# Spontaneous breaking of finite group symmetries at all temperatures

Pedro Liendo1, Junchen Rong2, Haoyu Zhang3 1DESY Hamburg, Theory Group,
Notkestraße 85, D-22607 Hamburg, Germany
2Institut des Hautes Études Scientifiques, 91440 Bures-sur-Yvette, France
3George P. & Cynthia Woods Mitchell Institute for Fundamental Physics and
Astronomy, Texas A&M University, College Station, TX 77843, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study conformal field theories with finite group symmetries with
spontaneous symmetry breaking (SSB) phases which 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 one-loop calculation in
the $4-\epsilon$ expansion shows that there exist infinitely many fixed points
that can host an all temperature 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.

###### Contents

  1. 1 Introduction
  2. 2 Coupled CFTs with finite group symmetries
    1. 2.1 Cubic(M) $\times$ Cubic(N)
    2. 2.2 Tetrahedral(M)$\times$Tetrahedral(N)
    3. 2.3 Cubic(M)$\times$Tetrahedral(N)
  3. 3 Discussion
  4. A large N analysis
    1. 1.1 Cubic(M) $\times$ Cubic(N)
    2. 1.2 Cubic(M)$\times$Tetrahedral(N)

##  1 Introduction

The study of spontaneous symmetry breaking phases that persist at all
temperatures has recently been revisited PhysRevLett.125.131603 ; Chai:2020zgq
, following the initial work of Weinberg almost fifty years ago
weinberg1974gauge . These SSB phases are described by conformal field theories
(CFTs) at finite temperature, and since the temperature is the only one scale
that these systems have, the physics at high temperature and low temperature
are related by scaling symmetry, which means that they are in the same phase.
Another interesting feature is that near the CFT fixed points, these systems
have phase diagrams with the ordered phase at high temperature, while the
disordered phase is at low temperature, which appear to violate the second law
of thermal dynamics. Such exotic inverted phase transitions were observed in
Rochelle or Seignette salt weinberg1974gauge ; jona1962ferroelectric .

In this note we consider scalar field theory with a CFT${}_{1}\times$CFT2
structure. More precisely, we consider two scalar field theories (CFT1 and
CFT2) coupled through their mass term.

$\mathcal{L}=\mathcal{L}_{CFT_{1}}(\vec{\phi}_{1})+\mathcal{L}_{CFT_{2}}(\vec{\phi}_{2})+g(\vec{\phi}_{1}\cdot\vec{\phi}_{1})(\vec{\phi}_{2}\cdot\vec{\phi}_{2}).$
(1.1)

For example, take CFT1 to be the O(m) invariant Wilson-Fisher CFT, and CFT2 to
be the O(N-m) invariant Wilson-Fisher CFT, we have the Lagrangian,

$\displaystyle\mathcal{L}_{CFT_{1}}(\vec{\phi}_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{\mu}\vec{\phi}_{1})\cdot(\partial_{\mu}\vec{\phi}_{1})+m_{1}\vec{\phi}_{1}\cdot\vec{\phi}_{1}+\lambda_{1}(\vec{\phi}_{1}\cdot\vec{\phi}_{1})^{2},$
$\displaystyle\mathcal{L}_{CFT_{2}}(\vec{\phi}_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{\mu}\vec{\phi}_{2})\cdot(\partial_{\mu}\vec{\phi}_{2})+m_{2}\vec{\phi}_{2}\cdot\vec{\phi}_{2}+\lambda_{2}(\vec{\phi}_{2}\cdot\vec{\phi}_{2})^{2}.$
(1.2)

Here $\vec{\phi}_{1}$ and $\vec{\phi}_{2}$ to are m-dimensional and
(N-m)-dimensional vectors respectively. This is precisely the
O(m)$\times$O(N-m) theory considered in Chai:2020zgq . Using both the
$4-\epsilon$ expansion and large-N methods, they show that these conformal
field theories have a spontaneous symmetry breaking (SSB) phase that persists
until infinite temperature. Setting $\epsilon=1$, the corresponding conformal
field theories are 2+1 dimensional quantum critical points. As noted already
in Chai:2020zgq , the all temperature SSB phase should be view as an SSB phase
in two spatial dimensions while the temporal dimension is compact with radius
$\beta=\frac{1}{T}$. The Coleman-Hohenberg-Mermin-Wagner theorem theorem
hohenberg1967existence ; coleman1973there ; mermin1966absence  forbids
spontaneous breaking of continuous symmetry in two dimension. In other words,
the all temperature SSB phase exist only in fractional dimensions.

