  We theoretically investigate a novel supersolid structure taking the form of
stacked, disk-shaped superfluid droplets connected via a dilute superfluid, in
an antidipolar condensate. A phase diagram is determined for varying the
particle number and scattering length, identifying the regions of a regular
dipolar superfluid, supersolid stacks, and isolated stacked disk-shaped
droplets in an experimentally realizable trapping potential. The collective
Bogoliubov excitation spectrum across the superfluid-supersolid phase
transition is studied, and the transition point is found to be associated with
the breaking of the degeneracy of the two lowest-lying modes. The dynamical
generation of the supersolid stacks is also investigated by ramping down the
scattering length across the phase transition. Moreover, we have studied the
impact of vortex-line penetration on the phase transition. We have found that
the presence of a vortex line causes the supersolid region to move towards
weaker contact interactions. Our detailed numerical simulations highlight that
an antidipolar condensate can create such supersolid stacks within an
experimentally reachable parameter regime.
