deletions | additions       diff --git a/textbf_Chern_number_With_some__.tex b/textbf_Chern_number_With_some__.tex  index a8f510f..70e8e31 100644  --- a/textbf_Chern_number_With_some__.tex  +++ b/textbf_Chern_number_With_some__.tex 
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Q^n=-\frac{1}{2\pi}\oint_{S}\textbf{b}_ndS  \end{equation}  Chern number is topological invariant. Which means under smooth deformations of the Hamiltonian or the surface. The only way its value can change is to close the gap. This means the state $|n(k)\rangle $ becomes degenerate with $|n-1\rangle $ or $|n+1\rangle $ at certain point on the surface $S$.  \textbf{Bernevig Hughes Zhang model 2D lattice:}Now we expand our discussions to 2 dimensional lattice called "Bernevig Hughes Zhang Model".\cite{Shen_2012} It can be used to describe the most common topological insulator HgTe quantum well system. Now we consider the half BHZ model instead of the full version. Its Hamiltonian can be written down as  \begin{equation}  \textit{H(k)}=[\Delta+cos\textit{k}_x+cos\textit{k}_y]\sigma+A(sin\textit{k}_x\sigma_x+sin\textit{k}_y\sigma_y)  \end{equation}  The term $\Delta$ is just like a Zeeman splitting term. The $A$ is the spin-orbit coupling term. The last term in the equation describes a hopping with spin flips. With the Hamiltonian, we can calculate the energy dispersion. The results are in Fig.7. Tuning the parameters in the Hamiltonian, we can control the system to open or close a bandgap. For example, the $\Delta=2$ case is closing the gap, and the vicinity of the closing point is called "Dirac point".