In order to bypass the Coleman-Hohenberg-Mermin-Wagner theorem, one way to
proceed is to work with long range models which is the approach taken in
Chai:2021djc ; Chai:2021tpt . Another possible direction is to work with four
dimensional gauge theories Chaudhuri:2021dsq ; Chaudhuri:2020xxb .

In this note, we work with short-range models with finite symmetry groups. It
is known that there are families of Wilson-Fisher fixed points in $4-\epsilon$
dimensions with finite groups as their global symmetry groups. For a summary
of known fixed point see for example Rychkov:2018vya . The groups we will
consider here are the cubic groups $S_{N}\ltimes(Z_{2})^{N}$ PhysRevB.8.3323 ;
PhysRevB.8.4270 ; Wallace_1973  and the tetrahedral groups $S_{N+1}\times
Z_{2}$ Zia_1975 . We will call these CFTs “Cubic(N) CFTs” and “Tetrahedral(N)
CFTs”. We take the CFT1 and CFT2 in (1.1) to be these finite group CFTs. As
explained in weinberg1974gauge ; Chai:2020zgq , searching for all temperature
SSB phases corresponds to search for conformal fixed points, whose thermal
mass squared matrix

$M_{ij}^{2}=\frac{\beta^{-2}}{24}\lambda_{ijkk}\bigg{|}_{\lambda=\lambda^{*}}$
(1.3)

has negative eigenvalues. This formula comes from the one loop renormalization
of the zero mode mass due to the Kaluza-Klein modes along the temporal circle.
The coupling constant is defined through the relation

$V(\phi)=\frac{1}{4!}\lambda_{ijkl}\phi^{i}\phi^{j}\phi^{k}\phi^{l},$ (1.4)

where we have used $\phi^{i}$ to collectively denote both $\vec{\phi}_{1}$ and
$\vec{\phi}_{2}$.

In real life experiments, relevant operators of a conformal field theory that
are invariant under all flavor symmetry groups correspond to experimental
parameters that need to be fine-tuned Pelissetto:2000ek . It is therefore
desirable to search for conformal field theories with as little relevant
operators as possible. Notice the group O(m)$\times$O(N-m) has two quadratic
invariants, which are the two mass terms $(\vec{\phi}_{1}\cdot\vec{\phi}_{1})$
and $(\vec{\phi}_{2}\cdot\vec{\phi}_{2})$. These mass operators are relevant.
One may wonder if it is possible to consider scalar field theories whose
flavor symmetry allow only one mass term. It is shown in Chai:2020zgq  that
these models can not host all temperature SSB phases (at least
perturbatively). Additional relevant operator may come from the quadratic
$\phi^{4}$ terms. Following the convention of Chai:2020zgq , the one loop beta
function for the quartic couplings is given by

$\beta_{ijkl}(\lambda_{ijkl})=-\epsilon\lambda_{ijkl}+\frac{1}{16\pi^{2}}(\lambda_{ijmn}\lambda_{mnkl}+\textrm{2
permutations}).$ (1.5)

The conformal fixed points are given by

$\beta_{ijkl}(\lambda_{ijkl})=0.$ (1.6)

Due to symmetry, the quartic couplings $\lambda_{ijkl}$ are not independent.
Denote the independent couplings as $\lambda_{i}$, then the stability matrix
of the RG flow can be written as

$H_{ij}=\frac{\partial{\beta_{i}}}{\partial\lambda_{j}}\bigg{|}_{\lambda=\lambda^{*}}.$
(1.7)

