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There is a book on the topic: B. Andrei Bernevig, Taylor L. Hughes: Topological Insulators and Topological Superconductors, Princeton University Press (2013).

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I think two great reviews on the topic (cited by almost any paper on the matter) are: M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010); X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011);

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We just posted a paper http://arxiv.org/abs/1507.04673 addressing this issue. For fermion topological orders, the fermionic version of this formula is $\Theta=\sum_a d_a^2 \theta_a=0$. See eq. 14 of the paper. So we cannot use eq. 14 to compute the chiral central charge of the fermionic topological orders. We have to use the bosonic extension of the ...

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Here is a derivation: Let me focus on the sum in the expression of $\sigma_{xy}$. First notice that $\langle \alpha|(\partial_{k_x}|\beta\rangle)=-(\partial_{k_x}\langle \alpha|)|\beta\rangle$ because $\langle \alpha|\beta\rangle=0$ since they have different energies. We can rewrite the expression as ...

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