One-dimensional spinless fermion model with competing interactions   beyond half-filling
  An accurate numerical consideration of 1D spinless fermion model with next-nearest neighbour (NNN) interactions is carried out for the electron concentrations 4/7. It is shown that depending on the parameters of the model it can be either Luttinger liquid or bipolaron liquid. In the former case competing interactions can result in a smooth behavior of one-electron distribution function at the Fermi surface with a divergence in the second derivative with respect to the quasimomentum (and not in the first one, as usual). In this connection, uncommon photoemission spectra for some 1D conductors can be explained qualitatively. 
A. K. Zhuravlev and M. I. Katsnelson