The negative eigenvalues of $H_{ij}$ correspond to relevant operators. In
looking for CFTs with a lower number of relevant operators, we will calculate
the renormalization group stability matrix $H_{ij}$ of these fixed points. The
result, however, shows that for all the candidate CFTs that can host all
temperature SSB phases, their $H_{ij}$ matrix have at least one negative
eigenvalue. A similar result was already notice in Chai:2020zgq  for the
O(m)$\times$O(N-m) model. This means that the all temperature SSB phases
appear near tetracritical points of the their phase diagrams, which can make
the search for these phases in real life or computer experiments challenging.
One example of such tetracritical points is the tetracritical Ising model,
which is also the two dimensional minimal model $M_{6,5}$
francesco2012conformal ; Frohlich:2006ch ; Chang:2018iay  with central charge
$c=4/5$. Another interesting tetracritical point is the O(5) invariant Wilson
Fisher fixed point as discussed in kivelson2001thermodynamics ;
hu2001bicritical . In these studies, the lattice model preserves the
SO(2)$\times$SO(3) subgroup of SO(5). In view of this subgroup, the O(5)
Wilson Fisher fixed point contains three relevant operators that are invariant
under the lattice symmetry, they are $O_{1}=\phi_{1}^{2}+\phi_{2}^{2}$,
$O_{2}=\phi_{3}^{2}+\phi_{4}^{2}+\phi_{5}^{2}$ and
$O_{3}=(\phi_{1}^{2}+\phi_{2}^{2})(\phi_{3}^{2}+\phi_{4}^{2}+\phi_{5}^{2})$.
This model interestingly describes the intervening between super-fluid phase
and the anti-ferromagnetic phase. For a review of this model, see also Section
6.2 of Pelissetto:2000ek .

##  2 Coupled CFTs with finite group symmetries

For convenience, we scale out $\frac{1}{16\pi^{2}}$ and $\epsilon$ in the beta
function (2.2) by redefining

$\tilde{\lambda}=\frac{\lambda}{16\pi^{2}\epsilon}.$ (2.1)

The condition that $\beta_{ijkl}=0$ becomes

$\tilde{\lambda}_{ijkl}=\tilde{\lambda}_{ijmn}\tilde{\lambda}_{mnkl}+2\,\textrm{permutations}$
(2.2)

In the following we will drop the tilde for convenience.

###  2.1 Cubic(M) $\times$ Cubic(N)

As already mentioned, the Cubic(N) group is isomorphic to
$S_{N}\ltimes(Z_{2})^{N}$. The N copies of the $Z_{2}$ subgroup flip the signs
of the $\phi_{i}$ fields, while the $S_{N}$ group permutes them. There are two
invariant tensors $\delta_{\mu\nu}$ and

$\delta_{\mu\nu\rho\sigma}=\bigg{\\{}\begin{array}[]{l}1,\quad\text{if}\quad\mu=\nu=\rho=\sigma\\\
0,\quad\text{otherwise.}\\\ \end{array}$ (2.3)

We take both CFT1 and CFT2 to be the Cubic groups. The coupling constants are

$\displaystyle\lambda^{(1)}_{\mu\nu\rho\sigma}=\lambda_{1}(\delta_{\mu\nu}\delta_{\rho\sigma}+\delta_{\mu\rho}\delta_{\nu\sigma}+\delta_{\mu\sigma}\delta_{\nu\rho})+\lambda_{2}\delta_{\mu\nu\rho\sigma}$
(2.4)
$\displaystyle\lambda^{(2)}_{ijkl}=\lambda_{3}(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{ijl}\delta_{jk})+\lambda_{4}\delta_{ijkl}$
(2.5) $\displaystyle\lambda^{(c)}_{\mu\nu
ij}=\lambda_{5}\delta_{\mu\nu}\delta_{ij}$ (2.6)

where $\lambda_{\mu i\nu j},\lambda_{\mu ij\nu}\cdots$ are fixed by
permutations.

According to the group structure, the beta function $\beta_{IJKL}$ can be
written as

$\displaystyle\beta_{IJKL}=$
$\displaystyle\beta_{1}(\delta_{\mu\nu}\delta_{\rho\sigma}+\delta_{\mu\rho}\delta_{\nu\sigma}+\delta_{\mu\sigma}\delta_{\nu\rho})+\beta_{2}\delta_{\mu\nu\rho\sigma}$
(2.8)
$\displaystyle+\beta_{3}(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})+\beta_{4}\delta_{ijkl}+\beta_{5}(\delta_{\mu\nu}\delta_{ij}+\cdots)$

From (2.2), we can read off the $\beta_{i}$’s,

$\displaystyle\beta_{1}=-\lambda_{1}+8\lambda_{1}^{2}+2\lambda_{2}\lambda_{1}+\lambda_{1}^{2}M+\lambda_{5}^{2}N$
(2.9)
$\displaystyle\beta_{2}=-\lambda_{2}+3\lambda_{2}^{2}+12\lambda_{1}\lambda_{2}$
(2.10)
$\displaystyle\beta_{3}=-\lambda_{3}+8\lambda_{3}^{2}+2\lambda_{4}\lambda_{3}+\lambda_{3}^{2}N+\lambda_{5}^{2}M$
(2.11)
$\displaystyle\beta_{4}=-\lambda_{4}+3\lambda_{4}^{2}+12\lambda_{3}\lambda_{4}$
(2.12)
$\displaystyle\beta_{5}=\lambda_{5}\left(\lambda_{2}+2\lambda_{3}+\lambda_{4}+4\lambda_{5}+\lambda_{1}(M+2)+\lambda_{3}N-1\right).$
(2.13)

