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Q: Z2 topological index: Is this unconventional formula summed over just filled bands, or all bands?
The $\mathbb{Z}_2$ topological index is usually defined in terms of the Pfaffian of the overlap matrix, as defined by eq. 4 of Kane and Mele's paper: $$ P(k)=\text{Pf}[\langle u_i(k) | \Theta | u_j(k) \rangle], \tag{1}$$ where Pf is the Pfaffian and $\Theta$ is the time reversal operator. However...