Altermagnetism and Altermagnets: A Brief Review

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please see the attached file.

Comments for author File: Comments.pdf

Author Response

Thank you for your review. For specific responses, please refer to the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper "Altermagnetism and altermagnets: A brief review" aims to provide a short review of the current status in the field of altermagnetism.
I find such a review timely and the selected topics to be appropriate.
However, in my opinion, the paper lacks coherence. It would benefit from a minor structure change (like splitting the paragraphs overloaded with information into smaller ones), a possible rewrite of some paragraphs, and adding sentences that provide a smoother transition between subtopics (when possible).
Thus, I would be inclined to recommend it for publication, but I think extra changes to improve the clarity and coherence are still necessary.

Below are my comments and suggestions:
1) I find the simultaneous use of the terms "altermagnet" and "non-collinear" misleading since altermagnetism is defined as a collinear magnetic phase.
However, I agree that non-collinear orders could also produce a nonrelativistic spin splitting of the band structure.
Please consider rewriting the relevant paragraph.

2) Section 2, Spin group theory description, attempts to give a broad overview of the spin group theory, but it gets too specific in some parts.
I would suggest to consider reworking it for more clarity.

3) "In case of RuO2 the related spin group symmetry is": here this example is coming "out of nowhere":
if one wants to use it, I would suggest adding a short crystal structure description.
It might be enough to mention that there are two Ru atoms in the unit cell and that the related Ru-O octahedra are rotated by 90 degrees w.r.t to each other (one could cite Fig. 1a here).

4) "The explicit spin-only group is defined by": here it is unclear to me if this is a specific example or if there is only one spin-only group.

5) "The three types of nontrivial spin Laue groups can describe different types of magnetic materials": this is one example of cases where starting a new paragraph would be beneficial.

6) It is unclear what bars over the symmetry operations mean: \bar{E} is an inversion, but what \bar{C_2} is?
Without further clarification, this notation might confuse a reader familiar with the double group theory.

7) "experimental techniques and theoretical frameworks": the word "frameworks" is mentioned in plural, but only one, DFT, is mentioned.
Was there an intent to mention more?
For clarity, it might be better to separate experimental techniques and theoretical frameworks into two separate lists.

8) "the presence of nonequivalent Mn sites": I am not sure why Mn sites are considered nonequivalent here, since a symmetry operation connects them.
I would suggest rewriting this paragraph for clarity.

9) Subsection "4.4. Perovskites" is a huge paragraph: it would benefit from being split.
This applies to other parts of the text too.

10) In 5.2. Nernst effects much of attention is devoted to RuO2, however, earlier it is mentioned that the presence of magnetism in this system is debatable.
Would it be possible to present this section in a more general format without too much focus on RuO2?
Otherwise, I think one would then need to explain what could be a potential explanation for the observation of the mentioned effects if RuO2 is a paramagnet.

11) Table 1:
a) RuO2 is mentioned twice: it would be better to group the two items.
b) please consider grouping the elements in the table by the space group designation.
c) I think using the double quote symbol to designate a repetition is misleading and redundant here:
the tick symbol does not require much more space, and now one has three symbols (with one not defined explicitly) instead of two.

12) Figure 1: 
a) I would suggest removing the gray background: its function is unclear to me.
b) another suggestion is to align different elements in the Figure.
For example, the AFM lattice in subfigure a) is shifted upward w.r.t the FM one.
An improvement would be to have the centers of all three subelements on a single horizontal line.
c) in subfigure b) the bandstructure "background" of the AFM case has a different size than the other two:
there is no particular reason for that.
d) I would like to ask for a minor correction to the bandstructure "background" too:
please make the vertical line to go through the intersection of the two black lines at the bottom. Currently, it is slightly offset to the right.
e) various shades of blue and red are used in the Figure. Please make them the same color.
The most prominent example is the colors of the lattice sites of FM in subfigure a) and the spin-up arrow.
f) in the caption of Figure 1, "Altermagnetic RuO2 with d-wave symmetry", since the illustration is rather general, one does not need to mention the RuO2; I think it would be enough to mention that it is a rutile structure.

13) Figure 2: please unify and align all of the subelements: for example, the sectors are of slightly different shapes and not all point to the center of the circle.

