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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, this review gives an alternative definition of the $\mathbb{Z}_2$ index in Eq. 7 of the review, that seems rather unconventional: $$\tag{2} \mathbb{Z}_2 = \frac{1}{2\pi}\left[ \oint_{\partial HBZ} A dk - \int_{HBZ} \Omega_z d^2k \right] (\text{mod } 2), $$ where HBZ is the half-Brillouin zone, $A$ is the Berry connection and $\Omega$ is the Berry curvature.

I am not sure how the unusual definition is used in practice (is it summed over filled bands? or all bands?). The reason I am asking is because the latter, unusual form seems easier to compute numerically (unless I missed some easier trick).

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  • $\begingroup$ In the interest of saving space here, I moved the discussion here to chat $\endgroup$
    – Tyberius
    Dec 15, 2021 at 15:17

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