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

Exact Topological Flat Bands from Continuum Landau Levels