  Using a beyond-mean-field model including a Lee-Huang-Yang-type interaction,
we demonstrate a supersolid-like spatially-periodic square- and
triangular-lattice crystallization of droplets in a polarized dipolar
condensate confined by an appropriate three-dimensional (3D) box trap. In this
paper we consider a rectangular box (cuboid) trap, a square box (cuboid with
two equal sides) trap, a cylindrical box trap and a hexagonal box (hexagonal
prism) trap. The droplet lattice is always formed in the $x$-$y$ plane
perpendicular to the polarization $z$ direction of dipolar atoms. In contrast
to a harmonic trap, the box traps allow the formation of a large clean
supersolid-like spatially-periodic crystallization in free space without any
distortion. Moreover, a droplet lattice can be formed in a 3D box trap with a
significantly reduced number of atoms than in a harmonic trap, which could
facilitate the experimental observation of droplet lattice in a box trap. With
present know-how such a supersolid-like crystallization of dipolar droplets in
a 3D box trap can be realized in a laboratory thus allowing the study of a
large periodic lattice of dipolar droplets in free space bounded by rigid
walls.
