Calculation of velocity matrix element

[Fengyuan Xuan]

Dear Developers,

I am trying to understand how the velocity matrix element is calculated in the BSE code. In the Eq. 45 of Ref. [Computer Physics Communications 183, 1269-1289 (2012)], will a random phase factor be introduced to <vk|v|ck>? Because the wave function with a small q-shift |vk+q> has a random phase factor. Finally this random phase factor will affect the many-body velocity matrix <0|v|S> = \sum A_vck <vk|v|ck>.

Regards,

Fengyuan XUAN

[Brad Barker]

Hello Fengyuan Xuan,

The appearance of the "exp(i q.r)" factor from the q-shift of the occupied bands is an essential part for the practical calculation of the dipole operator. Note that the limit q-->0 is taken.

Best,

Brad Barker

UC Merced

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[Fengyuan Xuan]

Dear Brad Barker,

Thank you so much for your reply.

My understanding is that the factor of "exp(i q.r)" will cancel the difference of Bloch wave vector between |ck>=exp(i (k+q).r) u_{vk+q}(r)  and |ck>=exp(i k. r) u_{ck}(r), so that one can evaluate the following quantity using the periodic part of the Bloch state:

<vk+q|exp(i q.r)|ck> = <u_{vk+q}|u_{ck}>

The "random phase factor" I am thinking about is the random phase associated with the eigenvalue problem. If |u_k> is an eigenstate of Hamiltonian H_k, then exp(i theta)|u_k> is also an eigenstate of H_k. 

In the wave function calculation like QuantumESPRESSO, it seems to me that the  "random phase factor" ------ exp(i theta), cannot be determined. So the wave function coefficients saved in "WFN_fi" and "WFNq_fi" contain some random phase factors, and finally in practice after we choose a small but finite q, there is an unknown phase factor associated to <vk+q|exp(i q.r)|ck> = <u_{vk+q}|u_{ck}>. This will affect the numerical result of the velocity matrix element. 

So I am thinking if in the BerkeleyGW package, there is a technique to remove this phase factor or some specific gauge is chosen for the wave function coefficients.

Regards,
Fengyuan

[Diana Qiu]

Dear Fengyuan,

You are correct that the small q-shift introduces a random phase factor for both the velocity matrix elements and the eigenvectors of the BSE Hamiltonian (I.e. the Exciton wavefunction envelope).  When plotting the exciton wavefunction, you can either plot the amplitude squared or you can remove the phase factor by rotating each k-point to remove the phase. For instance, if you rotate the phase of the lowest energy excitons so that the wavefunction envelope is real, you can apply this same gauge to all other excitons by rotating them in the same way. There is no automatic way to do this within BGW.

Best,

Diana 

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Diana Y. Qiu

Assistant Professor

Department of Mechanical Engineering and Materials Science

Yale University

Pronouns: she/her/hers

[Diana Qiu]

Dear Fengyuan,

I should add that while the random phase affects the velocity matrix elements, it does not matter for optical absorption, since the absorption depends only on the square of the velocity matrix elements.

Best,

Diana

[Fengyuan Xuan]

Dear Diana,

I see. The random phase in WFNq_fi will also affect the eigenvectors of BSE Hamiltonian.

Thank you so much for the help! 

Regards,
Fengyuan

[Hong Tang]

Dear Diana and Fengyuan,

It is an interesting question. But I still don't understand completely.

1.     " If |u_k> is an eigenstate of Hamiltonian H_k, then exp(i theta)|u_k> is also an eigenstate of H_k. "

the  "random phase factor" ------ exp(i theta), cannot be determined.  

Those statements are making sense. However, in one run of qe, say, you get the WFN file, I think all the coefficients of the wavefunction recorded in the WFN file should have a same so-called "random phase factor". Otherwise, the wavefunction you get in the WFN will be meaningless. 

2.     I think the same analysis for the other qe run for the WFNq file.  you will have another "random phase factor" for WFNq file. The 2 "random phase factors" for WFN and WFNq would be different, this is true, I think. 

3.      then I think it is true that  "This will affect the numerical result of the velocity matrix element." My question is that we neither can determine each of the  2 "random phase factors", nor the difference between the 2 "random phase factors", unless we apply any gauge. Then how can the rotation of k space remove the random phase factors in exciton states? 

4.    " For instance, if you rotate the phase of the lowest energy excitons so that the wavefunction envelope is real, you can apply this same gauge to all other excitons by rotating them in the same way. " I did not get this point. the wavefunction envelope A_cvk itself is a complex number. Since one exciton has many A_cvk and each of them may have different phase itself, how can one make the whole wavefunction envelope real number by rotating in k space?  Even though one can make some of A_cvk real, does this mean the difference between the 2 "random phase factors" for WFN and WFNq shrink to zero? I am not convinced by this. 

5.    It looks that the phase information in the exciton wavefunction is not reliable.   

Please let me know what I miss.

Best,

Hong

[Fengyuan Xuan]

Dear Hong,

Thank you for your comments and questions. Below is my personal understanding:

1&2, " I think all the coefficients of the wavefunction recorded in the WFN file should have a same so-called "random phase factor" "

I have checked numerically that the random phase is not the same for two different k point with same band index in one QE run. For example in system like monolayer WSe2, if you specify two k points, let's say K and K+q (q is small), in the QE input file, the random phase for the valence band (|u_vk>) at K and K+q are not the same. The difference can be very large (discontinous against the small q shift) and it seems depending on some initialization setup in the QE diagonalisation process. We need to choose a gauge to tune the random phase between |u_vK> and |u_vK+q> so that the discontinuous gauge difference is compensated. For one example of a simple gauge, please see Eq. 12 of Ref. [PHYSICAL REVIEW RESEARCH 2, 033256 (2020)]. But this does not mean that the output wave function from QE is meaningless. As long as the physical quantity constructed by QE output WFN is gauge invariant, we would get the correct numerical result for that quantity, no matter what random phase QE gives us in the WFN. The fundamental question is that, by definition from quantum mechanics, there is a phase freedom. 

3&4, The wave function from WFNq_fi and WFN_fi are used to construct the electron-hole basis, which is used to construct BSE Hamiltonian. The BSE eigenstate, Avck or the envelope, is the excited state coefficients based on this electron-hole basis. So the basis and Avck are paired up. If we change basis, Avck will be changed. We could tune the phases in the basis such that the envelope Avck is real. However each component of the exciton wave function, Avck |vck>, is still complex. And the total exciton wave function is a many-body state which has a total random phase that cannot be determined and it is different for different excitons.

5, the exciton charge density is gauge invariant physical quantity, so that if we plot exciton charge density, no matter what phase is associated with WFNq_fi and WFN_fi, we get the same charge density distribution. 

Regards,

Fengyuan

[Hong Tang]

Hi Fengyuan,

Thanks for explanations.

I will read your paper and may come back and discuss with you.

Best,

Hong