[Wannier] Question: Hermition Issue [ wannier90_r.dat ]

[Iryan Na]

Dear Wannier90 developers,

My question is the following:
>From the wannier90_r.dat, I constructed the position matrix elements of
r_x, r_y and r_z,
*[ r_{\alpha} (R) ]_{m,n} = < 0 m | r_{\alpha=x,y,z} | R n > *where R = [
a, b, c ] = a * R_1 + b* R_2 + c* R_3.

Then, the position matrix should satisfy the relation, *[ r_{\alpha} (R)
]^{\dag} = **[ r_{\alpha} (-R) ].*
*In other words, **( [ r_{\alpha} (a,b,c) ]_{m,n} ) * = **[ r_{\alpha}
(-a,-b,-c) ]_{n,m} where * is conjugate. *

I was only able to confirm
1) diagonal components: *( [ r_{\alpha} (a,b,c) ]_{n,n} ) * = **[ r_{\alpha}
(-a,-b,-c) ]_{n,n}.*
2) hermitian matrix at R=0=[0,0,0]: *[ r_{\alpha} (R=0) ]^{\dag} = **r_{\alpha}
(R=0) *
but, could not confirm the relation, *( [ r_{\alpha} (a,b,c) ]_{m,n} )
* = **[ r_{\alpha}
(-a,-b,-c) ]_{n,m}.*

Eventually, what I need to construct is the non-abelian Berry connection
matrix which can be obtained from the following equation,
*[ A_{\alpha} ]_{n,m} (k) = \sum_{R} exp{ i k \dot R }  [  r_{\alpha} (R)
]_{n,m}.*

Physically, Berry connection matrix, [ A_{\alpha}(k) ] should be hermitian.
However, if* ( [ r_{\alpha} (a,b,c) ]_{m,n} ) * = **[ r_{\alpha} (-a,-b,-c)
]_{n,m} *is not satisfied,
the connection matrix should not be hermitian matrix.

Would you explain to me why the position matrix is not satisfying the above
relation?

Sincerely,
Iryan
Physics PhD student, UC Berkeley
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