Finite doping of a one-dimensional charge density wave: solitons vs.   Luttinger liquid charge density
  The effects of doping on a one-dimensional wire in a charge density wave state are studied using the density-matrix renormalization group method. We show that for a finite number of extra electrons the ground state becomes conducting but the particle density along the wire corresponds to a charge density wave with an incommensurate wave number determined by the filling. We find that the absence of the translational invariance can be discerned even in the thermodynamic limit, as long as the number of doping electrons is finite. Luttinger liquid behavior is reached only for a finite change in the electron filling factor, which for an infinite wire corresponds to the addition of an infinite number of electrons. In addition to the half filled insulating Mott state and the conducting states, we find evidence for subgap states at fillings different from half filling by a single electron or hole. Finally, we show that by coupling our system to a quantum dot, one can have a discontinuous dependence of its population on the applied gate voltage in the thermodynamic limit, similarly to the one predicted for a Luttinger liquid without umklapp processes. 
Yuval Weiss, Moshe Goldstein and Richard Berkovits