Author Response

Thank you for your review.  For specific responses, please refer to the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

I thank the Authors for their effort in answering my questions and incorporating the suggestions.
I find most of my comments to be addressed, but I still think it would be beneficial to make some extra minor changes, see my comments below.
Besides that, there is something strange with the pdf containing version 2 of the manuscript, as described in the following text.

Response 2-5:
First, I would like to point out that the "Spin group theory description" section in the second version of the manuscript (magnetism-3647734-peer-review-v2.pdf) is different from the one listed in the cover letter (magnetism-3647734-coverletter.pdf).
Thus, I'd like to advise the Authors to cross-check that the uploaded version of the manuscript contains the latest edits.
In my comments below, I'll follow the version from the cover letter.

Unfortunately, I still find the discussion in this section to be rather convoluted, and I think it would benefit the general audience if the section were more streamlined.
Of course, the content of the section is solely up to the Authors and I would not insist on applying major changes, but would kindly ask the Authors to consider the following comments and reshape the section for more clarity:
1) it could be worth mentioning that in the Landau theory, the order parameter for an antiferromagnet and an altermagnet is the same, thus within the theory, it's not possible to distinguish them.
Moreover, that is the motivation for McClarty et al. work to use a more sophisticated order parameter.
2) from the text, it is not clear what the benefit of employing the spin group theory would be to explain NRSS in contrast to the approach proposed by McClarty et al.
3) it would be useful to define what spin-only group and nontrivial spin group are.
4) one should mention that A is an antiunitary symmetry operation.

Response 6: I think this answer should be added to the main text.

Response 8: In the symmetry operation [C2||C4z], one should include the fractional translation too.

Response 11: Table 1 in the main text is the same as in the previous version of the manuscript.

Response 12-13: I'm not sure if I received the correct version of the pdf file, but Figures 1 and 2 are the same as in the previous version of the manuscript, although the Authors commented that they have been revised.

Author Response

COMMENTS TO THE AUTHOR AND RESPONSE FROM THE AUTHOR:

 Reviewer comment to the authors: (Round 2)

1) It could be worth mentioning that in the Landau theory, the order parameter for an antiferromagnet and an altermagnet is the same, thus within the theory, it's not possible to distinguish them. Moreover, that is the motivation for McClarty et al. work to use a more sophisticated order parameter.

2)  from the text, it is not clear what the benefit of employing the spin group theory would be to explain NRSS in contrast to the approach proposed by McClarty et al.

Response 1&2:  We thank the reviewer for carefully reviewing our manuscript. Following this comment, we have made changes in our manuscript, which now reads as follows:

“In Landau theory, traditionally applied to ferromagnets and antiferromagnets, the coupling between spin and real space is inherently a relativistic effect[12]. The primary order parameter is the Néel vector (difference between magnetic dipole moments of the opposite spin sublattice); it fails to distinguish AFM and AM[68]. As a result, higher-order multipoles may be suitable for the discussion of AMs, which break TRS. McClarty et al.[69] extended this framework by introducing a set of TRS-odd multipolar secondary order parameters that bridge the gap between ideas about spin symmetries and the observed NRSS. This approach has been effective in explaining NRSS and TRS breaking in rutile MnF2, owing to the ferroic ordering of magnetic octupoles[68](discussed in section 4.1).”

3) it would be useful to define what spin-only group and nontrivial spin group are.

4) one should mention that A is an antiunitary symmetry operation.

Response 3&4: Following these comments (3-4) we have made following changes.

“The spin-only group is defined as r_s=C∞+  C∞, here C∞ is the group containing all the spin space rotational transformations about the spins’ common axis, and  is the two-fold rotation about an axis orthogonal to the spin, followed by reversal of spin space. C∞ makes spin a good quantum number; hence, the band structure of the collinear magnets does not mix the spin-up and spin-down channels[5]. Time reversal symmetry (T ) operation is equivalent to spin space inversion followed by flipping of the crystal momentum(k) direction[70,71] in the nonrelativistic limit. The symmetry operation [ ||T ] forms the symmetry of collinear system. While the spin-only group rs is common to all the nonrelativistic collinear magnets. The nontrivial spin group(Rs) is constructed using an isomorphic coset decomposition between the spin-space transformation group and the real-space crystallographic transformation group.”

