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The Spin Point Groups and their Representations
by Hana Schiff, Alberto Corticelli, Afonso Guerreiro, Judit Romhányi, Paul McClarty
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Submission summary
| Authors (as registered SciPost users): | Hana Schiff |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2307.12784v2 (pdf) |
| Date submitted: | 2023-08-14 23:45 |
| Submitted by: | Schiff, Hana |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The spin point groups are finite groups whose elements act on both real space and spin space. Among these groups are the magnetic point groups in the case where the real and spin space operations are locked to one another. The magnetic point groups are central to magnetic crystallography for strong spin-orbit coupled systems and the spin point groups generalize these to the intermediate and weak spin-orbit coupled cases. The spin point groups were introduced in the 1960's and enumerated shortly thereafter. In this paper, we complete the theory of spin point groups by presenting an account of these groups and their representation theory. Our main findings are that the so-called nontrivial spin point groups (numbering 598 groups) have co-irreps corresponding exactly to the (co-)irreps of regular or black and white groups and we tabulate this correspondence for each nontrivial group. However a total spin group, comprising the product of a nontrivial group and a spin-only group, has new co-irreps in cases where there is continuous rotational freedom. We provide explicit co-irrep tables for all these instances. We also discuss new forms of spin-only group extending the Litvin-Opechowski classes. To exhibit the usefulness of these groups to physically relevant problems we discuss a number of examples from electronic band structures of altermagnets to magnons.
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Report
The paper addresses considerably interesting subject of spin groups. Generally, this is one of the nowadays frequent articles with exhausting group theoretical data. The main general remark is that the style is more appropriate for textbook: most of the methodological (mathematical) considerations are well known, making redundant many rederivations, proofs, constructions and long explanations. Instead, only basic information with references (which are correctly given) are expected. So, the body of the paper is to be essentially reduced.
Particular comments
1.Title "Spin Point Groups..." is missleading, as only 32 crystallographic point groups are considered, while infinitely many other ones are omitted.
In the introduction there are some remarks on the great role that symmetry has in physics, solid state in particular. Maybe reference to Poincare groups is a sort of show-off, basically out of context; also, group theory enters in physics through representation theory, and formulations like “The group theory and representation theory of crystals are the foundations…” is pleonastic.
2. Section 2 can be reduced, in particular 2.4. In fact, the standard definition is correctly referred to, but then also fully elaborated (without new details) in 2.1, while in the 2.2, 2.3 and 2.4 the first paragraph gives relevant references, and there is no need for long repetitions of the contained results. Also, a footnote 1 is amusing: Pin and SPin groups are hardly within the scope of the researchers reading this paper.
3. Section 3 is very detailed explanation of the theory of induction of (co-) representations from the index-two subgroups. It seems that the authors are not aware that this theory is well developed, as no reference is offered besides Bradley and Cracknell (e.g. Wigner's classical book, or Jansen and Boon, Theory of Finite Groups: Applications in Physics). So, this Section is to be drastically compressed (if not omitted). Also, characterization of the types of the representations is given in terms of Dimock's criterion, with comparison to the Frobenius-Schur (footnote), but Wigner's is not even mentioned (later on Frobenius-Schur test is used).
4. As far as formalism is considered, Section 4 contains some new results. Although it is correct, I am also here puzzled by the chosen mathematical terminology. Instead of only giving result for SO(3) (and SO(2)) integrals, here the Haar measure is mentioned in 4.2, but neither corresponding parameterization (Euler's angles?) nor measure itself is explicated; btw, in such “higher mathematical” framework, methodological hierarchy suggests to start with the compactness of the group, and two-sided measure. Simply, in this Section ordinary language is more adequate, in particular taking into account possible audience.
5. Section 5 is nice, and correctly written. I read it with interest.
6. In the last section, the summary is given. Besides emphasizing some results which are well known in the literature (e.g. doubling of co-representation for particular values of Dimock's indicator), it is correctly written.
7. Appendices B and C, and pure theoretical part (which is known) of D are not necessary, being mostly rephrasing of the known results. On the other hand, the remaining parts of D, with examples and concrete derivation are important, and instructive.
To conclude, in this form manuscript is not publicable. However, major revision can help, and I will be ready to reconsider it.
Recommendation
Ask for major revision
Strengths
1. The data produced is expected to be useful for researchers working in the field.
Weaknesses
1. The article should provide more specificity regarding the novelty of the presented data, especially concerning previous works on the subject.
Report
Presented here is a comprehensive review of spin point groups and their representations, featuring an exhaustive list of data. This resource is expected to be utilized by researchers in the field of condensed matter physics, especially those investigating systems with spin ordering that is either decoupled or weakly coupled to orbital degrees of freedom.
While the authors provide valuable insights, they could enhance clarity regarding the novelty of the presented data, particularly in comparison to prior works such as Litvin's study [22] on spin point groups (Acta Crystallographica Section A 33(2), 279, 1977) and Damnjanovic and Vujicic research [26] on subgroups of weak-direct products and magnetic axial point groups (Journal of Physics A General Physics 14, 1055, 1981).
Additionally, reference [28] by Damnjanovic, "Symmetry in Quantum Nonrelativistic Physics", Faculty of Physics Belgrade 2014, http://www.ff.bg.ac.rs/Katedre/QMF/SiteQMF/pdf/sqnp2e.pdf, should be fully specified and cited within the text of the article, i.e. within section 2 ("Introduction to the Spin Point Groups") and section 3 ("Review of Magnetic Representation Theory").
Requested changes
1. Within the Introduction enhance the clarity regarding the novelty of the presented data in comparison to prior works such as Litvin's study [22] on spin point groups (Acta Crystallographica Section A 33(2), 279, 1977) and Damnjanovic and Vujicic research [26] on subgroups of weak-direct products and magnetic axial point groups (Journal of Physics A General Physics 14, 1055, 1981).
2. The reference [28] Damnjanovic, "Symmetry in Quantum Nonrelativistic Physics" Faculty of Physics Belgrade 2014, http://www.ff.bg.ac.rs/Katedre/QMF/SiteQMF/pdf/sqnp2e.pdf, should be fully specified and cited within the text of the article, i.e. within section 2 ("Introduction to the Spin Point Groups") and section 3 ("Review of Magnetic Representation Theory").