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(1) A Hamiltonian, say, H(k), where k is the crystal momentum.

(2) An appropriate complete basis set {a_1,a_2,a_3…}.

(3) Some symmetric operators {A,B,…} which commute with H(k), i.e. [A,H]=[B,H]=...=0.

Of course, by calculation, I can get any matrix element of H(k), i.e. <a_i|H(k)|a_j>.

After numerical calculation, not that surprisingly, I get many zeros of matrix elements at high symmetric k points. However, is there any method for me to verify the results or discuss the physical meaning?

For example, why for

(1) k=Γ point: <a_1|H(k=Γ)|a_3>=0 and <a_1|H(k=Γ)|a_5>≠0

(2) k=X point: <a_1|H(k=X)|a_3>≠0 and <a_1|H(k=X)|a_5>≠0?

Why there exists difference between different k points?

I think those symmetric operators might provide some help, but I am not familiar with this approach. Or are there some other techniques which might do the job for me? Thank you very much guys for reading my question!

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# A Theory to argue whether matrix elements vanish or not?

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