Possible mislabeling of 1b, 1c EBRs in p3, p3m1

[AliGhorashiCMT]

Calculating the bandreps manually and through crystalline seems to indicate that what Crystalline has as 1b (regardless of site symmetry group) is 1c and vice versa. I believe the culprit is

χᴳₖ += cis(2π*dot(kv′, tα′α′)) * χs[site_symmetry_index]

in line 170 of https://github.com/thchr/Crystalline.jl/blob/master/build/setup_2d_band_representations.jl

[thchr]

When we discussed this, I think I asked if you could give some more info on why you think they are wrongly labelled? I don't quite follow at the moment.

Presumably, we should fix this before eventual publication of your paper, right? Also, did you compare with the associated 3D space groups? They should have the same EBRs (just with redundant irrep info for the $k_z = \pi$ plane).

[thchr]

I still do not follow your thinking on the above: how did you arrive at this conclusion?

From a quick test now, doing the comparison with the 3D space group (comparing plane group 13 and space group 143, specifically), I did indeed see a difference - but afaict, this difference is due to a different labelling of the KA₂ and KA₃ irreps in the two cases. (It seems we picked different representative k-points in 2D and 3D also, specifically, having KA = [-1/3, -1/3, 0] in 3D and KA = [2/3, -1/3] in 2D; I cannot recall why).

[thchr]

If anything, it seems like the oddly labelled irreps are those of SG 143:

julia> characters(pgirreps(7,2)) # irreps of point group 3 in 2D
CharacterTable{2}: ⋕7 (3)
────┬──────────────────────────────────────
    │ Γ₁               Γ₂               Γ₃
────┼──────────────────────────────────────
  1 │  1                1                1
 3⁺ │  1  -0.5+0.866025im  -0.5-0.866025im
 3⁻ │  1  -0.5-0.866025im  -0.5+0.866025im
────┴──────────────────────────────────────


julia> characters(pgirreps(16,3)) # irreps of point group 3 in 3D
CharacterTable{3}: ⋕16 (3)
───────┬──────────────────────────────────────
       │ Γ₁               Γ₂               Γ₃
───────┼──────────────────────────────────────
     1 │  1                1                1
 3₀₀₁⁺ │  1  -0.5+0.866025im  -0.5-0.866025im
 3₀₀₁⁻ │  1  -0.5-0.866025im  -0.5+0.866025im
───────┴──────────────────────────────────────


julia> characters(lgirreps(143)["KA"]) # irreps of little group at KA in P3 (3D)
CharacterTable{3}: ⋕143 (P3) at KA = [-1/3, -1/3, 0]
───────┬───────────────────────────────────────
       │ KA₁              KA₂              KA₃
───────┼───────────────────────────────────────
     1 │   1                1                1
 3₀₀₁⁺ │   1  -0.5-0.866025im  -0.5+0.866025im
 3₀₀₁⁻ │   1  -0.5+0.866025im  -0.5-0.866025im
───────┴───────────────────────────────────────

julia> characters(lgirreps(13,2)["KA"]) # irreps of little group at KA in p3 (2D)
CharacterTable{2}: ⋕13 (p3) at KA = [2/3, -1/3]
────┬───────────────────────────────────────
    │ KA₁              KA₂              KA₃
────┼───────────────────────────────────────
  1 │   1                1                1
 3⁺ │   1  -0.5+0.866025im  -0.5-0.866025im
 3⁻ │   1  -0.5-0.866025im  -0.5+0.866025im
────┴───────────────────────────────────────

Note that the KA 2 and 3 irreps in P3 (3D) do not agree with the labelling of its corresponding point group irreps.
On the hand, the KA irreps do match with the point group irreps in p3 (2D).

[AliGhorashiCMT]

Returning to this, my reasoning is as follows: we know that in re-centering to position 1b (1/3, 2/3). Changes the K and KA irreps as follows: K1->K2->K3, KA1->KA3->KA2. With the current list of plane group 13 EBRS, this actually maps the 1b EBRS to the 1c EBRs under re-centering. However, this doesn't make sense. If an orbital is placed at 1b, then re-centering to 1b (putting the origin at 1b, i.e. using the new dielectric function epsilon'(r) = epsilon(r+1b)), should give you orbitals at 1a.

I checked what you said about the 3d plane groups/EBRs. We seem to be consistent with them, which is confusing me even more. Is it possible that this could be an inconsistency in how the symmetry eigenvalues were defined in the source used for the 3D EBRS? i.e. defining the symmetry eigenvalues as the eigenvalues <psi|g|psi> vs <psi|g^{-1}|psi>?

[AliGhorashiCMT]

I copied over the code in the calc_bandreps function (and the attendant code in the other methods it uses), and our code reproduces exactly what Bilbao gives (what you get if you just get the bandreps through the bandreps method)...which makes sense since we probably cross-checked this long ago. So, I'm leaning on there being an inconsistency with the Bilbao definition

[AliGhorashiCMT]

But fixing the sign in https://github.com/thchr/Crystalline.jl/blob/master/build/setup_2d_band_representations.jl does indeed give BRS that make sense with regards to re-centering.

[thchr]

As part of finally getting calc_bandreps merged and comprehensively tested (#60), I looked at whether the cis sign should be swapped: my conclusion is that it cannot be swapped without simultaneously breaking calc_bandreps for very many cases (which then fail to subduce onto LGIrreps correctly).

Also, calc_bandreps agrees with the 3D reference band representations from Bilbao in also cases that are possible to test uniquely. The sign is also consistent with our swapped signs elsewhere (I've added a note about this in f86a299).

Because of this, I feel quite sure the sign is correct as-is.

Did you compare the the results to the 3D parent poitn group p3 and p3m1?

[thchr]

So, given our discussion a few weeks back, should this issue be closed?