“The G − H = AH contains the residual half of the transformation, where A is an antiunitary symmetry operation, involving a real-space rotation(proper or improper, symmorphic or non-symmorphic), but excludes the identity and inversion operations. Only real space transformations that exchange atoms between sublattices of the same spin are contained in H, and exclusive real space transformations that exchange atoms between sublattices of opposite spin are present in G − H. The non-trivial spin subgroup [E||H] determines the anisotropic spin densities in real space. In contrast, [C2||AH]E(k,σ)=E(k’,-σ); this implies that E(k,σ)≠E(k’,-σ), indicating that [C2||AH] is responsible for broken TRS and non-relativistic spin splitting of bands. Together, there are ten nontrivial spin Laue groups categorizing d-,g-, and i-wave AMs[ 4 , 5].”

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Reviewer comment to the authors (Round 1):

Response 11: Table 1 in the main text is the same as in the previous version of the manuscript.

Response 12-13: I'm not sure if I received the correct version of the pdf file, but Figures 1 and 2 are the same as in the previous version of the manuscript, although the Authors commented that they have been revised.

• (Round 1) In 5.2. Nernst effects much of attention is devoted to RuO2, however, earlier it is mentioned that the presence of magnetism in this system is debatable. Would it be possible to present this section in a more general format without too much focus on RuO2? Otherwise, I think one would then need to explain what could be a potential explanation for the observation of the mentioned effects if RuO2 is a paramagnet.

Response 9: We have merged the two sections into one, which now reads as follows:

“Magneto-transport phenomena

The anomalous Hall effect(AHE) arises from breaking time-reversal symmetry in magnetic materials, typically due to the intrinsic (owing to Berry curvature) and extrinsic(due to skew scattering and side jump ) contributions, and is observed in FMs[ 192 ]. The AHE is also coined as the spontaneous Hall effect, which surprisingly was predicted and observed to occur in non-collinear antiferromagnets[193 – 199 ] and, particularly, in AMs[120 , 168 ,189 , 200 – 208 ]. The spontaneous Hall voltage is a phenomenon in which the application of an electric field causes the electrons to gain transverse velocity without an external magnetic field. This effect is related to the antisymmetric dissipation-free portion of the conductivity tensor, which is represented by the Hall pseudovector and controls the Hall current[109 ]. In collinear AFMs, the simplified magnetic structure defined solely by the spin orientations and spatial arrangement of magnetic atoms does not produce a spontaneous Hall conductivity. The necessary asymmetry emerges only when additional atoms, often nonmagnetic, occupy noncentrosymmetric sites. In AMs, the arrangement of non-magnetic atoms induces an asymmetry in the magnetization density between the two opposite spin sublattices. The inclusion of relativistic SOC generates anisotropic Berry curvature[ 120 ]. Anomalous transport coefficients can be obtained by integrating the Berry curvature within the BZ[ 109 ,190 ]. Smejkal et al. [ 109 ] initially calculated anomalous transport coefficients in RuO2 using first-principles and coined the term crystal Hall effect (CHE). After initial predictions of an AHE in RuO2[109], experiments followed quickly with a confirmation [189, 203] shown in Fig.7. Furthermore, AHE has been observed in altermagnetic thin films of Mn5Si3[202] and hexagonal MnTe[209 ]. In the thin film of RuO2, one can observe the reorientation of the Néel vector from [001] easy plane to [110] hard plane caused by the application of an electric field, allowing for the observation of AHE [188,198,203]. The Nernst effect, analogous to the AHE, is a fundamental phenomenon in which a longitudinal temperature gradient in a material gives rise to a transverse voltage, without the application of an external magnetic field. Similar to the AHE, in conventional collinear antiferromagnets, the anomalous Nernst effect (ANE) and the anomalous thermal Hall effect (ATHE) are expected to vanish. However, theoretical work conducted by Zhou et al. [ 190 ] revealed that significant thermal transport effects exist in RuO2 (magnetotransport measurements are shown in Fig.7), specifically known as the crystal Nernst effect (CNE) and the crystal thermal Hall effect (CTHE). These effects resemble the ANE and ATHE but are unique to the crystal structure of RuO2. The study suggests that the sporadic thermal and electrical transport coefficients in RuO2 adhere to an extended Wiedemann-Franz law in a wide temperature range of (0-150K)[ 190 ], a range much wider than what is typically expected for traditional magnetic materials. The detection of the AHE and CTHE in antiferromagnets, and consequently in AM, offers new possibilities for designing spintronic and spin caloritronics devices that leverage these materials’ robustness and unique properties. Potential applications include high-speed, low-power, and low-dissipative electrical devices as well as innovative quantum computer elements[1–3,210].”

