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Effect of quartic-quintic beyond-mean-field interactions on a self-bound dipolar droplet

We study the effect of beyond-mean-field quantum-fluctuation (QF) Lee-Huang-Yang (LHY) and three-body interactions, with quartic and quintic nonlinearities, respectively, on the formation of a stable self-repulsive (positive scattering length $a$) and a self-attractive (negative $a$) self-bound dipolar Bose-Einstein condensate (BEC) droplet in free space under the action of two-body contact and dipolar interactions. Previous studies of dipolar droplets considered either the LHY interaction or the three-body interaction, as either of these interactions could stabilize a dipolar BEC droplet against collapse. We find that the effect of three-body recombination on the formation of a dipolar droplet could be quite large and for a complete description of the problem both the QF LHY and three-body interactions should be considered simultaneously, where appropriate. In the self-repulsive case for small $a$ and in the self-attractive case, no appropriate LHY interaction is known and only three-body interaction should be used, otherwise both beyond-mean-field interactions should be used. We consider a numerical solution of a highly-nonlinear beyond-mean-field model as well as a variational approximation to it in this investigation and present results for size, shape and energy of a dipolar droplet of polarized $^{164}$Dy atoms. The shape is filament-like, along the polarization direction, and could be long, for a large number of atoms $N$, short for small $N$, thin for negative $a$ and small positive $a$, and fat for large positive $a$.