Quantum-Mechanical Position Operator and Localization in Extended Systems

TL;DR

This paper introduces a complex quantity, $z_{L}$, that distinguishes conducting from non-conducting phases in extended quantum systems by relating its phase to the position operator and its modulus to localization length.

Contribution

It presents a novel, general framework using $z_{L}$ to analyze phase transitions and localization in quantum many-body systems at arbitrary fillings.

Findings

$z_{L}$ effectively characterizes insulator to superconductor transitions.

The method applies to models with infinite Coulomb repulsion.

Localization length is derived from the modulus of $z_{L}$.

Abstract

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.