  A mathematical lattice, called the von Neumann lattice, is a subset of
coherent states and exists periodically in the phase space. It is unlike solids
or Abrikosov lattices that are observable in physical systems. Abrikosov
lattices are vortices closely packed into a lattice with a flux quantum through
a unit cell. Although Abrikosov lattices appear generally in various physical
systems, vortex lattices with multiple-flux quantums through a unit cell are
more stable than Abrikosov lattices in some physical regimes of the systems
with non-local interactions between particles. No theory is able to describe
these vortex lattices today. Here, we develop a theory for these vortex
lattices by extending von Neumann lattices to the coordinate space with a unit
cell of area that is proportional to flux quantums through a unit cell. The von
Neumann lattices not only show the same physical properties as the Abrikosov
lattice, but also describe vortex lattices with multiple-flux quantums through
a unit cell. From numerical simulations of a rapidly rotating dipole-blockaded
gas, we confirm that vortex lattices showed in our simulations are the
representation of von Neumann lattices in the coordinate space. We anticipate
our theory to be a starting point for developing more sophisticated
vortex-lattice models. For example, the effect of Landau-level mixing on vortex
lattice structures, vortices formed inside superfluid droplets and structural
phase transitions of vortex matter in two-component Bose-Einstein condensates
will be relevant for such developments.