Revised table 1

Space Group	Materials	[C2||A]
P63/mmc	CrSb [82,90–92] MnTe [75,96,98] CoNb4Se8 [113,114] Co1/4NbSe2 [115,116]	[C2||C6z]
P4/mmm	KV2Se2O [101] Rb1−δV2Te2O [100]	[C2||C4z]
I41/md	GdAlSi [117]	[C2||C4z]
P42/nmm	RuO2 [99,118,119]	[C2||C4z]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Revised Figure 1

Figure 1. a)Schematic comparison of spin densities in FM, AFM, and AM. FM and AFM have isotropic spin densities. Rutile AM with anisotropic spin density of magnetic atoms(red and blue), the double-headed arrow indicates that two opposite spin sub-lattices are connected by rotation only([C₂||C_4zt]). b) Comparison of the schematic band structures of FMs, AFMs, and AMs followed the iso-surface depicted at the bottom of the diagram. FMs show constant spin-splitting in the momentum space, whereas AFMs show no spin-splitting. However, the Band structure corresponding to AMs showssignificant alternating spin-splitting. These figures are reproduced from Ref.[4, 5] under CC BY 4.0 International license. Published by APS, copyright 2022.

 

 

 

 

 

Revised Figure 2

Figure 2. Schematic d-, g- and i- even-parity waveform of AMs depicted in 2D. The figure is adapted from ref[5] under CC-BY 4.0 International license. Published by American Physical Society(APS)

Additionally, we have revised Figure 5a. A simplified schematic illustrating the induction of altermagnetism via the twisting technique has been added, with permission obtained from the American Physical Society. The permission license has been included in the manuscript file.

Figure 5. Strategies for inducing altermagnetism: Fig a) Twisting of Van der Waals bilayer enabled identification of new AMs(e.g., VoBr, Co2S2, MnBi2Te4). Figure a) is reproduced with permission[134 ]. Copyright 2024, American Physical Society. b),c) Janus AM V2SeTeO formed by replacing the bottom Se layer with Te atoms. Figures b),c) are reproduced with permission[135 ]. Copyright 2023, AmericanChemical Society. d) Monolayer and bilayer of PtBr3. With suitable sliding of the top layer, either an altermagnetic(e) or ferroelectric(f) materials can be obtained. Figures d)-f) is reproduced with permission [136]. Copyright 2024, American Physical Society.

 

