Dirac spin liquid as an "unnecessary" quantum critical point on square lattice antiferromagnets

Dear Editor, Thank you for the updated referee reports and editorial recommendation on our manuscript entitled “Dirac spin liquid as an “unnecessary” quantum critical point in square lattice antiferromagnets”, which we hereby resubmit to SciPost Physics Core. Having addressed all issues raised, we believe that the revised manuscript is now suitable for publication in SciPost Physics Core. Yours sincerely, Yunchao Zhang, Xue-Yang Song, and T. Senthil

Point-by-point response to the comments made by Referee # 1: Referee: I thank the authors for responding to my remarks (comments made by Referee # 1 in their reply). I am happy with most replies. Unfortunately I am still not satisfied about the discussion in response to one of my questions which led to Change 12. First, I am not following the usage of the word slower. If operator O1 has a higher scaling dimension than O2, in which sense is the two-point correlation function “slower” ? If anything it’s faster, since it’s decreasing faster. I think the use of “slower” is quite confusing. Can you replace the use of the word “slower” by just saying which operator has a larger scaling dimension? Response: We thank the referee for the careful reading of our manuscript. Instead of using the word “slower”, we have clarified that the flavor mass operator has a lower scaling dimension than the singlet mass operator in Change 3. Referee: Second, if these two operators did have unequal scaling dimensions, I don’t see why this should imply anything about breaking symmetry, even at a heuristic level. Has this argument ever been invoked already in the literature? If yes, a reference would be welcome. If no, a more detailed explanation of heuristic would be very interesting to have, perhaps in a footnote. Response: The two operators do have unequal scaling dimensions as the flavor mass operator has a lower scaling dimension than the singlet mass operator. Therefore, at the QED3 fixed point, the dominant long-wavelength correlations will come from the flavor mass operator and, one can imagine that slightly away from the fixed point, a flavor mass may be likely to condense, thus breaking the flavor symmetry. The intuition is that the operator with the slower correlations at the CFT fixed point is more “almost ordered” and if there is a relevant perturbation, then it is more likely to freeze in the fluctuations of this operator. A more concrete example of this heuristic argument is to consider an array of spin-1/2 chains coupled together by antiferromagnetic interactions between nearest neighbor chains. Each spin chain has power law Neel and VBS correlations; the Neel correlations are enhanced over the VBS by a log factor, so the Neel is (slightly) more slowly fluctuating than the VBS. Now the inter- chain coupling is known to be relevant. The belief is that the relevant flow takes you to the Neel ordered state, rather than the VBS ordered state (at least so long as the interchain interaction is not frustrating). This line of reasoning should only be taken as a rough heuristic in the absence of methods to probe the parameter space around the fixed point. We have added these comments in Changes 3 and 4. Referee: Third, Ref. [10] states that the flavor and singlet mass operators actually have the same scaling dimension in the large Nf limit, to all orders in 1/Nf expansion ([10], App.D). This seems to contradict to what is written in the current paper in the second paragraph of Section IV, which seems to imply that the dimensions are actually unequal. (Also since Ref. [10] does not compute the dimensions but cites another paper for the calculation, that paper also needs to be cited.) Response: Ref. [10] has an erratum which shows the flavor and singlet mass operators have different scaling dimensions. We have added the correct reference to this erratum in Changes 3 and 13. We have also added a reference to Ref. [24], which does the original calculation for the large Nf scaling dimensions.

Point-by-point response to the comments made by Referee # 2: Referee: The authors have addressed my questions, I am happy to recommend it for publica- tion. Response: We are grateful for the positive evaluation of our work and for recommending publication in SciPost Physics Core.