Solving for fixed points and plugging the fixed points couplings into the
thermal mass matrix, we can examine the eigenvalues of $M_{ij}$ case by case .
For a specific choice of M, there is an infinite family of choices of N, which
leads to CFTs with all temperature SSBs. The results are summarized in Fig. 1.

Figure 1: Cubic(M)$\times$Cubic(N). The right panel is a zoom in of the left
panel plot.

The linear growth of the plot at large N can be seen from a large N analysis
(with M$/$N fixed). The details are summarized in Appendix A. It’s always the
symmetry group with smaller order that gets spontaneously broken. The RG
stability matrix $H_{ij}$ of these fixed points have at least one negative
eigenvalue. Together with the two mass terms, this means that these CFTs have
at least three relevant operators (which are invariant under the flavor
symmetry groups).

It is not particularly illuminating to present all the details of the
solutions of (2.9). To demonstrate our result, we list in Table 1 the coupling
constants, mass matrix eigenvalues and the RG stability matrix eigenvalues in
the Cubic(60)$\times$Cubic(4) case. We have omitted the fixed points when at
least one of the coupling constants vanishes. They correspond either to CFTs
when the symmetry gets enhanced to continuous groups, or CFTs when CFT1 and
CFT2 are decoupled.

fixed points | $\lambda_{1}$ | $\lambda_{2}$ | $\lambda_{3}$ | $\lambda_{4}$ | $\lambda_{5}$ | $m_{1}/(\frac{2}{3}\pi^{2}\epsilon\beta_{\text{th}}^{2})$ | $m_{2}/(\frac{2}{3}\pi^{2}\epsilon\beta_{\text{th}}^{2})$
---|---|---|---|---|---|---|---
F1 | 0.00420718 | 0.316505 | 0.0631077 | 0.0809026 | -0.00922458 | 0.540451 | -0.0939264
F2 | 0.00330785 | 0.320102 | 0.0496178 | 0.134862 | 0.0105606 | 0.567431 | 1.0662
F3 | 0.00520833 | 0.3125 | 0.00520833 | 0.3125 | 0.00520833 | 0.65625 | 0.65625

fixed points | $H_{1}$ | $H_{2}$ | $H_{3}$ | $H_{4}$ | $H_{5}$
---|---|---|---|---|---
F1 | 1. | 0.929986 | 0.259747 | -0.205562 | 0.0527271
F2 | 1. | 0.914259 | 0.24244 | -0.23492 | 0.0359878
F3 | 1. | 0.979167 | -0.3125 | 0.3125 | 0

Table 1: Coupling constants, mass matrix eigenvalues and the RG stability
matrix eigenvalues of the Cubic(60)$\times$Cubic(4) fixed points. Here
$\lambda_{i}$ are the couplings at the critical point, $m_{i}$ and $H_{i}$ are
the eigen-values of $M_{ij}$ and $H_{ij}$ respectively. The F1 fixed point has
negative $m_{2}$, therefore can host all temperature SSB phase.

###  2.2 Tetrahedral(M)$\times$Tetrahedral(N)

The Tetrahedral group is simply the product of the permutation group $S_{N+1}$
and the cylic group $Z_{2}$. The scalar fields transform in the standard (N
dimensional) representation of the group $S_{N+1}$. It is well known that in
this representation, the group $S_{N+1}$ has two invariant tensors
$\delta_{\mu\nu}$ and $d_{\mu\nu\rho}$ which is fully symmetric and traceless.
The explicit construction of $d_{\mu\nu\rho}$ can be found in Zia_1975 . The
independent coupling constants can be defined as

