Group Structure of Wilson Loops in 2D Models with 2- and 4-Band Energy Spectra

TL;DR

This paper investigates the non-Abelian gauge structure of Wilson loops in 2D tight-binding models with 2- and 4-band spectra, revealing their group structure by analyzing Berry curvature singularities and employing the non-Abelian Stokes theorem.

Contribution

It introduces a novel approach to study Wilson loops in multiband 2D models by leveraging the singularities of Berry curvature and the non-Abelian Stokes theorem, simplifying the analysis.

Findings

Wilson loops form specific group structures in 2D models.

Berry curvature is singular at isolated points in the Brillouin zone.

The approach simplifies the study of Wilson loops without path ordering.

Abstract

We consider a tight-binding model defined by a matrix Hamiltonian over 2D Brillouin zone. Multiband energy spectrum gives rise to a non-Abelian gauge structure set by the Berry connections. The corresponding curvature $F_{\mu\nu}$ vanishes throughout the Brillouin zone except an isolated points where $F_{\mu\nu}$ is singular. Combining the singular behaviour of $F_{\mu\nu}$ with non-Abelian Stokes theorem allows to avoid the path ordering procedure in studying the structure of Wilson loops. 2D models with 2-band and 4-band energy spectra are considered as a demonstrative examples and the group structure of the corresponding Wilson loops is revealed.