Crystallography, Group Cohomology, and Lieb-Schultz-Mattis Constraints

This paper systematically studies the mod 2 cohomology ring of 3D crystallographic space groups, presents a detailed classification, and applies it to 3D LSM constraints with additional on-site SO(3) symmetry. The authors obtained significant and fruitful results, which are particularly important for this field and potentially impactful beyond it.

The paper is well organized and well written. Its extensive and informative treatment of the mod 2 cohomology ring for 3D space groups, along with the related LSM classes presented in Appendix F and the accompanying open-source code, will be highly valuable for future research and deserves recognition. Although some general theories regarding the topological aspects of the LSM theorem have appeared in previous literature, this work provides additional insights—especially in 3D—by offering explicit lattice quantities that facilitate concrete calculations.

I strongly recommend that this paper be published in SciPost Physics.

Although optional, it would be very helpful if the authors could address the following questions and comments:
1. Generalization to Magnetic Groups: How can the results be generalized to magnetic groups?
2. UV Anomaly and Lattice Degrees of Freedom: It is stated that symmetric moving lattice degrees of freedom do not change the UV anomaly. Why is this the case? Since moving lattice sites typically alter the underlying lattice structure, how does one relate the UV anomaly for different UV lattices? I did not find a physical justification for this point in the referenced literature (if I did not overlook it).
3. Nontrivial Action in Group Cohomology: Is there a case where the group cohomology expression
H^p(P, H^q(T, Z_2)) involves a nontrivial action of P on H^q? It appears that the p4g group exhibits such a situation. How can this be derived explicitly?
4. Typographical Errors in Equations: I do not understand Equation 17 (as well as Equations 26 and 27). Are there any typographical errors?
5. Typos in Equation 20: It appears that there are typos in Equation 20, where the leftmost column of the table contains two terms with q=0.
6. Calculations in the LHS Spectral Sequence: How should one perform the calculations for d^r
with r≥3 in the LHS spectral sequence? I noticed some results in Ref. [81] that are challenging to interpret. Do you have any comments regarding this?
7. Definitions in Equation 21: Could you clarify the definitions of B_\beta
(with B_\beta = w_{11}) and Cγ (with C_\gamma = \tilde{w}_{12}) as stated in Equation 21?
8. Lattice Homotopy Proposal: Since this paper and some related previous works by one of the authors are based on the idea of lattice homotopy introduced in Ref. [34] (and possibly on the anomalous texture in Ref. [35]), does the lattice homotopy proposal capture all possible LSM-type constraints?
9. Definitions Around Equation 66: What are the definitions of \mu_1 and X_1 as presented around Equation 66?
10. Cohomology Class Form in Equation 67: Is there any justification for expressing the cohomology class of the LSM anomaly in the form presented in Equation 67?
11. Discrepancy Between Equations 67 and 80: In Equation 67, the authors propose that the topological class for the 3D LSM anomaly can be expressed in a simple form—a result justified by Equation 79 (from Ref. [81]) for the bosonic electric charge in a 3D U(1) quantum spin liquid. However, they later propose a generalization in Equation 80 that does not appear as simplified. Could you explain the reason for this discrepancy?
12. Association of IWP with H^3(G,Z_2 ): Regarding the 3D LSM anomaly, any IWP is associated with an element in H^3(G, Z_2), but I did not find a clear justification for this association. Although such reasoning is provided for the 2D LSM case in Ref. [15]—relying on an argument involving an SO(3) monopole braiding around a 1D Haldane chain and applicable only to SO(3) cases—I am concerned that extending this reasoning to 3D may involve new subtleties.
13. 3D Fermionic Systems: Do you have any comments regarding the implications of your work for 3D fermionic systems?

Publish (easily meets expectations and criteria for this Journal; among top 50%)

In the manuscript “Crystallography, Group Cohomology, and Lieb–Schultz–Mattis Constraints”, the authors computed the Z_2-coefficient cohomology for all the 3D space groups and obtained a complete set of Lieb-Schultz-Mattis (LSM) constraints for “SU(2) spins” on general 3D lattices. The authors use the U(1) quantum spin liquid on a 3D pyrochlore lattice as an example to demonstrate how the LSM constraints restrict the patterns of symmetry fractionalization on the electric and magnetic charges.

This work is an important addition to the literature on the general LSM constraints. The result is very systematic and thorough. I believe it will be a very useful reference for many future studies of highly correlated spin systems in 3D. I recommend the publication of this work on SciPost.

Here are some questions I hope the authors can comment on if possible:

As is mentioned in the manuscript, the lattice homotopy method by Ref. [34] and the topological theory of the LSM constraint by Ref. [35] yield the same result for “SU(2) spins” (which is tied to the Z_2 coefficient cohomology). With the method developed in the current manuscript, can the authors comment on whether the two approaches by Ref. [34] and [35] always produce the same answer, even for more general pseudo spins? If not, what type of pseudo spin will lead to a mismatch between these two approaches?

Is there an example of a 3D LSM anomaly that cannot be saturated by 3D U(1) spin liquid with any symmetry fractionalization pattern?

Is there a typo in (12)? Should the coefficient on the left-hand side be F instead of Z_2?

Publish (easily meets expectations and criteria for this Journal; among top 50%)