A theoretical question about anisotropy in yambo

[milesj]

Hello all,

Forgive me if this is not the place to ask a theoretical question, but I'm making my way through the yambo school videos on youtube, and one point from the linear response theory is confusing me.

In both the video from this year and last year (https://youtu.be/ulEXZ-aUA-s, https://youtu.be/0VN43BcWOHg), the point is made that, in the optical limit, the transverse and longitudinal components of the dielectric tensor are equal. Is this point specific to the random phase approximation, or is it a general fact about materials? Moreover, does it imply that our calculated absorption spectrum will be independent of the direction and polarization of incident light? In this case we would not be able to simulate polarization-dependent absorption, unless this statement does not apply to more advanced computations like BSE.

I would also appreciate any suggestions about where to read about this kind of thing in the literature or textbooks.

Best,
Miles

[Davide Sangalli]

Is this point specific to the random phase approximation, or is it a general fact about materials?

Yes.

does it imply that our calculated absorption spectrum will be independent of the direction and polarization of incident light?

No, it means that, for example

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lim_{q_x -> 0} epsilon_xx(q_x) = lim_{q_y -> 0} epsilon_xx(q_y) = lim_{q_z -> 0} epsilon_xx(q_z)

where epsilon_xx(q_x) is the longitudinal dielectric function, while epsilon_xx(q_y) and epsilon_xx(q_z) are two component of the 2x2 transverse part of the dielectric function.
This means that the dielectric function is independent from the direction of the limit, and we can define it as epsilon_xx

However in general any component of the tensor epsilon_ij can be different.
This describes polarization effects.

Best,
D.

[milesj]

Hi Davide,

Your explanation here seems to be at odds with this slide in the lecture, which is saying epsilon_xx(0)=epsilon_yy(0)=epsilon_zz(0). I don't think I understand what you're saying here - the equation you wrote just looks like a trivial consequence of the continuity of epsilon as a function of the vector q, and it seems to me from the slide attached that epsilon_yy(q) and epsilon_zz(q) are the transverse components of the dielectric tensor, not epsilon_xx(anything). I ask because it appears the entire point of this slide is to justify ignoring the tensorial nature of epsilon, which would imply RPA cannot treat anisotropic effects.

If you know of a textbook which addresses this subject I believe that would be helpful. In every reference I've been able to find (for example, the thesis of FRANCESCO SOTTILE, or this paper linked from the yambo wiki https://www.cond-mat.de/events/correl16 ... eining.pdf), this point is either not addressed or the material is assumed to be sufficiently isotropic.

Best,
Miles

[Davide Sangalli]

First of all, please keep in mind that what is transverse or longitudinal depends on the direction of the momentum.

Now, in the slide, to simplify the notation,
- I fix the direction of the momentum along z (as you see in the equation below there is q_z),
- when I write epsilon_{ii}(0) I imply

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epsilon_{ii}(0) = lim_{q_z -> 0} epsilon_{ii}(q_z)

This means epsilon_{zz}(0) is longitudinal, while epsilon_{xx} (0) and epsilon_{yy}(0) are transverse
- finally I chose a case which holds for cubic symmetry only (e.g. what you refer to as "isotropic"). This is the reason of the ~ symbol, where the dielectric tensor is diagonal, and all the diagonal components are identical, besides the distinction of being longitudinal or transverse, which (I stress it again) depends on the direction of the momentum.

The explanation given in my previous post, instead, is more general, and holds for arbitrary symmetries. To this end, I fix the matrix element of the dielectric tensor (epsilon_xx), and consider different directions for the limit along which q->0

Last, you are right, the fact that the epsilon(0) does not depend on the direction of the limit "implies that" (/"results from the fact that") the dielectric tensor is analytical at q=0.
In other words epsilon_{ii}^L(0)=epsilon_{jj}^T(0). However this is not trivial. Indeed the dielectric tensor can be expressed in terms of the density-density response functions which is non analytical at q=0. This results in the so called L-T splitting in the excitonic dispersion.

For references I can suggest:
- section 8 of the paper by Strinati: https://link.springer.com/article/10.1007/BF02725962
- this paper by Andreani, Bassani, and Quattropiani: https://link.springer.com/article/10.1007/BF02454213

D.

[milesj]

This makes sense, thanks! And thanks for the references as well!