I was wondering if there are any examples or if you are aware of anyone who has used spgrep to find the irreducible force constants for a given crystal? I saw the examples on lattice vibrations which are very similar, however, not quite what I was after.

I am no expert in group theory, so best I understand it is that the block diagonalization of the dynamical matrix is somehow related to the irreducible form (maybe it is the irreducible form), but then my questions are:

• How does this extend to tensors (e.g. third order force constants)
• Where can I find resources combining the irreducible representation idea with lattice vibrations (phonons). Many of the group theory papers are too dense and most of the lattice vibration papers gloss over these steps.

Thanks!

Hi there, I'm reviewing this repository for JOSS. Everything else looks great to me, but I am unable to run examples/lattice_vibration.py as the file phonopy_mp-2998.yaml.xz is missing, and I couldn't find it in the raw data from http://phonondb.mtl.kyoto-u.ac.jp/ph20180417/d002/mp-2998.html. Is this an oversight, or am I missing something?

I noticed the following note in the source code:

Can you give an example to illustrate this situation? How large a rank will trigger this problem, and is there a way to deal with it, such as performing some sort of decomposition to lower the rank?

Regards,
Zhao

See my following example:

gap> SGGenSet227me;
[ [ [ 0, -1, 0, 1/2 ], [ 0, 0, -1, 1/2 ], [ -1, 0, 0, 1/2 ], [ 0, 0, 0, 1 ] ], [ [ -15/4, 29/4, -15/4, -15/16 ], [ -33/8, 55/8, -25/8, -25/32 ], 
      [ -25/8, 55/8, -33/8, -41/32 ], [ 0, 0, 0, 1 ] ] ]
gap> S1:=SpaceGroupOnLeftIT(3,227);
SpaceGroupOnLeftIT(3,227,'2')
gap> S2:=AffineCrystGroupOnLeft(SGGenSet227me);
<matrix group with 2 generators>
gap> conj:=AffineIsomorphismSpaceGroups(S2, S1);
[ [ 5/2, 3, 5/2, -5 ], [ 5/2, 5/2, 3, -21/4 ], [ 3, 5/2, 5/2, -5 ], [ 0, 0, 0, 1 ] ]
gap> S2^(conj^-1)=S1;
true

Here, S1 is the 3d space group 227 given in ITA, then how can I use your package to compute the irreducible representations of S2?

Regards,
Zhao

In recent years, topological materials science is a great revolutionary progress in the field of materials science, which is closely related to group theory, in which, a typical representative emerging discipline is Topological Quantum Chemistry. So, add some related features are also intriguing.

Related literature and studies:

https://doi.org/10.1038/nature23268
https://doi.org/10.1103/PhysRevB.102.115117
https://github.com/oashour/AbInitioTopo.jl

Regards,
Zhao

In the joss paper of this project, I noticed the following description:

1. You said "Although these packages do not rely on tabulations of irreps"

According to my understanding, the irvsp program also relies on table lookups. More specifically, it depends on the necessary information from the appropriate BCS database to execute its tasks, and its calculations are restricted to trace information exclusively.

2. You said "spgrep provides unique irreps for given space groups"

If the basis/wave function is specified first, then the representation matrix is completely determined; however, if the representation matrix is given first, the basis/wave function is naturally not unique.

So, I'm still unclear about what you're trying to express by saying "spgrep provides unique irreps for given space groups". It's unclear what you're referring to.

The representation matrix you provided naturally corresponds to a specific basis, and after changing the basis, it simply changes to an equivalent representation too.

3. What's "a deterministic symmetry-adapted basis" and why must it be unique?

Regards,
Zhao

It's well known that one of the most important applications of group representation theory in quantum mechanics is the analysis of irreducible representation of wave function of the system under study. Indisputably, it's a very meaningful thing to interface this package with other DFT codes or their auxiliary analysis tools, say, py4vasp, so that we can calculate and analysis the irreducible representations of electronic states based on the result of DFT calculation.

See here for some related discussion.

