  We construct and characterize tight binding Hamiltonians which contain a
completely flat topological band made of continuum lowest Landau level
wavefunctions sampled on a lattice. We find an infinite family of such
Hamiltonians, with simple analytic descriptions. These provide a valuable tool
for constructing exactly solvable models. We also implement a numerical
algorithm for finding the most local Hamiltonian with a flat Landau level. We
find intriguing structures in the spatial dependence of the matrix elements for
this optimized model. The models we construct serve as foundations for
numerical and experimental studies of topological systems, both non-interacting
and interacting.
