How localized is an extended quantum system ?

TL;DR

This paper introduces a complex geometric quantity, z_L, to characterize the localization and conducting properties of extended quantum systems, linking it to physical observables like polarization and Berry phase.

Contribution

It proposes a novel geometric framework using z_L to distinguish quantum phases and analyzes its scaling in the Hubbard model, advancing understanding of quantum localization.

Findings

z_L discriminates between conducting and non-conducting phases.

The phase of z_L relates to the Berry phase and position expectation.

The modulus of z_L is connected to polarization fluctuations.

Abstract

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.