Regards,
Zhao

Based on the generators of space group 141 listed here, the following GAP checking can be performed:

gap> gensSGITA141O1:=[
>  [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0,0,0,1]],
>  [[-1, 0, 0, 1/2], [0, -1, 0, 1/2], [0, 0, 1, 1/2], [0,0,0, 1]],
>  [[0, -1, 0, 0], [1, 0, 0, 1/2], [0, 0, 1, 1/4], [0,0,0, 1]],
>  [[-1, 0, 0, 1/2], [0,  1, 0,  0], [0,  0,  -1, 3/4], [0,0,0, 1]],
>  [[-1, 0, 0, 0], [0, -1, 0, 1/2], [0, 0, -1, 1/4], [0,0,0, 1]],
>  [[1, 0, 0, 1/2], [0, 1, 0, 1/2], [0, 0, 1, 1/2], [0,0,0, 1]]
> ];;
gap> gensSGITA141O2:=[
>  [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0,0,0, 1]],
>  [[-1, 0, 0, 1/2], [0, -1, 0, 0], [0, 0, 1, 1/2], [0,0,0, 1]],
>  [[0, -1, 0, 1/4], [1, 0, 0, 3/4], [0, 0, 1, 1/4], [0,0,0, 1]],
>  [[-1, 0, 0, 1/2], [0, 1, 0, 0], [0, 0, -1, 1/2], [0,0,0, 1]],
>  [[-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0,0,0, 1]],
>  [[1, 0, 0, 1/2], [0, 1, 0, 1/2], [0, 0, 1, 1/2], [0,0,0, 1]]
> ];;
gap> SGITA141O1:=AffineCrystGroupOnLeft(gensSGITA141O1);
<matrix group with 6 generators>
gap> SGITA141O2:=AffineCrystGroupOnLeft(gensSGITA141O2);
<matrix group with 6 generators>
gap> ConjugatorSpaceGroups(SGITA141O1, SGITA141O2);
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 1/4 ], [ 0, 0, 1, 7/8 ], [ 0, 0, 0, 1 ] ]

Can the package perform the above or similar work to identify a possible isomorphism between two space groups?

Regards,
Zhao

Here gives the following example:

from spgrep import get_spacegroup_irreps
from spgrep.representation import get_character

# Rutile structure (https://materialsproject.org/materials/mp-2657/)
# P4_2/mnm (No. 136)
a = 4.603
c = 2.969
x_4f = 0.3046
lattice = [
    [a, 0, 0],
    [0, a, 0],
    [0, 0, c],
]
positions = [
    [0, 0, 0],  # Ti(2a)
    [0.5, 0.5, 0.5],  # Ti(2a)
    [x_4f, x_4f, 0],  # O(4f)
    [-x_4f, -x_4f, 0],  # O(4f)
    [-x_4f + 0.5, x_4f + 0.5, 0.5],  # O(4f)
    [x_4f + 0.5, -x_4f + 0.5, 0.5],  # O(4f)
]
numbers = [0, 0, 1, 1, 1, 1]

kpoint = [0.5, 0, 0]  # X point
irreps, rotations, translations, mapping_little_group = get_spacegroup_irreps(
    lattice, positions, numbers, kpoint
)

# Symmetry operations by spglib
assert len(rotations) == 16
assert len(translations) == 16

# At X point, the little co-group is isomorphic to mmm (order=8)
assert len(mapping_little_group) == 8
print(mapping_little_group)  # [ 0,  1,  4,  5,  8,  9, 12, 13]

# Two two-dimensional irreps
for irrep in irreps:
    print(get_character(irrep))
# [2.+0.j 0.+0.j 0.+0.j 2.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
# [2.+0.j 0.+0.j 0.+0.j -2.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]

In this example, the space group is P4_2/mnm (No. 136), and the selected k point is [0.5, 0, 0] (X point). I checked this information from BCS KVEC's convention represented here, as shown below:

As you can see, the two coordinates given above are not consistent. Any hints for this problem?

Regards,
Zhao

According to my understanding, the little group in nature is the site symmetry group in reciprocal space. On the other hand, the site symmetry group and the Wyckoff position has the following relationship:

In crystallography, a Wyckoff position is a point belonging to a set of points for which site symmetry groups are conjugate subgroups of the space group.

So, I think it should be possible to build/construct the little group based on Wyckoff positions.

Any hints/comments/corrections on this idea will be appreciated.

N.B.: I also noticed the related - to some extent, if not all - application here.

Regards,
Zhao