Bootstrapping frustrated magnets: the fate of the chiral ${\rm O}(N)\times {\rm O}(2)$ universality class

With this resubmission we have taken into account the feedback given by the referees and addressed their concerns. A full list of changes is given below.

In addition we would like to provide the following replies to some of the remarks in the reports:

• Report 2 mentions "if C+ becomes a saddle point, local minima must appear at the same time (in agreement with Morse theory)." We are not aware of mathematical results which could prevent one of two minima in the negative region turning into a saddle in a one-parameter family of functions. Perhaps the following 1d example could help: The function f(x)=-1-x^2+x^4 has two symmetric minima, but deforming it by a linear term +h x the minimum at positive x turns into a saddle at h~1, while the minimum at negative x remains there. This is roughly what we believe happens as we decrease N, with C+ minimum turning into a saddle and disappearing, C- minimum remaining, and no new minima appearing.

• Report 2 mentions "If the minimum C+ disappears around N=7, I would expect that, for $3.76<N<7, the island found by the authors corresponds to an unstable fixed point in the renormalization-group sense." While the C+ minimum disappears, we do not interpret this as the fixed point disappearing here. Note that there is no exact correspondence between navigator minima and fixed points. What is more important is the conformal data values that such a stable CFT would take cannot be excluded. I.e. they are in the allowed region. It is quite possible that at higher derivative order the C+ minima would persist (and would be separated from other allowed regions such as the C- island and the peninsula). However, proving this expectation would require further analysis beyond what is reported in the paper. We mostly think our conservative language already reflects the fact that the identification is only tentative, but we clarify this further with the following edit:

"We observe several different features compared to Fig. 7. First, at N = 8 the navigator function now has two well separated minima, close to the respective expected positions of C± within the island. We track both minima which we therefore tentatively call the C+ navigator minimum (diamonds) and the C− minimum (dots)."

and the addition of the footnote:

"While it is very interesting to note that all local minima of the navigator function discovered up to now seem to have their origin in physical theories it is important to keep in mind that the absence of the "C+" minimum below N<7 does not imply this theory does not exist. What is more important is that the conformal data values that such a stable CFT would have are not excluded by our current analysis for N>3.78, but are rigorously excluded for N<3.78."

-Report 2 asks whether we understand the origins of the features at small N in figure 8. We restructured the paragraph and added a footnote that clarifies that there is not sufficient evidence to interpret these features:

"Instead we see in \cref{fig:trackingD3} that the $\mathcal{C}_+$ minimum disappears too soon, followed by a rather large region between $N_c^\text{CB}(3) = 3.78$ and $N\approx 6.5$ without good agreement with the $\epsilon$-expansion. All of this could however be numerical artifacts, and in future work it would be interesting to see how the \cref{fig:trackingD3} evolves when increasing $\Lambda$ and whether the features at small $N$ remain or the bounds converge closer to the ideal scenario.\footnote{At this point it is for example unclear whether there is physical significance to the change of slopes in the conformal data that is observed around $N=4.2$. This would be interesting to investigate further.}"

• Report 2 suggests to mention two references earlier which we have done.

- p3: Added a footnote mentioning related previous numerical conformal bootstrap studies earlier when the numerical conformal bootstrap is first introduced.
- p17: Added the word tentatively and restructured the sentence:
"We observe several different features compared to Fig. 7. First, at N = 8 the navigator function now has two well separated minima, close to the respective expected positions of C± within the island. We track both minima which we therefore tentatively call the C+ navigator minimum (diamonds) and the C− minimum (dots)."
- p17: Added the footnote: "While it is very interesting to note that all local minima of the navigator function discovered up to now seem to have their origin in physical theories it is important to keep in mind that the absence of the "C+" minimum below N<7 does not imply this theory does not exist. What is more important is that the conformal data values that such a stable CFT would have are not excluded by our current analysis for N>3.78, but are rigorously excluded for N<3.78."
- p 18: Restructured a paragraph: "Instead we see in \cref{fig:trackingD3} that the $\mathcal{C}_+$ minimum disappears too soon, followed by a rather large region between $N_c^\text{CB}(3) = 3.78$ and $N\approx 6.5$ without good agreement with the $\epsilon$-expansion. All of this could however be numerical artifacts, and in future work it would be interesting to see how the \cref{fig:trackingD3} evolves when increasing $\Lambda$ and whether the features at small $N$ remain or the bounds converge closer to the ideal scenario." and added the footnote: "At this point it is for example unclear whether there is physical significance to the change of slopes in the conformal data that is observed around $N=4.2$. This would be interesting to investigate further."
-p 25: Added an acknowledgement to the Resnick High Performance Computing Center, a facility supported by Resnick Sustainability Institute at the California Institute of Technology were some of the computations were performed.