Weak-coupling to unitarity crossover in Bose-Fermi mixtures: Mixing-demixing and spontaneous symmetry breaking in trapped systems
The usual treatment of a Bose-Fermi mixture relies on weak-coupling Gross-Pitaevskii (GP) and density-functional (DF) Lagrangians, often including the more realistic {perturbative} Lee-Huang-Yang (LHY) corrections. We suggest analytic {non-perturbative} beyond-mean-field Bose and Fermi Lagrangians valid along the crossover from weak- to strong-coupling limits of intra-species interactions consistent with the LHY corrections and the strong-coupling (unitarity) limit for small and large scattering lengths $|a|$, respectively, and use these to study the Bose-Fermi mixture. We study numerically mixing-demixing and spontaneous symmetry breaking in Bose-Fermi mixtures in spherically-symmetric and quasi-one-dimensional traps while the intra-species Bose and Fermi interactions are varied from weak-coupling to strong-coupling limits.The LHY correction is appropriate for medium to weak atomic interactions and diverges for stronger interactions (large scattering length $|a|$), whereas the present beyond-mean-field Lagrangian is finite in the unitarity limit ($|a|\to \infty$). We illustrate our results using the Bose-Fermi $^7$Li-$^6$Li mixture under a spherically-symmetric and a quasi-one-dimensional trap. The results obtained with the present model for density distribution of the Bose-Fermi mixture along the crossover could be qualitatively different from the usual GP-DF Lagrangian with or without LHY corrections. Specifically, we identified spontaneous symmetry breaking and demixing in the present model not found in the usual model with the same values of the parameters.