Quantum-Mechanical Position Operator and Localization in Extended   Systems
  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. 
A. A. Aligia (1) and G. Ortiz (2) ((1) Centro At\'omico Bariloche,   Bariloche, Argentina, (2) Theoretical Division, Los Ala...