Hi David,
Thank you for the nice explanation and suggestions. I tried to increase
the mesh spacing, but the problem still persists. The time-dependent
observable that I calculate should according to symmetry be 60-degree
periodic in ZnO , but there is a small error. I also tried using
symmetry-adapted Wannier, which yields similar results. Of the different
crystals that I tried (MoS2, Si, ZnO), the problem only arises in ZnO. I
believe this is perhaps too specific an issue for this mailing list, but
thanks again for your suggestions.
Best,
Lun
On 6/4/23 9:28 AM, David Vanderbilt wrote:
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
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