  We study the stability of zero-vorticity and vortex lattice quantum droplets
(LQDs), which are described by a two-dimensional (2D) Gross-Pitaevskii (GP)
equation with a periodic potential and Lee-Huang-Yang (LHY) term. The LQDs are
divided in two types: onsite-centered and offsitecentered LQDs, the centers of
which are located at the minimum and the maximum of the potential,
respectively. The stability areas of these two types of LQDs with different
number of sites for zerovorticity and vorticity with S = 1 are given. We found
that the u-N relationship of the stable LQDs with a fixed number of sites can
violate the Vakhitov-Kolokolov (VK) criterion, which is a necessary stability
condition for nonlinear modes with an attractive interaction. Moreover, the u-N
relationship shows that two types of vortex LQDs with the same number of sites
are degenerated, while the zero-vorticity LQDs are not degenerated. It is worth
mentioning that the offsite-centered LQDs with zero-vorticity and vortex LQDs
with S = 1 are heterogeneous.
