How Localized is an Extended Quantum System?

Abstract

We elaborate on a geometric characterization of the electromagnetic properties of matter. A fundamental complex quantity, zL, is introduced to study the localization properties of extended quantum systems. zL, 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 zL in the one-dimensional repulsive Hubbard model.