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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.

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  • $\begingroup$ Is $Tr(\mathbb{x})$ supposed to represent the trace of that messy tensor, $\mathbb{x}$? If so, then the result should be a scalar by definition, I believe. $\endgroup$ Nov 25, 2015 at 14:16
  • $\begingroup$ Welcome to SE. Perhaps you could give us the reference where you found this expression. It's an integer because you correctly define the prefactor :-) You just walk on the surface of a sphere of dimension $4\pi^{2}$ and the walk is weighted by a complicated tensor of dimension $G\cdot G^{-1} \cdots$=1 and a permutation symbol with total number of permutation $3!$... or at least that's what I conclude from your incomplete presentation of the formula. You can also think in terms of winding number, when you accumulate some phase: if $G\propto e^{i\phi x}$ then $G\partial G^{-1}\propto\phi$ ... $\endgroup$
    – FraSchelle
    Nov 25, 2015 at 19:14
  • $\begingroup$ Thanks. If there is only one group of $G \partial G^{-1}$, this is clearly a winding number around a vortex. But here, with three groups of $G \partial G^{-1}$, I don't know how to understand it as some kind of 'winding number'. $\endgroup$
    – wonderboy
    Nov 26, 2015 at 3:19

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