Relation between winding numbers and energy dispersions

Abstract

Two-band Hamiltonians provide a typical description of topological band structures, in which the eigenfunctions can be characterized by a %Bloch vector field whose winding number that defines an integer topological invariant. This winding number is quantized and protected against continuous deformations of the Hamiltonian. Here we show that the Bloch vector and its winding number can be directly related to the gradient of the energy dispersion. Since the energy gradient is proportional to the group velocity, our result establishes an experimentally accessible correspondence between the Bloch vector field and angle-resolved photoemission spectroscopy measurements. We discuss a mapping between the gradient of the energy dispersion and the Bloch vector. This implies a direct and measurable relation between two-band Hamiltonians and their underlying topological structures.

Figures (1)

• Figure 1: 2D projection of the BV field: a) The BV field of a tight-binding Hamiltonian on a honeycomb lattice has several nodes. The marked loops around these nodes have the winding numbers $\pm 1$, indicated by the clockwise or counter-clockwise winding, respectively. The actual length of the $s_1$-$s_2$ projected BV is color encoded. b) The winding trajectory of the BV of the BdG Hamiltonian with $w=n_w=3$. The continuous color change indicates the evolution of the trajectory with $t$ for $0\le t<2\pi$.