  The persistent current in a lattice model of a one-dimensional interacting
electron system is systematically studied using a complex version of the
density matrix renormalization group algorithm and the functional
renormalization group method. We mainly focus on the situation where a single
impurity is included in the ring penetrated by a magnetic flux. Due to the
interplay of the electron-electron interaction and the impurity the persistent
current in a system of N lattice sites vanishes faster then 1/N. Only for very
large systems and large impurities our results are consistent with the
bosonization prediction obtained for an effective field theory. The results
from the density matrix renormalization group and the functional
renormalization group agree well for interactions as large as the band width,
even though as an approximation in the latter method the flow of the
two-particle vertex is neglected. This confirms that the functional
renormalization group method is a very powerful tool to investigate correlated
electron systems. The method will become very useful for the theoretical
description of the electronic properties of small conducting ring molecules.
