deletions | additions       diff --git a/textbf_Time_reversal_invariant_topological__.tex b/textbf_Time_reversal_invariant_topological__.tex  index 89e316b..fac73fa 100644  --- a/textbf_Time_reversal_invariant_topological__.tex  +++ b/textbf_Time_reversal_invariant_topological__.tex 
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\textbf{Time reversal invariant topological insulator:} Now we turned our discussion into another type of topological insulator, this kind of system respects the time reversal symmetry. First, we see some characteristics of time reversal operator, $T$. "$T$" is an antiunitary operator, which means it can represented by a unitary operator and a complex conjugation operator. To be more specific, its action is transforming $\hat{x}\rightarrow\hat{x}$ and $ \hat{p}\rightarrow-\hat{p}$ It can be shown that $T^2=\pm1$. Due to this unique property of $T$ operator, there comes the Kramer's theorem. It states that for every eigenstate $|\psi\rangle$, its time reversal partner $T|\psi\rangle$,which is also an energy eigenstate with the same energy, is orthogonal. This is also known as Kramer's degeneracy. For a system with time reversal symmetry, it requires that $\textit{H(k)}=T\textit{H(k)}T^-1$.