  We study the persistent current and the Drude weight of a system of spinless
fermions, with repulsive interactions and a hopping impurity, on a mesoscopic
ring pierced by a magnetic flux, using a Density Matrix Renormalization Group
algorithm for complex fields. Both the Luttinger Liquid (LL) and the Charge
Density Wave (CDW) phases of the system are considered. Under a Jordan-Wigner
transformation, the system is equivalent to a spin-1/2 XXZ chain with a
weakened exchange coupling. We find that the persistent current changes from an
algebraic to an exponential decay with the system size, as the system crosses
from the LL to the CDW phase with increasing interaction $U$. We also find that
in the interacting system the persistent current is invariant under the
impurity transformation $\rho\to 1/\rho $, for large system sizes, where $\rho
$ is the defect strength. The persistent current exhibits a decay that is in
agreement with the behavior obtained for the Drude weight. We find that in the
LL phase the Drude weight decreases algebraically with the number of lattice
sites $N$, due to the interplay of the electron interaction with the impurity,
while in the CDW phase it decreases exponentially, defining a localization
length which decreases with increasing interaction and impurity strength. Our
results show that the impurity and the interactions always decrease the
persistent current, and imply that the Drude weight vanishes in the limit $N\to
\infty $, in both phases.
