Group structure of Wilson loops in 2D tight-binding models with 2-band and 4-band energy spectra

Abstract

Tight-binding models represent one of the basic approaches for studying the topological states of a matter. In the given account, we consider a tight-binding model set by a matrix Hamiltonian over 2D Brillouin zone. The corresponding multiband energy spectrum gives rise to non-Abelian gauge construction formed by the non-Abelian Berry connection [Formula: see text]. The main gauge invariant quantities in such cases are the eigenvalues of Wilson loops. The purpose followed is the search for any special (on top of general) properties of Wilson loops for 2D tight-binding models which impose certain restrictions on the structure of its eigenvalues. The non-Abelian Berry connection is shown to be pure gauge with point-like singularities. The corresponding curvature tensor [Formula: see text] vanishes throughout the Brillouin zone except the isolated points where [Formula: see text] is singular. Combining such behavior of [Formula: see text] with non-Abelian Stokes theorem allows to avoid the path-ordering procedure in calculating the Wilson loops. In this approach, we show that Wilson loops are endowed by the group structure isomorphic to the fundamental group of the Brillouin zone. The latter is the Abelian group [Formula: see text] forcing the set of eigenvalues of Wilson loops to comprise of two fixed phases associated with the two primitive elements (generators) of [Formula: see text].