Point-by-point response to the comments made by Referee # 3: Referee: I thank the authors for addressing my comments! I noticed a few small things in the revision: Very minor: there is an instance of “there is are” in Section III. Response: We thank the referee for a careful reading of the manuscript. This typo is corrected in Change 1. Referee: A couple of lines afterwards, the authors say “we will focus on the ordering of the particular set of 5 operators”. I find the meaning of the word “ordering” a bit unclear; it would be great to rephrase. For instance, does it mean “condensation”/“acquiring an expecation value”? Response: We use the word ordering to mean condensation or acquiring an expectation value. This is clarified in Change 2. Referee: I find some aspects of the description of the various SO(6) representations in the Appendix still confusing/ incorrect. In particular: -in Table III, the expressions for the operators transforming in the 105, 175 are definitely incorrect. The operator that is supposed to transform in 105 is not symmetric in (i,j,k,l), which is a requirement of this representation. The operator that is supposed to transform in 175 does not vanish when contracted with δij , which I think should be the case. Response: We thank the referee for an attentive reading of the appendix. The expressions for 175 and 105 are incorrect as the traces being subtracted were not properly symmetrized. We have corrected this error in Change 8. Referee: because it is not symmetric in (i,j,k,l). - in Table IV, the expression for the operator transforming in the 50 is incorrect - in the text it is mentioned that the operator transforming in the 50 is the traceless part. This formula is incorrect because it is not symmetric in (i,j,k,l) Response: The expression for 50 is incorrect as the trace part being subtracted was not properly symmetrized. This is corrected in Change 11. The corresponding expression for the 50 irrep in the text is also corrected in Change 6. Referee: The phrase “minus traces and symmetric part” used in Tables III and IV is unclear. It is not clear whether one should symmetrize or remove the symmetric part. It would be better to give an actual formula like for the other cases. Response: We have added the explicit formulas for these cases in Changes 10 and 12. Referee: I would be very happy to recommend the paper for publication after these issues are addressed. Response: We are appreciation the positive evaluation of our work and for recommending publication in SciPost Physics Core.

List of changes made
1. Section III., Paragraph 2
There is are corrected to There are.
2. Section III., Paragraph 2
However, we will focus on the condensation of the particular set of 5 operators that couple
to the adjoint fermion billinear mass.
3. Section IV. Paragraph 2
A further clue on the nature of this condensate comes from examining the particle-hole
operator with the lowest scaling dimension at the QED3 fixed point: it is presumably these
operators that will acquire an expectation value once the monopole fugacity flows to strong
coupling. It is known from calculations of scaling dimensions[10,11,24](from the large-Nf
expansion) that the flavor mass operator¯
ψMψ, where M is an adjoint SU(4) matrix, has
a lower scaling dimension than the singlet mass operator¯
ψψ. Therefore, the dominant,
slowly decaying, long-wavelength correlations at the QED3 fixed point will arise from the
flavor mass operator¯
ψMψ. Due to these enhanced correlations at the QED3 fixed point, we
heuristically expect flavor symmetry to be broken.
4. Section IV. Paragraph 2, footnote
The intuition is is that the operator with the slower correlations at the CFT fixed point
is more “almost ordered” and if there is a relevant perturbation, then it is more likely to
freeze in the fluctuations of this operator, though a more thorough analysis would require a
consideration of the parameter space surrounding the fixed point. A more concrete example
of this heuristic argument is to consider an array of spin-1/2 chains coupled together by
antiferromagnetic interactions between nearest neighbor chains. Each spin chain has power
law Neel and VBS correlations; the Neel correlations are enhanced over the VBS by a log
factor, so the Neel is (slightly) more slowly fluctuating than the VBS. Now as the inter-chain
coupling is known to be relevant, the belief is that the relevant flow leads to the Neel ordered
state, rather than the VBS ordered state (at least so long as the interchain interaction is not
frustrating).
5. Appendix A.1., Paragraph 1
Note the (105,0) is the fully symmetric, traceless representation, while (84,0) contains all
the rest of the operators symmetric with respect to O→O†, minus the fully symmetric and
trace components.
6. Appendix A.2., Paragraph 2
Corrected the expression for the form of the 50 irrep.
7. Appendix, Table II
Corrected minor typo in the second column for the 15 irrep in Table II.
8. Appendix, Table III
Corrected the explicit tensor forms for the 105 and 175 irreps.
9. Appendix, Table III
Corrected minor typo in the second column for 20’ irrep in Table III.
10. Appendix, Table III
Provided explicit tensor form for the 84 irrep.
11. Appendix, Table IV
Corrected the explicit tensor form for the 50 irrep.
12. Appendix, Table IV
Provided explicit tensor form for the 64 irrep.
13. References
Added reference to Hermele, Senthil, and Fisher, Phys. Rev. B 76, 149906 (2007).