References

4. Šmejkal, L.; Sinova, J.; Jungwirth, T. Emerging research landscape of altermagnetism. Physical Review X 2022, 12, 040501.
5. Šmejkal, L.; Sinova, J.; Jungwirth, T. Beyond conventional ferromagnetism and antiferromagnetism: A phase with nonrelativistic spin and crystal rotation symmetry. Physical Review X 2022, 12, 031042
6. Fender, S.S.; Gonzalez, O.; Bediako, D.K. Altermagnetism: A chemical perspective. Journal of the American Chemical Society 2025, 147, 2257–2274.
7. Bhowal, S.; Spaldin, N.A. Ferroically Ordered Magnetic Octupoles in d-Wave Altermagnets. Phys. Rev. X 2024, 14, 011019. https://doi.org/10.1103/PhysRevX.14.011019
8. McClarty, P.A.; Rau, J.G. Landau theory of altermagnetism. Physical Review Letters 2024, 132, 176702.
9. Litvin, D.B. Spin point groups. Acta Crystallographica Section A: Crystal Physics, Diffraction, Theoretical and General Crystallography 1977, 33, 279–287.
10. Litvin, D.B.; Opechowski, W. Spin groups. Physica 1974, 76, 538–554.
11. Šmejkal, L.; González-Hernández, R.; Jungwirth, T.; Sinova, J. Crystal time-reversal symmetry breaking and spontaneous Hall effect in collinear antiferromagnets. Science advances 2020, 6, eaaz8809.
12. Mazin, I.I.; Koepernik, K.; Johannes, M.D.; González-Hernández, R.; Šmejkal, L. Prediction of unconventional magnetism in doped FeSb2. Proceedings of the National Academy of Sciences 2021, 118, e2108924118.
13. Naka, M.; Motome, Y.; Seo, H. Anomalous Hall effect in antiferromagnetic perovskites. Physical Review B 2022, 106, 195149.
14. Sattigeri, R.M.; Cuono, G.; Autieri, C. Altermagnetic surface states: towards the observation and utilization of altermagnetism in thin films, interfaces and topological materials. Nanoscale 2023, 15, 16998–17005.
15. Feng, Z.; Zhou, X.; Šmejkal, L.; Wu, L.; Zhu, Z.; Guo, H.; González-Hernández, R.; Wang, X.; Yan, H.; Qin, P.; et al. An anomalous Hall effect in altermagnetic ruthenium dioxide. Nature Electronics 2022, 5, 735–743.
16. Zhou, X.; Feng, W.; Zhang, R.W.; Šmejkal, L.; Sinova, J.; Mokrousov, Y.; Yao, Y. Crystal thermal transport in altermagnetic RuO 2. Physical review letters 2024, 132, 056701.
17. Nagaosa, N.; Sinova, J.; Onoda, S.; MacDonald, A.H.; Ong, N.P. Anomalous hall effect. Reviews of modern physics 2010, 82, 1539–1592.
18. Chen, H.; Niu, Q.; MacDonald, A.H. Anomalous Hall effect arising from noncollinear antiferromagnetism. Physical review letters 2014, 112, 017205.

 Kübler, J.; Felser, C. Non-collinear antiferromagnets and the anomalous Hall effect. Europhysics Letters 2014, 108, 67001.

194. Nakatsuji, S.; Kiyohara, N.; Higo, T. Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature. Nature 2015, 527, 212–215.
195. Sürgers, C.; Kittler, W.; Wolf, T.; Löhneysen, H.v. Anomalous Hall effect in the noncollinear antiferromagnet Mn5Si3. AIP Advances 2016, 6.
196. Boldrin, D.; Samathrakis, I.; Zemen, J.; Mihai, A.; Zou, B.; Johnson, F.; Esser, B.D.; McComb, D.W.; Petrov, P.K.; Zhang, H.; et al. Anomalous Hall effect in noncollinear antiferromagnetic Mn 3 NiN thin films. Physical Review Materials 2019, 3, 094409.
197. Šmejkal, L.; MacDonald, A.H.; Sinova, J.; Nakatsuji, S.; Jungwirth, T. Anomalous hall antiferromagnets. Nature Reviews Materials 2022, 7, 482–496.
198. Ghimire, N.J.; Botana, A.; Jiang, J.; Zhang, J.; Chen, Y.S.; Mitchell, J. Large anomalous Hall effect in the chiral-lattice antiferromagnet CoNb3S6. Nature communications 2018, 9, 3280.
199. Naka, M.; Hayami, S.; Kusunose, H.; Yanagi, Y.; Motome, Y.; Seo, H. Anomalous Hall effect in κ-type organic antiferromagnets. Physical Review B 2020, 102, 075112.
200. Attias, L.; Levchenko, A.; Khodas, M. Intrinsic anomalous Hall effect in altermagnets. Physical Review B 2024, 110, 094425.
201. Leiviskä, M.; Rial, J.; Bad’ura, A.; Seeger, R.L.; Kounta, I.; Beckert, S.; Kriegner, D.; Joumard, I.;

Schmoranzerová, E.; Sinova, J.; et al. Anisotropy of the anomalous Hall effect in thin films of the altermagnet candidate Mn 5 Si 3. Physical Review B 2024, 109, 224430.

