The one-dimensional extended t-V model of fermions on a lattice is a model with repulsive interactions of finite range that exhibits a transition between a Luttinger liquid conducting phase and a Mott insulating phase. It is known that by tailoring the potential energy of the insulating system, one can force a phase transition into another insulating phase. We show how to construct all possible charge-density-wave phases of the system at low critical densities in the atomic limit. Higher critical densities are investigated by a brute-force analysis of the possible finite unit cells of the Fock states.
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