Friedel oscillations and charge density waves in chains and ladders

The density matrix renormalization (DMRG) group method for ladders works much more efficiently with open boundary conditions. One consequence of these boundary conditions is ground-state charge density oscillations that often appear to be nearly constant in magnitude or to decay only slightly away from the boundaries. We analyze these using bosonization techniques, relating their detailed form to the correlation exponent and distinguishing boundary induced generalized Friedel oscillations from true charge density waves. We also discuss a different approach to extracting the correlation exponent from the finite size spectrum which uses exclusively open boundary conditions and can therefore take advantage of data for much larger system sizes. A general discussion of the Friedel oscillation wave vectors is given, and a convenient Fourier transform technique is used to determine it. DMRG results are analyzed on Hubbard and $t-J$ chains and 2 leg $t-J$ ladders. We present evidence for the existence of a long-ranged charge density wave state in the $t-J$ ladder at a filling of $n=0.75\mathrm{}$ and near $J/t\approx \mathrm{0.25.}$