203. Tschirner, T.; Keßler, P.; Gonzalez Betancourt, R.D.; Kotte, T.; Kriegner, D.; Büchner, B.; Dufouleur, J.; Kamp, M.; Jovic, V.; Smejkal, L.; et al. Saturation of the anomalous Hall effect at high magnetic fields in altermagnetic RuO2. APL Materials 2023, 11.
204. Jin, H.; Tan, Z.; Gong, Z.; Wang, J. Anomalous Hall effect in two-dimensional vanadium tetrahalogen with altermagnetic phase. Physical Review B 2024, 110, 155125.
205. Gonzalez Betancourt, R.; Zubáˇc, J.; Gonzalez-Hernandez, R.; Geishendorf, K.; Šobá ˇn, Z.;

Springholz, G.; Olejník, K.; Šmejkal, L.; Sinova, J.; Jungwirth, T.; et al. Spontaneous anomalous Hall effect arising from an unconventional compensated magnetic phase in a semiconductor. Physical Review Letters 2023, 130, 036702.

206. Nguyen, T.P.T.; Yamauchi, K. Ab initio prediction of anomalous Hall effect in antiferromagnetic CaCrO 3. Physical Review B 2023, 107, 155126.
207. Hou, X.Y.; Yang, H.C.; Liu, Z.X.; Guo, P.J.; Lu, Z.Y. Large intrinsic anomalous Hall effect in both Nb 2 FeB 2 and Ta 2 FeB 2 with collinear antiferromagnetism. Physical Review B 2023, 107, L161109.
208. Wang, M.; Tanaka, K.; Sakai, S.; Wang, Z.; Deng, K.; Lyu, Y.; Li, C.; Tian, D.; Shen, S.; Ogawa,N.; et al. Emergent zero-field anomalous Hall effect in a reconstructed rutile antiferromagnetic metal. Nature Communications 2023, 14, 8240.
209. Kluczyk, K.; Gas, K.; Grzybowski, M.; Skupi ´nski, P.; Borysiewicz, M.; F ˛as, T.; Suffczy ´nski, J.; Domagala, J.; Grasza, K.; Mycielski, A.; et al. Coexistence of anomalous Hall effect and weak magnetization in a nominally collinear antiferromagnet MnTe. Physical Review B 2024, 110, 155201.
210. Šmejkal, L.; Mokrousov, Y.; Yan, B.; MacDonald, A.H. Topological antiferromagnetic spintronics. Nature physics 2018, 14, 242–251.

Author Response File: Author Response.pdf

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

I thank the Authors again for their effort in answering my questions and incorporating the suggestions.
I find most of my comments to be addressed, albeit I still think that Figure 2 could benefit from being more symmetric:
for example, the schematic g-parity waveform has a gap between the circle and inner content on the top left part, but there is no gap at the bottom right.

I would like to kindly ask the Authors to revert the change regarding "one should mention that A is an antiunitary symmetry operation":
I overlooked the notation, and that's not the case.
Sorry for the inconvenience caused.

Author Response

Response to Reviewer 2 Comments

 

Ref. Manuscript Title: Altermagnetism and altermagnets: A brief review

Manuscript ID: magnetism-3647734

We would like to express our gratitude to the members of the editorial board and the appointed subject experts for carefully reviewing our manuscript and providing valuable comments for improvement. As communicated to us, the following revisions have been made to our manuscript.

 

Point-by-point response to Comments and Suggestions for Authors

Comment 1: I thank the Authors again for their effort in answering my questions and incorporating the suggestions. I find most of my comments to be addressed, albeit I still think that Figure 2 could benefit from being more symmetric: for example, the schematic g-parity waveform has a gap between the circle and inner content on the top left part, but there is no gap at the bottom right.

Author’s response: We thank the reviewer for carefully reviewing our manuscript. Following this comment we have revised figure 2.

Previous Figure 2

 

 

 

 

 

Revised Figure 2

Figure 2. Schematic d-, g- and i-even-parity waveform of AMs depicted in 2D. The figure is adapted from ref[5] under CC-BY 4.0 International license. Published by American Physical Society(APS)

 

 

 

 

Comment 2: I would like to kindly ask the Authors to revert the change regarding "one should mention that A is an antiunitary symmetry operation".

 

Author’s response: Following this comment we have removed the sentence “A is an antiunitary operator”. The correction is highlighted in page 5 which now reads as follows:

 

“The G − H = AH contains the residual half of the transformation, where A is a real-space rotation(proper or improper, symmorphic or non-symmorphic), but excludes the identity and inversion operations.”

Author Response File: Author Response.pdf