Timeline for Is Chern-number for free fermion system always limited by total band number, i.e. number of orbits with a unit cell?
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| Nov 28, 2021 at 8:39 | history | edited | Urb | CC BY-SA 4.0 |
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| Dec 22, 2019 at 3:19 | comment | added | Meng Cheng | @Histoscienology if you use the formula of Chern number in terms of Berry curvatures of the Bloch bands it is fairly straightforward to see that the sum of Chern numbers of all bands is 0. Physically, once you occupy all bands this is just filling all orbitals on each site, so a completely trivial situation. | |
| Nov 27, 2019 at 14:57 | comment | added | Histoscienology | Sorry for bringing up this old post again but is there a simple way to see why the sum of Chern numbers is zero mathematically? It's not clear to me why the winding number characterizes the map from $T^2$ to the hamiltonian space $S^2$ has to correspond to the Chern number of each band... | |
| May 20, 2015 at 21:41 | comment | added | Meng Cheng | For complex fermions, if the hoppings are real then it is time-reversal invariant, so the Chern number must vanish. | |
| May 20, 2015 at 20:02 | comment | added | An Zhou | OK. I believe you. That would be the interesting case actually :). By the way, do you think it is necessary for the hopping to be complex to achieve higher Chern-number? Said differently, is it also possible to realize Chern-number larger than 2 for a two orbits free Majorana fermion? (Since the hopping of Majorana fermion would always be imaginary) | |
| May 20, 2015 at 19:46 | vote | accept | An Zhou | ||
| May 20, 2015 at 17:28 | history | edited | Meng Cheng | CC BY-SA 3.0 |
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| May 20, 2015 at 17:17 | comment | added | Meng Cheng | Forget about how they get their model (through 3-layers, whatever), there is a clear, well-defined question: how many orbitals per unit cell? And the answer is 2. The model is purely 2D. I don't think there is anything ambiguous about these statements. So this is a counterexample to your conjecture. | |
| May 20, 2015 at 17:05 | comment | added | An Zhou | To make a cleaner discussion, let's add a further restriction: a 2D free fermion model. Are there examples that violate the statement then? | |
| May 20, 2015 at 17:03 | comment | added | An Zhou | Sorry the 'finite' in the last reply should be infinite. In such stacking trick, stacking layers behave as a internal degree of freedom essentially is adding orbitals.Keeping unit cell having only 2 cites is more or less an meaningless appearance for Chern-number purpose since it is not defined for 3D material. We're forced to view this stacked system as a three band tight-binding and each of them I guess could still have Chern-number 2 at most. | |
| May 20, 2015 at 16:59 | comment | added | An Zhou | I still think this is kind of artificial. Suppose we have a finite system in the 2D plane. So then our B.Z actually also has 3 layer. It would be strange to say which is the total Chern-number, rather, it only makes sense to say what is the Chern-number for each layer of them. And I guess each one of them does not has C>=3. | |
| May 20, 2015 at 16:56 | comment | added | An Zhou | When I say total number of orbit, I mean the sum of number of orbits of all sites in a unit cell. Then the 3 layered Haldane model has the appearence of 6 orbits per unit cell. However, indeed if they're able to restore the translation invariance in the stacked direction, then indeed the unit cell is reduced and again possess 2 orbits again. However, the our problem becomes 3D, and there are 3 available momentums in Brillouin Zone in the stacked direction. | |
| May 20, 2015 at 14:33 | history | answered | Meng Cheng | CC BY-SA 3.0 |