Quantum Mechanical Position Operator and Localization in Extended Systems

We introduce a fundamental complex quantity, $z_L$, which allows us to discriminate between conducting and nonconducting thermodynamic phases 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 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.