This manuscript theoretically investigates the possibility of self-bound Bose-Fermi droplets. It begins with an infinite uniform system analysis then moves to the finite self-bound droplet. Later, dynamic simulations are considered to test droplet stability, and droplet preparation. I do not recommend this manuscript for publication because the methods employed are inappropriate to describe the system studied.
Self-bound droplets typically support only a small number of bound states, usually a tiny percentage of the total atom number, see Petrov, PRL 115, 155302 (2015), Wenzel et al., Physica Scripta 93, 104004 (2018), and Baillie et al., PRL 119, 255302 (2017). While this is not a problem for the bosons, only one fermion can occupy each bound state. If the number of fermions exceeds the number of bound states then the excess fermions must be unbound, which will affect the stability of the entire droplet. The model in the present manuscript doesn’t account for this crucial physics, nor do they consider how the ratio of fermion number to bound state number behaves in the thermodynamic limit. Therefore, the authors cannot claim that their results are physically relevant for either the finite or uniform systems. Such considerations must be a central focus for any paper considering the self-bound nature of Bose-Fermi droplets, as was demonstrated in the paper on Bose-Fermi droplets with dipolar interactions by the Pfau group: Physica Scripta 93, 104004 (2018).
Furthermore, for the section on finite systems, the small numbers of fermions expected (typically 10s of bound states) should instead be treated as discrete modes, as was done in the paper by the Pfau group: Physica Scripta 93, 104004 (2018). Even if the fermion number was large, any continuous treatment (e.g. local density approximation) of the fermions would require a high energy cutoff (at zero energy) so that only the bound fermion states are included within the droplet.
Another significant problem is that Eq. (10) is inappropriate to describe the dynamics of fermions. Equation (10) attempts to describe all fermions within a single pseudo wavefunction. However, it is well-known that this is unphysical for describing dynamics. What would be the physical interpretation of the ‘phase’ of the fermion wavefunction? The unphysical effects on the dynamics from assigning a ‘phase’ to a single wavefunction to describe many fermions has already been clearly addressed in the literature. For example, the paper by Girardeau et al. PRL 84, 5239 (2000), which considers fermionized bosons, focusses quite a lot on this point. For another example see Kunal et al. PRA 65, 063603 (2002).