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1. Are there two definitions of TIs, one for usual TIs, and another (wider) including HgX? No, there is only one definition a 3D band topological insulator (TI): the $\mathbb{Z}_{2}$ classification scheme proposed by Fu, Kane, and Mele: Liang Fu, Charles L. Kane, and Eugene J. Mele. “Topological insulators in three dimensions.” Physical Review ...

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The short answer is, yes, the chiral edge state is determined by bulk topological property. It is known as bulk-edge correspondence. The paper you should read is: Protected edge modes without symmetry - 1301.7355. Generically, to determine whether the edge states are robust is by determining whether the edge states are protected'' by any of the three ...

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Band inversion is a necessary but not sufficient condition for topological insulators (TIs). For band TIs you need to evaluate the topological (or $\mathbb{Z}_{2}$) invariant defined by Fu, Kane and Mele in Eq. (2) of: Liang Fu, Charles L. Kane, and Eugene J. Mele. “Topological insulators in three dimensions.” Physical Review Letters 98, no. 10 (2007): ...

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I wouldn't say so. The "topological" part of this kind of insulators come from a different Chern number. For a simple naive picture I suggest the paper of Haldane published in prl 61.2015.

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Simple, combine both real- and $\mathbf{k}$-space pictures! The basic idea is to split up your $n$-dimensional system into multiple $(n-1)$-dimensional systems. For example, say you have a 2D square lattice and you define your edges along the $x$-direction. Then you need to break the 2D lattice into 1D lattices pointing in the $x$-direction. In other words, ...

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