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Is it possible to define and calculate chern number for two bands while they're crossing each other?

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  • $\begingroup$ If you're referring to the cumulative Chern number of the two bands together, the answer is yes. All you need is that the two bands are spectrally isolated from all other bands. $\endgroup$ – PPR Aug 19 at 4:01
  • $\begingroup$ thank you, I found the answer and I agree with you $\endgroup$ – Mr.Fox Aug 20 at 13:05
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Just for completeness, here is the formula. To appeal to a wider audience, let us assume translation invariance so that we work in momentum space. Say the projection onto the two bands is $P(k)$, then the Chern number associated to $P$ is $$ \frac{\mathrm{i}}{2\pi}\int_{\mathrm{BZ}}\operatorname{tr}(\mathrm{d}P \wedge P\mathrm{d}P) = \frac{\mathrm{i}}{2\pi}\int_{k \in [0,2\pi]^2}\operatorname{tr}(P(k)[\partial_{k_1}P(k),\partial_{k_2}P(k)])\mathrm{d}k_1\mathrm{d}k_2 \,.$$

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