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

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.