Hidden-vortex quantum droplets in an optical lattice

We explore discrete quantum droplets with hidden -vortex structures in a deep two-dimensional optical lattice containing a Bose -Bose mixture. Using the extended Gross-Pitaevskii equation, and considering local interactions and Lee -Huang -Yang corrections, we affirm the stable existence of these droplets, which are identified by identical density patterns and opposing vortices in both components. We extensively study the droplets with lower topological charges, focusing on their morphological changes. Despite the expansion of internal boundaries within the vortex mode with increased charge, the lattice symmetry and phase change limit the boundary growth, leading to a unique mode pattern degeneracy exclusive to discrete systems. By fixing the norm of the states (representing the particle number of the system), we identify multiple degenerate modes with topological charges up to 5, plotting their chemical potential curves and providing a theoretical explanation for the energy changes linked to topological charge. Our simulations confirm these outcomes, validating our discoveries.