim1$, where $p_F$ is the Fermi momentum. In the regime of large particle numbers in mixtures such as in traps, the Fermi momentum can be estimated as $p_F\simeq (3\pi^2\rho)^{1/3}$ with $\rho$ the average number density of the mixture. Our prediction [M.Ya. Amusia, A.Z. Msezane, and V.R. Shaginyan, Phys. Lett. A{\bf 293}, 205 (2002)] that two component systems composed of Fermi and Bose gases, which retain the significant features of the considered two component Fermi system, collapse is in good agreement with recent facts [C. Ospelkaus {\it et al.,} Phys. Rev. Lett. {\bf 96}, 020401 (2006)]. Our results suggest that the equation of state of a low density neutron matter has peculiarities at $|a_{nn}|p_F\sim 1$ ($a_{nn}$ is the neutron-neutron scattering length), which can lead to the equilibrium state of neutron matter.