[Wannier] Hermiticity of Berry potential matrix

[David Vanderbilt]

Lun Yue,

As far as the implementation is concerned, I hope some of my
collaborators who are closer to the code will answer you.

My feeling is that, once the Hermiticity is enforced, you are
just dealing with the intrinsic presence of the mesh
discretization error, and if this error is a problem for you,
maybe the only path is to increase the mesh density.

However, I think there could be a way to make the A(q) matrix
Hermitian from the start.  Let M_nm(q) = <u_nq|u_m,q+b> be the
overlap matrix between the two sets of states, then obtain its
"unitary part" U_mn(q), defined by doing the singular value
decomposition M = V Sigma W^dagger and setting U = V W^dagger.
Then I guess set A = i ln(U), where this is the matrix log.
I think this is the same at leading order in b, but is guaranteed
to be Hermitian.  It's a bit heavier computationally, though,
as several matrix operations are needed.

In the end, however, I'm not sure if this would solve your
problem; it could be that you really just have to reduce the
mesh spacing if you need higher accuracy.

David


On Wed, 31 May 2023, Lun Yue wrote:

> Dear all,
>
> I am writing a real-time propagation code that requires high accuracy of the 
> the Berry connection matrix A_{nm}(k) = i <u_n|d_k u_m> in the Hamiltonian 
> gauge, which is e.g. given in WYSV2006 Eq.(25) and depends on A in the 
> Wannier gauge. However, in the calculation of the Berry connection matrix in 
> the Wannier gauge (get_oper.F90), it is mentioned that this quantity is not 
> Hermitian and Hermiticity is explicitly forced:
>
> /"//Since Eq.(44) WYSV06 does not preserve the Hermiticity of the Berry 
> potential matrix, take Hermitean part (whether this makes a difference or not 
> for e.g. the AHC, depends on which expression is used to evaluate the Berry 
> curvature.//See comments in berry_wanint.F90)"/
>
> I believe that this step introduces some small error, which is reflected in 
> my final results. I also cannot find the mentioned file berry_wanint.F90. I 
> am wondering if there are some ways to solve this problem, i.e. by making the 
> Berry connection matrix naturally Hermitian? Any help or references is 
> appreciated.
>
> Best regards,
>
> Lun Yue
>
> Louisiana State University
>
>
>