$\displaystyle\lambda^{(1)}_{\mu\nu\rho\sigma}=\lambda_{1}(\delta_{\mu\nu}\delta_{\rho\sigma}+\delta_{\mu\rho}\delta_{\nu\sigma}+\delta_{\mu\sigma}\delta_{\nu\rho})+\lambda_{2}(d_{\mu\nu\kappa}d_{\kappa\rho\sigma}+d_{\mu\rho\kappa}d_{\kappa\nu\sigma}+d_{\mu\kappa\sigma}d_{\sigma\rho\nu})$
(2.14)
$\displaystyle\lambda^{(2)}_{ijkl}=\lambda_{3}(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})+\lambda_{4}(d_{ijm}d_{mkl}+d_{ikm}d_{mjl}+d_{ilm}d_{mkj})$
(2.15) $\displaystyle\lambda^{(c)}_{\mu\nu
ij}=\lambda_{5}\delta_{\mu\nu}\delta_{ij}$ (2.16)

where $\lambda_{\mu i\nu j},\lambda_{\mu ij\nu}\cdots$ are fixed by
permutation. From

$\displaystyle\beta_{IJKL}=$
$\displaystyle\beta_{1}(\delta_{\mu\nu}\delta_{\rho\sigma}+\delta_{\mu\rho}\delta_{\nu\sigma}+\delta_{\mu\sigma}\delta_{\nu\rho})+\beta_{2}(d_{\mu\nu\kappa}d_{\kappa\rho\sigma}+d_{\mu\rho\kappa}d_{\kappa\nu\sigma}+d_{\mu\sigma\kappa}d_{\kappa\rho\nu})$
(2.19)
$\displaystyle+\beta_{3}(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})+\beta_{4}(d_{ijm}d_{mkl}+d_{ikm}d_{mjl}+d_{ilm}d_{mkj})$
$\displaystyle+\beta_{5}(\delta_{\mu\nu}\delta_{ij}+\cdots),$

and (2.2), we can read out the beta functions

$\displaystyle\beta_{1}=-\lambda_{1}+\frac{4\lambda_{2}^{2}(M-1)(M+1)^{4}}{M^{6}}+\lambda_{1}\left(\frac{4\lambda_{2}(M-1)(M+1)^{2}}{M^{3}}\right)+\lambda_{1}^{2}(M+8)+\lambda_{5}^{2}N$
(2.20)
$\displaystyle\beta_{2}=-\lambda_{2}+12\lambda_{1}\lambda_{2}+\frac{3\lambda_{2}^{2}(M+1)^{2}(3M-7)}{M^{3}}$
(2.21)
$\displaystyle\beta_{3}=-\lambda_{3}+\lambda_{5}^{2}M+\frac{4\lambda_{4}^{2}(N-1)(N+1)^{4}}{N^{6}}+\lambda_{3}\left(\frac{4\lambda_{4}(N-1)(N+1)^{2}}{N^{3}}\right)+\lambda_{3}^{2}(N+8)$
(2.22)
$\displaystyle\beta_{4}=-\lambda_{4}+12\lambda_{3}\lambda_{4}+\frac{3\lambda_{4}^{2}(N+1)^{2}(3N-7)}{N^{3}}$
(2.23)
$\displaystyle\beta_{5}=-\lambda_{5}+4\lambda_{5}^{2}+\frac{2\lambda_{2}\lambda_{5}(M-1)(M+1)^{2}}{M^{3}}+\frac{2\lambda_{4}\lambda_{5}(N-1)(N+1)^{2}}{N^{3}}$
(2.24)
$\displaystyle\quad\quad+\lambda_{3}\lambda_{5}(N+2)+\lambda_{1}\lambda_{5}(M+2).$
(2.25)

From our numerical search, we find that all temperature SSB phases appear only
when $M=3$ and $N\geq 51$ (or when $N=3$ and $M\geq 51$). Notice the
Tetrahedral(3) group is isomorphic to the Cubic(3) group. We will consider the
Cubic(M)$\times$Tetrahedral(N) CFTs in the next section. Overall, all the CFTs
we found to exhibit persistent symmetry breaking have at least one symmetric
group which is the Cubic group. The RG stability matrix $H_{ij}$ of these
fixed points have at least one negative eigen-values.

