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Perturbative and Nonperturbative Studies of CFTs with MN Global Symmetry
by Johan Henriksson, Andreas Stergiou
Submission summary
As Contributors:  Johan Henriksson · Andreas Stergiou 
Arxiv Link:  https://arxiv.org/abs/2101.08788v1 (pdf) 
Date submitted:  20210308 11:06 
Submitted by:  Henriksson, Johan 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Fixed points in three dimensions described by conformal field theories with $MN_{m,n}= O(m)^n\rtimes S_n$ global symmetry have extensive applications in critical phenomena. Associated experimental data for $m=n=2$ suggest the existence of two nontrivial fixed points, while the $\varepsilon$ expansion predicts only one, resulting in a puzzling state of affairs. A recent numerical conformal bootstrap study has found two kinks for small values of the parameters $m$ and $n$, with critical exponents in good agreement with experimental determinations in the $m=n=2$ case. In this paper we investigate the fate of the corresponding fixed points as we vary the parameters $m$ and $n$. We find that one family of kinks approaches a perturbative limit as $m$ increases, and using large spin perturbation theory we construct a large $m$ expansion that fits well with the numerical data. This new expansion, akin to the large $N$ expansion of critical $O(N)$ models, is compatible with the fixed point found in the $\varepsilon$ expansion. For the other family of kinks, we find that it persists only for $n=2$, where for large $m$ it approaches a nonperturbative limit with $\Delta_\phi\approx 0.75$. We investigate the spectrum in the case $MN_{100,2}$ and find consistency with expectations from the lightcone bootstrap.
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Anonymous Report 1 on 2021414 Invited Report
Strengths
1 Very interesting method (bootstrap), which might provide an answer to longstanding controversies
Weaknesses
2  Some confusion on the applications
Report
The paper discusses the so called mn model from the bootstrap perspective. The paper contains interesting physics, but the presentation requires some significant changes. I suggest publication after revision.
Requested changes
1) The model discussed here is usually called "mn model" in the literature, while the results for helimagnets and XY stacked triangular antiferromagnets are usually discussed using the "chiral" model. To avoid confusion, it is important to mention that for m=n=2 the two models can be mapped one onto the other. It should also be mentioned that none of the "epsilon"expansion fixed points can describe such models (the only stable fixed point in the mn model cannot attract models in the chiral region) for such values of m and n.
2) After Eq. (1.2) the authors mention epsilonexpansion results. Within this approach, there are four fixed points. One can show nonperturbatively that the O(n) fixed point is stable (see A. Aharony,in Phase Transitions and Critical Phenomena, edited by C. Domb and J. Lebowitz Academic, New York, 1976, Vol. 6, p. 357). I think that a brief summary of the epsilonexpansion results would be needed, quoting the results of Aharony.
3) The authors report exponent estimates in Eq. (2.2) and then they compare them with "epsilonexpansion" results. I think the latter results should be reported. Second, it would be important to mention which fixed point is considered (if I'm note mistaken, it should be the mixed/cubic fixed point) among the four present.
4) Because of the mapping mnmodel > chiral model for m=n=2, in the literature there are also estimates of "nonepsilonexpansion" exponents, which are the object of the controversy mentioned at the beginning of p. 2. I think that the paragraph below Eq. (1.2) would be the proper place to present them. Are they close/very different from the results quoted in Eq. (1.2) ? It should be remarked that these
"nonepsilonexpansion" exponents are the only ones that can be compared with those for XY STAs.
5) The chiral model has been studied in two papers that I would suggest to quote:
Nakayama, Ohtsuki, Phys. Rev. D 89, 126009 (2014)
Nakayama, Ohtsuki, Phys. Rev. D 91, 021901(R) (2015)
Also in these two papers some "nonepsilon expansion" CFTs were found. Given that chiral and mn models agree for n=2,m=2, there should be a relation between the results obtained in the chiral setting and those obtained here. Some comments would be welcome. For instance, are the CFTs found for n=3,m=2 in the chiral model by Nakayama and Ohtsuki related (by smoothly changing n from 3 to 2) to the CFTs obtained here?
As an aside: is the approach of Nakayama, Ohtsuki (from a technical point of view) the same as the one used here? Do they implement the boostrap approach in the same way? Some comment is needed.
6) Fig. 1. Delta_phi and Delta_X are related to the standard critical exponents. It would be useful to report this relation in the caption. Probably, it would also help the reader to anticipate Eqs. (3.17) and (3.18) in the introduction .
7) The comparison made in Table 2 is not clear as it is mixing results of different nature. The experimental data refer to the chiral fixed point, and so do the MC results of Kawamura [I should mention that more accurate MC results are presented in Calabrese et al, Phys. Rev. B 70, 174439 (2004)]
On the other hand, the epsilon expansion results refer to the MN mixed point, which cannot describe the critical behavior of the XY STA. So they should not be compared with the STAs.
The bootstrap result quoted corresponds to kink 1. If I understand the analysis of the authors, it should not describe the STAs.
The only result result that might be relevant for the STAs is kink 2 which is not reported in the Table.
8) In the conclusions, the authors mention the failure of the epsilonexpansion. This is discussed at length by Calabrese et al. in the paper mentioned above. Another case is scalar electrodynamics: for N=2 there is a fixed point that cannot be obtained by using the epsilon expansion in the corresponding field theory [see Bonati et al., Phys. Rev. B 103, 085104 (2021)]. I think the discussion should be extended with more examples of systems where this phenomenon occurs.