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Kevin Lee edited In_the_manner_we_can__.tex
over 8 years ago
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In the same manner, we can visualize the problem in the $h(k_x,k_y)$ form. We can rewrite the spectrum in the following way.
\begin{equation}
h(k_{x},h_{y})=\begin{pmatrix} Asink_x \\ Asink_y \\ \Delta+cos(k_{x})+cos(k_{y}) \\ \end{pmatrix}
\end{equation}
The Chern number of the system can be thought as a monopole at the $h$ space origin. And as $k_x$ and $k_y$ go from $-\pi$ to $\pi$ ,which corresponds to the 1st Brillouin zone, it is a torus walking in the $h$ space. As long as the torus passing through origin, it contributes 1 Chern number. The evolution of the torus is shown in Fig.8. The $\Delta$ parameter controls the initial center of the torus. Therefore, different $\Delta$ values determines different topological order of the system.
