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Question 1. eq. 3.102 defines a matrix in "band-space", e.g. it's NxN where N = number of bands being considered. The right hand side is a regular vector inner product for fixed m,n. The vectors on the right hand side are in any basis you want, so long as they span the same Kohn-Sham/band subspace. Question 2. I'm not sure where you're lost. You ...

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I think my understanding on this problem has improved somewhat. Let me try to clarify my confusions. If anyone sees issues with this reasoning, please let me know. The idea of phases in this context is related to the Chern number $c$, in the context of topological/Chern insulators. The answer to this question would require computing the Chern number and ...

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The Bernevig-Hughes-Zhang (BHZ) model for topological insulators comes to mind, see e.g. this pedagogical introduction or the review by Qi and Zhang (arXiv link here). It's often justified to set the block-off-diagonal terms zero, as was done in the paper introducing the model (arXiv version here), which simplifies analytical calculations considerably (it ...

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As a good starting, you may take a look at this classical paper and its supplementary material: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.117.097601 Here I just show some general steps: Calculate the polarization (I assume that you can use the VASP package): Calculate the $A, B, C, D$ of Landau-Ginzburg expansion: $$E= \sum_i \dfrac{A}{2}(... 3 The TD-Schrodinger equation is$$H(\lambda)|\psi(t)\rangle=i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle$$So, we just need to plug in \eqref{4} and remove all terms that aren't first-order in \dot\lambda. Let's start with the left-hand side, since its a little simpler.$$\begin{align}H(\lambda)|\psi(t)\rangle&=e^{i\phi(\lambda(t))}e^{-i\gamma(t)}\...

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I don't think I can use this Hamiltonian to recover the band structure. The band structure should use a tight-binding Hamiltonian with several unit cells. A colleague told me that this Hamiltonian is just for their discussion in Figure 4.

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I think I figured this out. For most practical purposes, I think it is fine to just choose $-\pi\leq k_x\leq \pi$ and $0\leq k_y\leq \pi$ (half-BZ, with exact ranges depending on the model). I think I over-complicated the authors' work. It is unlikely that the vortices/singularities will occur near the boundary of the chosen region. So, I can just take the ...

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I found the following slides helpful: https://physics.princeton.edu//~haldane/talks/dirac.pdf Here are my answers: Yes. From what I understood, the above is not necessarily a 3D effect. The wormholes appear only at gapless points, so we can expect only one Dirac string in my example. Partly correct. We have only one wormhole from the gapless point. However, ...

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