Quantum metric on the Brillouin Zone in correlated electron systems and its relation to topology for Chern insulators

TL;DR

This paper introduces a generalized quantum metric for correlated electron systems based on optical conductivity, establishing its properties and relation to topology, and confirming its validity through numerical analysis.

Contribution

It defines a new quantum metric for interacting systems, linking it to the Chern number and validating it with numerical models.

Findings

The GQM is equivalent to the noninteracting quantum metric.

The GQM is positive semi-definite.

Numerical confirmation in the Qi-Wu-Zhang model.

Abstract

Geometric aspects of physics play a crucial role in modern condensed matter physics. The quantum metric is one of these geometric quantities which defines the distance on a parameter space and contributes to various physical phenomena, such as superconductivity and nonlinear conductivity. Despite its importance, the quantum metric in interacting systems is poorly understood. In this paper, we introduce a generalized quantum metric(GQM) on the Brillouin zone for correlated electron systems. This quantum metric is based on the optical conductivity that is written by single-particle Green's functions. We analytically prove that this definition is equivalent to the existing definition of the quantum metric in noninteracting systems and that it is positive semi-definite as necessary for a metric. Furthermore, we point out the relationship between the GQM and the Chern number in interacting systems. We then numerically confirm these properties of the GQM in the Qi-Wu-Zhang model with and without interaction. We believe that the GQM will be a step toward generalizing the quantum metric to the interacting regime.