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

This paper proposes the existence of quantum critical points in the phase diagram of lattice anti-ferromagnets described by $N_f = 4$ quantum electrodynamics in $2+1$ dimensions. An interesting aspect of this proposal is that the quantum critical points are between two identical phases of matter, a situation the authors refer to as that of an "unnecessary quantum critical points." I believe this is an interesting proposal, and that the paper should be published, but I find various parts of it a bit hard to follow in the current form. I have some concrete suggestions for how to improve it below.

1. In I.B, first paragraph: it would be helpful to spell out what $G_{UV}$ is explicitly, in particular to define ${\cal T}$ and to list the generators of the square lattice space group. I assume these generators are listed in Table I in Appendix A, but a brief explanation of what these generators correspond to (lattice translations, rotations, etc.) would be very helpful. Maybe a drawing of the lattice would be useful.

2. Paragraph containing eq. (5): it would be helpful to include a brief argument why the operator $\bar \psi_i \psi_j \Phi$ has dimension $\Delta_1 + 2 \sqrt{2}$ in the large $N_f$ limit. I assume it is because the lowest excitation of $\psi$ in the unit charge monopole background on the sphere has energy $\sqrt{2}$ (in units of the inverse radius of the sphere), but it would be good to see this explained more clearly.

3. It would be helpful to connect the discussion at the end of Section II (paragraphs 4, 5, 6) about $q=0$ operators to tables II and III in Appendix A. In particular, which operators in these tables are fermion bilinears, which operators are quartic in the fermions, etc.?

4. In the second paragraph of Section III, what does it mean for the two sets of 5 monopole operators to "couple" to four fermion terms or to fermion bilinear mass terms? Does the word "couple" mean that there's a term in the Lagrangian that contains products of monopole operators and two-fermion or four-fermion operaators? This doesn't make much sense, so the authors must mean something else by the word "couple". It would be great to clarify this point.

5. Section III, end of 2nd paragraph: In the sentence "We label these operators as $n^a$ with $a = 1, \ldots , 5$", what do the words "these operators" refer to? Do they refer to the monopole operators or to the fermion bilinears?

6. If the $n^a$ are fermion bilinears (as it is suggested in the 4th paragraph of Section III), which $SO(6)$ representation are the operators $n^a$ part of when $\lambda = 0$? Are they part of the ${\bf 15}$, which under the decomposition $SO(6) \to SO(5)$ becomes ${\bf 10} + {\bf 5}$, with the ${\bf 5}$ being the $n^a$? It would be great to clarify this point.

7. Section IV, second paragraph: it would be nice to give more details supporting the idea that, in the absence of monopoles, the fermions of QED$_3$ are the $2 \pi$ vortices of an order parameter carrying charge-1 under $U(1)_\text{top}$. Where does this come from? Why are such vortices not gauge-invariant? Why do they transform in a projective representation of $SO(6)$ (namely the ${\bf 4} + \bar {\bf 4}$)? I think more explanation is needed.

8. When writing $\bar \psi M \psi$ in the 2nd paragraph of Section IV, do the authors mean that $M$ is an $\mathfrak{su}(4)$ matrix instead of an $SU(4)$ matrix (i.e.~Lie algebra vs.~group element)? This should be made clear.

9. In the second paragraph of Section IV, regarding the second to last sentence, if I'm understanding it correctly: why is it that if the leading operator that breaks a symmetry has a lower scaling dimension than the leading operator that doesn't break the symmetry, then one should expect the symmetry to be broken? It would be great to have an explanation, or to clarify the statement if I misunderstood it.

10. Is there a difference between ${\bf n}$ in eq. (8) and $\hat n$ in Section IV.A? If there is, the authors should explain the difference; if not, it would be better I think to use the same notation everywhere.

11. After eq. (10) it is mentioned that the operator in (10) is part of the symmetric traceless representation of SO(6). Which symmetric traceless representation? The rank-two one, namely the ${\bf 20}'$? It would be great to make this explicit.

12. Shouldn't eq. (10) have $2(n_1^2 + n_2^2 + n_3^2) - 3 (n_4^2 + n_5^2)$ instead of $n_1^2 + n_2^2 + n_3^2 - n_4^2 - n_5^2$ so that it is a traceless polynomial?

13. Third paragraph of Section IV.A: what is the evidence that the RG flow diagram is that in Figure 2? This fact is just stated with no evidence provided as far as I can tell, so I think the authors should improve the explanation here.

14. Throughout the paper, the authors should change the notation for the rank-two symmetric traceless tensor representation of $SO(6)$ from ${\bf 20}$ to the more common notation ${\bf 20}'$. (The irrep the authors refer to is usually called the ${\bf 20}'$; the ${\bf 20}$ is usually a different irrep of $SU(4)$.)

15. Appendix A: It would be good to explain in more detail where the assumptions for the triangular and Kagome lattices come from. As an alternative, one can remove the mention of these lattices from the Appendix since it does not seem immediately relevant to the present paper.

