  We study lattice models of charged particles in uniform magnetic fields. We
show how longer range hopping can be engineered to produce a massively
degenerate manifold of single-particle ground states with wavefunctions
identical to those making up the lowest Landau level of continuum electrons in
a magnetic field. We find that in the presence of local interactions, and at
the appropriate filling factors, Laughlin's fractional quantum Hall
wavefunction is an exact many-body ground state of our lattice model. The
hopping matrix elements in our model fall off as a Gaussian, and when the flux
per plaquette is small compared to the fundamental flux quantum one only needs
to include nearest and next nearest neighbor hoppings. We suggest how to
realize this model using atoms in optical lattices, and describe observable
consequences of the resulting fractional quantum Hall physics.
