Wigner crystal \emph{vs.} Friedel oscillations in the 1D Hubbard model*

We investigate the ground state density distribution of the Hubbard model in a finite one-dimensional wire. For weak interactions we find the expected Friedel oscillations, but for low filling a destinct Wigner crystal state can be observed. Although there cannot be a phase transition in a 1D system we observe a well-defined crossover into a Wigner crystal region with different physical behavior even for relatively weak short range interactions. The combination of Luttinger liquid theory and numerical Density Matrix Renormalization Group (DMRG) calculations allows a quantitative analysis of the crossover as a function of system length, lattice filling, and interaction strength.