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Fourier transformation of the creation/annihilation operators is no different than any other Fourier transform; the spin index just comes along for the ride. For instance, $$c^\dagger_{i \alpha} = \frac{1}{\sqrt{2\pi}} \int_\text{BZ} d^2k\,c^\dagger_{\mathbf{k} \alpha}e^{i\mathbf{k}\cdot\mathbf{x}_i}$$ With that operator in hand, your second term (ignoring ...

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A topological invariants is a continuous map $n$: $$\mathfrak{H}\ni H\mapsto n\left(H\right)\in S$$ where $H$ is the Hamiltonian of your system and $S$ is some topological space. $\mathfrak{H}$ is the space of all admissible Hamiltonians. Unfortunately the exact definition of $\mathfrak{H}$ is still a matter of current research. To keep things short, we ...

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Both examples you ask about are for gapless systems, but in your edit you then bring up some notions related to the gapped systems, so let me address both. Moreover it will turn out that understanding the gapped case well (which is simpler) almost naturally implies the gapless case. Topological invariants for band insulators Suppose we have our Brillouin ...

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As I know there is a notion of "bulk boundary correspondence" if you terminate the infinite slab of graphene along an axis in an arbitarily direction then you get an edge and surface states. It will support chiral movement of electrons provided that its bulk(I.e unterminated graphene) be in a topological phase characterized by some topological quantity. This ...

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You should use parity analysis in wien2k by group theoretical considerations. You must first converge your material then run "x irrep -so" for spin orbit inclusion or without it. You should just notice to use case.vector file from the k-path of the band to have a comparison between case.band.agr and case.outputir[so] or case.irrep. don't forget to use ...

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