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We analyze the existence and characteristics of discrete vortex quantum droplets with intrinsic topological charges S of up to 5 in two-component Bose-Einstein condensates in a deep two-dimensional (2D) optical lattice. The 2D discrete Gross-Pitaevskii equation with a Lee-Huang-Yang (LHY) correction term is used to model the condensate, which incorporates local contact interactions and LHY corrections induced by quantum fluctuations. The characteristics of vortical modes with S = 1 and S = 2 are studied in detail. The chemical potential μ and peak density ρ max of the modes are calculated as functions of the norm N. The μ − N curves of on-site-centered and off-site-centered modes with the same number of lattice sites are nearly identical. We make an analysis and reach a conclusion consistent with these simulation results. This system contains a nontrivial type of vortex mode known as the mixed-vortex mode, in which the two components have the same density pattern and different topological charges. To the best of our knowledge, we are the first to report vortex modes with isomorphic density patterns and heteromorphic vortices in the field of quantum droplets. • Two-dimensional discrete vortex quantum droplets (QDs) are considered for the first time. • A novel discrete vortex QD is found, in which two components with different topological charges have the same pattern. • The existence and characteristics of discrete vortex QDs with topological charges of up to 5 are investigated. [ABSTRACT FROM AUTHOR]

• The model of one-dimensional (1D) discrete quantum droplets (QDs) in optical lattice are given. • The characteristics and dynamics of 1D discrete QDs in optical lattice are studied systematically, and the numerical results are consistent with the theoretical analysis. • Stable staggered discrete QDs in 1D system are proposed. We consider the dynamical model of quantum droplets (QDs) launched in a deep optical lattice. This setting is modeled by a one-dimensional discrete Gross-Pitaevskii equation with Lee-Huang-Yang corrections. We find that the hopping rate, C , plays a dominant role in characterizing the properties of the system. The system can be divided into two regions: the quasicontinuum (QC) and tightly-bound (TB) regions. In the QC region, where the hopping rate C > 0.21 , the discrete QDs can behave similar to their counterparts in the continuous system. In the TB region, where C < 0.21 , the presence of the Peierls-Nabarro (PN) potential barrier induces multistablity and discreteness. In this region, the curves for 3 characteristics (chemical potential μ , peak values ρ , and effective width W) are no longer continuous, being split into many branches, and most of the solutions on the μ (N) (N is the total norm of the QD) curves violate the Vakhitov-Kolokolov criterion. Analyses are performed to explain these effects, with the results agreeing well with the numerical simulations. By introducing synthetic gauge fields, we create for the first time stable staggered discrete QDs in the current system. The mobilities and collisions of discrete QDs are also discussed, showing that the phenomena of the dynamics in the two regions (TB and QC) are different. [ABSTRACT FROM AUTHOR]