When I am studying quantum Hall effect, the quantum Hall conductance can be represented by Green function $\left(\text{up to}\ \frac{e^2}{h}\large \right)$:
I cannot understand why it is an integer?
(This expression appears in many papers, e.g. Jackiw and Rebbi, PRL, 1976; Niu, Thouless and Wu, PRB, 1985; Aoki and Ando, PRL, 1986; Volovik, The universe in a helium droplet, P136, 2003; Wang, Qi and Zhang, PRL, 2010; Gurarie, PRB, 2011...)
I have learned that using wave function, the Hall conductance can be expressed as the 1st Chern number. But here, via Green's function, I got lost.