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Tyberius
Tyberius
3:15 PM
4
Q: Z2 topological index: Is this unconventional formula summed over just filled bands, or all bands?

TribalChiefThe $\mathbb{Z}_2$ 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...

condensed-matter topological-insulators berry-phase
+1. Indeed the review paper cites a Kane and Mele paper, from 2005, published in PRL, right before presenting that "unusual" definition of the $\mathbb{Z})2$ index, it was this 2005 PRL by Kane and Mele as opposed to this 2005 PRL by Kane and Mele which you cited in your question. The former has 6000+ citations on Google Scholar as opposed to the latter which has only 5928. I don't see Eq. 7 of the review paper in either of PRL papers though. — Nike Dattani Jun 15 at 18:03
The PRL paper from your question says "The $Z_2$ index can thus be determined by counting the number of pairs of complex zeros of $P$" so it's no surprise that there's a contour integral, but this specific equation given in the review paper isn't explicitly given in either of the PRL papers. — Nike Dattani Jun 15 at 18:04
I don't understand much of it, so I can't provide an answer, but the equivalence of these two expressions is proven in the Appendix of Fu&Kane2006 (Also available through arxiv) — Tyberius ♦ Jun 15 at 20:00
Thank you Nike and Tyberius. I guess the only question that remains is the issue of whether the latter definition has to be calculated per band, or over all filled bands. This wasn't clear to me after a brief read of the paper by Fu and Kane, but I may have missed it. — TribalChief Jun 15 at 20:29
Great find @Tyberius ! How did you come across that paper? TribalChief, is there any part of that proof in the Appendix that you aren't able to follow? If so, perhaps that could be asked as a separate question, while the aspect about whether or not the sum is over only filled bands versus over all bands, can be addressed in this thread. — Nike Dattani Jun 21 at 23:18
@NikeDattani, thanks for checking in. I think I am having trouble writing out the 'transition matrix' in eq A2 of the arxiv version, and understanding the sum over $n_A$, in the definition of $\mathcal{A}^A$, above eq A5. I think the latter might contain the solution to my outstanding question. When they work using Wannier centers, they consider all bands. — TribalChief Jun 22 at 3:11
I had emailed the authors about this some time ago and don't seem to have ever got a reply. @Tyberius seems to have made some progress by finding the proof of the equivalence of those two expressions, and since you (TribalChief) have now clarified more about what part of the problem is bothering you, perhaps one of the three of us can figure out what the issue is. Unfortunately I might not be able to look at it much until later in December. You can ping me / remind me at that time if it looks like I've forgotten (and/or if Tyberius isn't able to come up with anything!). — Nike Dattani 8 hours ago
@NikeDattani wow. Thank you!!! — TribalChief 8 hours ago
 

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