[Wannier] Hermiticity of Berry potential matrix

[Lun Yue]

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
>>
>>
>>