Connection between the winding number and the Chern number

TL;DR

This paper explores the relationship between the winding number and the Chern number in topological insulators, demonstrating how winding numbers predict edge states and linking them to the Chern number through numerical and theoretical analysis.

Contribution

It explicitly shows the connection between 1D winding numbers and edge states in the extended SSH model and relates the 2D Chern number to differences in winding numbers across momentum space.

Findings

Winding numbers predict the number of edge states in the extended SSH model.

A numerical demonstration confirms the correspondence between winding numbers and edge states.

An identity relating the 2D Chern number to differences in 1D winding numbers is established.

Abstract

Bulk-edge correspondence is one of the most distinct properties of topological insulators. In particular, the 1D winding number $\n$ has a one-to-one correspondence to the number of edge states in a chain of topological insulators with boundaries. By properly choosing the unit cells, we carry out numerical calculation to show explicitly in the extended SSH model that the winding numbers corresponding to the left and right unit cells may be used to predict the numbers of edge states on the two boundaries in a finite chain. Moreover, by drawing analogy between the SSH model and QWZ model, we show that the extended SSH model may be generalized to the extended QWZ model. By integrating the ``magnetic field'' over the momentum strip $0\le p_2 \le \pi, 0\le p_1 2\pi$ in the Brillouin zone, we show a identity relating the 2D Chern number and the difference between the 1D winding numbers at $p_2=0 $ and $p_2 =\pi$.