  We analytically and numerically analyze the one-dimensional "thin-torus"
limit of Fractional Topological Insulators in a series of simple models
exhibiting exactly flat bands with local hopping. These models are the
one-dimensional limit of two dimensional Chern Insulators, and the Hubbard-type
interactions projected into their lowest band take particularly simple forms.
By exactly solving the many-body interacting spectrum of these models, we show
that, just like in the Fractional Quantum Hall effect, the zero modes of the
thin-torus limit are CDW states of occupation numbers satisfying generalized
Pauli principles. As opposed to the FQH where the thin-torus CDW appear in
orbital space, in the thin-torus FCI states, the CDW states are in real-space.
We show the counting of the quasihole excitations in the energy spectrum cannot
distinguish between a CDW state and a FQH state. However, by exactly computing
the entanglement spectrum for the thin-torus states, we show that it can
qualitatively and quantitatively distinguish between a CDW and a fractional
topological state such as the FCI. We then discover a previously unknown
separation of energy scales of the full FQH energy spectrum in the thin torus
limit and find that Chern insulator models exhibiting strong isotropic FCI
states have a similar structure in their thin-torus limit spectrum. We close by
numerically computing the evolution of energy and entanglement spectra from the
thin-torus to the isotropic limit. Our results can also be interpreted as an
analysis of one-body, 1-dimensional topological insulators stabilized by
inversion symmetry in the presence of interactions.
