We present a detailed numerical study of the orthogonality catastrophe exponent for a one-dimensional lattice model of spinless fermions with nearest-neighbor interaction using the density-matrix renormalization-group algorithm. Keeping up to $2000$ states per block we achieve a very great accuracy for the overlap, which is needed to extract the orthogonality exponent reliably. We discuss the behavior of the exponent for three different kinds of a localized impurity. For comparison we also discuss the noninteracting case. In the weak impurity limit our results for the overlap confirm scaling behavior expected from perturbation theory and renormalization-group calculations. In particular we find that a weak backward scattering component of the orthogonality exponent scales to zero for attractive interaction. In the strong impurity limit and for repulsive interaction we demonstrate that the orthogonality exponent cannot be extracted from the overlap for systems with up to 100 sites, due to finite-size effects. Neverthless we find indirect evidence that the backward scattering contribution to the exponent scales to $1/16$ based on predictions of boundary conformal field theory.

• Received 8 July 1997

DOI:https://doi.org/10.1103/PhysRevB.57.8878