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I assume you refer to something like a BHZ-$\mathbb{Z}_2$-insulator (with the experimental realization of a 2d-$\text{HgTe}$-quantum well). This model can be understood as two Chern insulators. Chern insulators have a chiral edge mode (that is, an edge mode, that propagates only in one turning sense, thus breaking time reversal). This is analogous to the ...

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Firstly, I think you've confused several points in your analysis of the situation. Working with the first picture, by standard electrostatic analysis, we see that there must be a standard Coulomb force between the plates for all times $t\ge t_2$, not just at $t=t_2$. This point is not too important, but I thought I'd mention it. Continuing to work with the ...

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It is the total $H = \sum_k H_k$ invariant under time reversal

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I guess you are confused by the action on wave functions and operators. 1) The second approach to your first concern is "more correct." Notice that $t(k)$ or $w(k)$ is a $c$-number( or scalar, not an operator). When you write $w(k) \rightarrow w(-k)$, I guess you treat it as a wave function (I am not sure, because if it is a wave function, it should ...

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