Friedel oscillations in disordered quantum wires: Influence of electron-electron interactions on the localization length

The Friedel oscillations caused by an impurity located at one edge of a disordered interacting quantum wire are calculated numerically. The electron density in the system's ground state is determined using the density-matrix renormalization-group method, and the Friedel oscillation data are extracted using the density difference between the case in which the wire is coupled to an impurity and the case where the impurity is uncoupled. We show that the power-law decay of the oscillations occurring for an interacting clean one-dimensional sample described by the Luttinger liquid theory is multiplied by an exponential decay term due to the disorder. Scaling of the average Friedel oscillations by this exponential term collapses the disordered sample data on the clean results. We show that the length scale governing the exponential decay may be associated with the Anderson localization length and thus be used as a convenient way to determine the dependence of the localization length on disorder and interactions. The localization length decreases as a function of the interaction strength, in accordance with previous predictions.