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A mathematical lattice, called the von Neumann lattice, is a subset ofcoherent states and exists periodically in the phase space. It is unlike solidsor Abrikosov lattices that are observable in physical systems. Abrikosovlattices are vortices closely packed into a lattice with a flux quantum througha unit cell. Although Abrikosov lattices appear generally in various physicalsystems, vortex lattices with multiple-flux quantums through a unit cell aremore stable than Abrikosov lattices in some physical regimes of the systemswith non-local interactions between particles. No theory is able to describethese vortex lattices today. Here, we develop a theory for these vortexlattices by extending von Neumann lattices to the coordinate space with a unitcell of area that is proportional to flux quantums through a unit cell. The vonNeumann lattices not only show the same physical properties as the Abrikosovlattice, but also describe vortex lattices with multiple-flux quantums througha unit cell. From numerical simulations of a rapidly rotating dipole-blockadedgas, we confirm that vortex lattices showed in our simulations are therepresentation of von Neumann lattices in the coordinate space. We anticipateour theory to be a starting point for developing more sophisticatedvortex-lattice models. For example, the effect of Landau-level mixing on vortexlattice structures, vortices formed inside superfluid droplets and structuralphase transitions of vortex matter in two-component Bose-Einstein condensateswill be relevant for such developments.

Physical Realization of von Neumann Lattices in Rotating Dipole-blockaded Bose Gases