16. It would be good to explain in more detail how Table I was derived. It seems very mysterious in its current form.

17. Appendix A, 2nd paragraph, regarding ''$({\bf 20}, 0)$ and $({\bf 84}, 0)$ are singlets in any lattice QED$_3$ simulation.'' Do the authors mean "are singlets" or "contain singlets"?

18. The allowed operators in the third columns of Tables II, III, IV do not obey the right symmetry and tracelessness conditions. For example, in Table II, the operator $O_1^\dagger O_1 + O_3^\dagger O_3$ is not traceless; instead, the traceless operator should be $O_1^\dagger O_1 + O_3^\dagger O_3 - \frac 12 (O_2^\dagger O_2 + O_4^\dagger O_4 + O_5^\dagger O_5 + O_6^\dagger O_6)$.

19. If I understand correctly, I think the "allowed" operators in the third columns of Tables II, III, IV are not all the allowed operators. For instance, in the bottom right box of Table II, it seems to me that we can also have $O_2^\dagger O_2 - \text{trace}$, $O_4^\dagger O_4 - \text{trace}$, etc. It would be great if the authors could comment on this point (or correct me if I'm wrong).

20. At the end of the Appendix, various estimates for scaling dimensions are given. The authors should give a reference or explain how they are derived.

21. A general comment: the authors propose that $N_f = 4$ QED$_3$ can be found in the phase diagram of square lattice anti-ferromagnets. But as far as I can tell, the arguments in Sections III and IV involve continuum field theory, in particular QED$_3$ and its deformations. Do I understand correctly that the continuum field theory picture is a good approximation only if the parameters $\lambda$ and $\kappa$ are close to the origin in Figure 2? If so, is it obvious that this range of parameters can be accessed from the lattice Hamiltonian? If not, where is the continuum approximation valid?

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The manuscript proposed Dirac spin liquid (DSL) as an unnecessary quantum critical point in square lattice antiferromagnets. The main idea is based on the conjecture that, deforming DSL with the relevant $2\pi$ monopole will trigger a RG flow towards the SO(5) DQCP, which will further flow to an ordered phase due to some dangerously irrelevant operator. Therefore, DSL can in principle appear as a critical point, inside the same ordered phase, i.e. Neel order or valence bond solid.

The proposal is interesting, and would be helpful for future experimental or numerical search of DSL inside the more conventional ordered phase. I am happy to recommend this paper.

1. In the second paragraph of Sec. IV, the authors wrote "Then the fermions of QED3 can be viewed as the basic $2\pi$ vortices...". I am not sure I understand this statement, maybe it is good to elaborate it a bit.

2. Page 6, first paragraph, there is a typo, "The QED3 CFT will then describe a sexond..." should be second.

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

This paper deals argues that QED$_3$ with $N_f=4$ fermions can be realized as an "unnecessary" critical point sitting within the Neel phase of 2+1 D quantum antiferromagnets on the square lattice (and also within the VBS phase). "Unnecessary" here means that on both sides of the codimension one critical manifold we find the same phase. Their argument starts by identifying the relevant operator which has to be tuned at criticality (charge-1 monopole). Then they argue that for any size of the perturbing monopole coupling the RG flow terminates in the same theory, SO(5) sigma model with a WZW term, Eq. (8). Then they discuss how this picture changes in presence of (dangerously) irrelevant perturbations arising from the microscopic symmetry.

This is an interesting paper and I think it should be published. I have however a few questions and requests.

- Consider the RG flows (6) with the irrelevant terms ... set to zero. Is the RG flow with $\lambda$ positive and negative supposed to be identical for all distances or only the IR fixed point (8) is supposed to be identical? The first possibility would be realized e.g. when perturbing the Ising model by the magnetic field perturbation, as is trivial to show since the sign of the coupling is flipped by a Z2. The second possibility is much more nontrivial. Whichever it is, it would be worth pointing out explicitly.

- p.3 last but one paragraph "hence must be irrelevant". "Must be" sounds confusing here. Can the authors rephrase it more explicitly, e.g. that they believe it to be irrelevant based on this evidence and you will assume it so? (This is in line with the description of $\Delta_2$ above)

- p.4 second paragraph. "One set of 5 will couple to four fermion terms" "orthogonal set of 5 operators that couple to the adjoint" I don't understand what the authors mean by these phrases. Could be some jargon that I'm not aware of. I'd be grateful for more details here.

- p.5 first paragraph "has slower correlations" Can the authors explain what "slower" means here and why this implies the expectation that flavor symmetry is broken?

- second column "supported by searches using the conformal bootstrap". Here and elsewhere the authors demonstrate familiarity with the conformal bootstrap results, which is great. Can they provide some references here, and they do in other places of the paper?

- p.6 sexond->second

- Appendix A. Are results in tables I,II,III,IV,V new? A citation or , alternatively, at least some details on the derivation would be needed.

- Ref. [52] - "to a tricritical point" Several recent papers providing evidence for tricritical point include 2405.06607, 2405.04470, 2307.05307. I believe a reference these and other relevant works here would be appropriate, to properly balance the cited evidence for the complexed fixed points.

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