  We introduce a fundamental complex quantity, $z_{L}$, which allows us to
discriminate between a conducting and non-conducting thermodynamic phase in
extended quantum systems. Its phase can be related to the expectation value of
the position operator, while its modulus provides an appropriate definition of
a localization length. The expressions are valid for {\it any} fractional
particle filling. As an illustration we use $z_{L}$ to characterize insulator
to ``superconducting'' and Mott transitions in one-dimensional lattice models
with infinite on-site Coulomb repulsion at quarter filling.
