Luttinger liquid versus charge density wave behaviour in the   one-dimensional spinless fermion Holstein model
  We discuss the nature of the different ground states of the half-filled Holstein model of spinless fermions in 1D. In the metallic regime we determine the renormalised effective coupling constant and the velocity of the charge excitations by a density-matrix renormalisation group (DMRG) finite-size scaling approach. At low (high) phonon frequencies the Luttinger liquid is characterised by an attractive (repulsive) effective interaction. In the charge-density wave Peierls-distorted state the charge structure factor scales to a finite value indicating long-range order. 
H. Fehske, G. Wellein, G. Hager, A. Weisse, K. W. Becker, A.R. Bishop