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Spontaneous breaking of finite group symmetries at all temperatures
by Pedro Liendo, Junchen Rong, Haoyu Zhang
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Submission summary
Authors (as registered SciPost users):  Junchen Rong 
Submission information  

Preprint Link:  https://arxiv.org/abs/2205.13964v3 (pdf) 
Date submitted:  20221018 08:27 
Submitted by:  Rong, Junchen 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study conformal field theories with finite group symmetries with spontaneous symmetry breaking (SSB) phases that persist at all temperatures. We work with two $\lambda \phi^4$ theories coupled through their mass terms. The two $\lambda \phi^4$ theories are chosen to preserve either the cubic symmetry group or the tetrahedral symmetry group. A oneloop calculation in the $4\epsilon$ expansion shows that there exist infinitely many fixed points that can host an alltemperature SSB phase. An analysis of the renormalization group (RG) stability matrix of these fixed points, reveals that their spectrum contains at least three relevant operators. In other words, these fixed points are tetracritical points.
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Reports on this Submission
Anonymous Report 2 on 20221129 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2205.13964v3, delivered 20221129, doi: 10.21468/SciPost.Report.6229
Strengths
1 The work is wellmotivated.
2 The analysis is largely selfcontained.
Weaknesses
1 There is a focus on a few specific examples but a lack of a general treatment of biconicaltype models.
2 The different fixed points and the ranges of $M$, $N$ for which they exist as unitary fixed points are not discussed at all. This is particularly relevant when the tetrahedral theory is considered.
Report
In this manuscript the authors perform an analysis of biconicaltype fixed points in the $\varepsilon=4d$ expansion in the context of persistent symmetry breaking. More specifically, they consider models of the type $\text{CFT}_1\times\text{CFT}_2$ coupled via their quadratic invariants, with $\text{CFT}_{1,2}$ taken to have either hypercubic or hypertetrahedral global symmetry, and examine the possibility that these biconicaltype CFTs have spontaneously broken global symmetry for arbitrarily high temperature. The authors indeed find that these models display persistent spontaneous symmetry breaking for a variety of $M,N$, thus enlarging the space of fixed points with such behavior in the $\varepsilon$ expansion. Since the symmetry groups considered are discrete, the authors' results may continue to hold all the way down to $d=3$.
The motivation for this work is clear, and the paper is well written and contains new results. Despite the fact that in the end this is a limited examplebyexample analysis, it is still a valuable addition to the literature around persistent symmetry breaking and should be published. A minor addition before publication is requested below.
Requested changes
1 In section 2.2 the authors should mention that for $M=3$ or $N=3$ the corresponding CFT's symmetry group is the same as that of the cubic theory. This relation is mentioned in section 5.3 but in relation to the results of section 5.1. In section 5.2 this makes their examples of persistent symmetry breaking of the type $\text{Cubic}(3)\times \text{Tetrahedral}(\geq 51)$. A corresponding comment should then be added in section 5.3, for $M=3$, $N\geq 51$ there. It should perhaps be pointed out in the discussion that all their examples of persistent symmetry breaking involve at least one cubic CFT in the product $\text{CFT}_1\times\text{CFT}_2$.
Anonymous Report 1 on 20221123 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2205.13964v3, delivered 20221123, doi: 10.21468/SciPost.Report.6186
Report
This paper demonstrates the existence of CFTs in $(4\epsilon)$ dimensions where certain finite group symmetries are spontaneously broken at any nonzero temperature. It builds upon the ideas developed in the last few years where the question of symmetry nonrestoration in the high temperature limit of QFTs has been recast as that of symmetry breaking at nonzero temperatures in CFTs. Since a CFT lacks any intrinsic scale, if any of its symmetries is broken at a nonzero temperature, it will remain broken at all nonzero temperatures, thereby providing an example of symmetry nonrestoration. The novelty of this work lies in finding new examples of CFTs with finite group symmetries (cubic and tetrahedral symmetries) which exhibit this phenomenon. This is particularly important as the MerminWagnerHohenbergColeman theorem rules out the possibility of spontaneous breaking of a continuous symmetry in a 3dimensional local QFT at any nonzero temperature. However, it imposes no such constraint on finite group symmetries. Therefore, I expect the new $(4\epsilon)$dimensional models introduced in this work to provide fresh clues for exploring whether persistent symmetry breaking is possible in 3dimensional local conformal theories. For this reason I would recommend the paper for publication after a few minor revisions.
Requested changes
I suggest the following minor revisions:
1 I think the authors should clearly mention which of the two symmetry groups is broken in each of the examples that they study and also clarify if there is any general pattern.
2 The penultimate line in the first paragraph of the introduction should be
"Another interesting feature is that these systems have “inverted” phase diagrams near the CFT fixed point, with the disordered phase at low temperature, while the ordered phase is at high temperature."
instead of
"Another interesting feature is that these systems have “inverted” phase diagrams near the CFT fixed point, with the ordered phase at low temperature, while the disordered phase is at high temperature."
3 Based on the form of the potential given in eq. 1.4 and the couplings given in the first line of eq. 2.4, there should be an additional factor of 3 multiplying $\lambda_1$ in the Lagrangian of the cubic(N) theory given at the beginning of section 2.1 in page 4.
4 The symmetry group for the fixed points that are given in table 1 should be Cubic(60)$\times$ Cubic(4) rather than Cubic(4) $\times$ Cubic(60).
5 In the last term of the first line of eq. 2.7 (and the same of eq. 2.8), the index $\sigma$ is written thrice. I think a couple of them should be replaced by a dummy index.
6 In the third term of the second line of eq. 2.7 (and the same of eq. 2.8 and eq. 2.10), there is an extra index $j$ in one of the delta functions which should be removed.
7 The above two typos are also there in eq. 2.11. Moreover, in its present form, eq. 2.11 is inconsistent with eq. 2.10 . To make these two equations consistent $\beta_1$ and $\beta_2$ in eq. 2.11 should be exchanged with $\beta_3$ and $\beta_4$ respectively. Furthermore, in eq. 2.11 the indices ($\mu,\nu,\rho,\sigma,\kappa$) should be exchanged with the indices ($i,j,k,l,m$).
8 In the second last line of section 2 (page 7), the figure that is referred to should be Fig. 1 instead of Fig. 2.4.