  We elaborate on a geometric characterization of the electromagnetic
properties of matter. A fundamental complex quantity, z_{L}, is introduced to
study the localization properties of extended quantum systems. z_L, which
allows us to discriminate between conducting and non-conducting thermodynamic
phases, has an illuminating physical (and geometric) interpretation. Its phase
can be related to the expectation value of the position operator (and a Berry
phase), while its modulus is associated with quantum electric polarization
fluctuations (and a quantum metric). We also study the scaling behavior of z_L
in the one-dimensional repulsive Hubbard model.
