  We study an effective Hamiltonian for the standard $\nu=1/3$ fractional
quantum Hall system in the thin cylinder regime. We give a complete description
of its ground state space in terms of what we call Fragmented Matrix Product
States, which are labeled by a certain family of tilings of the one-dimensional
lattice. We then prove that the model has a spectral gap above the ground
states for a range of coupling constants that includes physical values. As a
consequence of the gap we establish the incompressibility of the fractional
quantum Hall states. We also show that all the ground states labeled by a
tiling have a finite correlation length, for which we give an upper bound. We
demonstrate by example, however, that not all superpositions of tiling states
have exponential decay of correlations.
