Theory to argue whether matrix elements vanish or not?

[Hao Hsu]

I have known:
(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!

 

[DrDu]

That's where you need group theory! I.e. you can determine the irreducible presentation each operator belongs to under the given little group corresponding to the specified symmetry point. You can do so also for your states a_i. Then you can also determine which irreducible representations are present in the decomposition the product of the states with the operator. The product has to contain the totally symmetric representation for the matrix element to be non-zero.
Here is some nice reference:
http://web.mit.edu/course/6/6.734j/www/group-full02.pdf

 

[Hao Hsu]

Thank you very much! I will study them this weekend ^^