###  2.3 Cubic(M)$\times$Tetrahedral(N)

In the cases of Cubic(M)$\times$Tetrahedral(N), we define the coupling
constant as

$\displaystyle\lambda^{(1)}_{\mu\nu\rho\sigma}=\lambda_{1}(\delta_{\mu\nu}\delta_{\rho\sigma}+\delta_{\mu\rho}\delta_{\nu\sigma}+\delta_{\mu\sigma}\delta_{\nu\rho})+\lambda_{2}\delta_{\mu\nu\rho\sigma}$
(2.26)
$\displaystyle\lambda^{(2)}_{ijkl}=\lambda_{3}(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})+\lambda_{4}(d_{ijm}d_{mkl}+d_{ikm}d_{mjl}+d_{ilm}d_{mkj})$
(2.27) $\displaystyle\lambda^{(c)}_{\mu\nu
ij}=\lambda_{5}\delta_{\mu\nu}\delta_{ij}$ (2.28)

where $\lambda_{\mu i\nu j},\lambda_{\mu ij\nu}\cdots$ are fixed by symmetry.
From

$\displaystyle\beta_{IJKL}=$
$\displaystyle\beta_{1}(\delta_{\mu\nu}\delta_{\rho\sigma}+\delta_{\mu\rho}\delta_{\nu\sigma}+\delta_{\mu\sigma}\delta_{\nu\rho})+\beta_{2}\delta_{\mu\nu\rho\sigma}+\beta_{3}(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$
(2.30) $\displaystyle+\beta_{4}(d_{\ij
m}d_{klm}+d_{ikm}d_{jkm}+d_{ilm}d_{jkm})+\beta_{5}(\delta_{\mu\nu}\delta_{ij}+\cdots),$

and (2.2), we get

$\displaystyle\beta_{1}=-\lambda_{1}+2\lambda_{1}\lambda_{2}+\lambda_{1}^{2}(M+8)+\lambda_{5}^{2}N$
(2.31)
$\displaystyle\beta_{2}=-\lambda_{2}+\lambda_{2}\left(12\lambda_{1}+3\lambda_{2}\right)$
(2.32)
$\displaystyle\beta_{3}=-\lambda_{3}+\lambda_{5}^{2}M+\frac{4\lambda_{4}^{2}(N-1)(N+1)^{4}}{N^{6}}+\lambda_{3}\left(\frac{4\lambda_{4}(N-1)(N+1)^{2}}{N^{3}}\right)+\lambda_{3}^{2}(N+8)$
(2.33)
$\displaystyle\beta_{4}=-\lambda_{4}+\lambda_{4}\left(12\lambda_{3}+\frac{3\lambda_{4}(3N-7)(N+1)^{2}}{N^{3}}\right)$
(2.34)
$\displaystyle\beta_{5}=-\lambda_{5}+\lambda_{5}\left(2\lambda_{1}+\lambda_{2}+2\lambda_{3}+4\lambda_{5}+\lambda_{1}M+\frac{2\lambda_{4}(N-1)(N+1)^{2}}{N^{3}}+\lambda_{3}N\right).$
(2.35)

When $N$ is small, there are no negative mass eigenvalues no matter how large
$M$ is. For every $M$, there is always an infinite family of Tetrahedral(N)
CFTs such that they can couple together and the spontaneous symmetry breaking
happens. The results are summarized in Fig. 2. The linear behavior of the plot
at large N can be seen from a large N analysis, the details are summarized in
Appendix A. Notice when $N=3$, $M\geq 44$ is allowed. This agrees with the
result in Fig. 1, after considering the group isomorphism
Cubic(3)$=$Tetrahedral(3). These fixed points, as before, have at least one
renormalization group flow unstable direction. When $M\neq 3$, it’s always the
Cubic group that gets spontaneously broken at finite temperature.

Figure 2: Cubic(M)$\times$Tetrahedra(N). The right panel is a zoom in of the
left panel plot. Notice that $N=3$ and $M\geq 44$ is allowed, as indicated by
the straight line close to the N-axis.

##  3 Discussion

It was noticed in Chai:2020zgq  that the O(m)$\times$O(N-m) fixed point has
one RG flow unstable direction. Analyzing the stability matrix of several
other CFTs proposed to exhibit all temperature SSB phases, we notice that they
all share the same feature. These include the long range models in
Chai:2021djc , and the 3+1 dimensional gauge theories proposed in
Chaudhuri:2021dsq . Together with the two mass terms, this means that in real
life or computer experiments, one needs to fine tune at least three parameters
to reach the fixed point. This type of fixed points are accessible through
numerical experiments hu2001bicritical .

Ideally though, one would like to find CFTs which can host an all temperature
SSB phase, and whose operator spectrum contains as few relevant operators as
possible. This could be achieved by looking at more general models. Notice we
have only considered a sort of restricted set of fixed points, where two CFTs
are coupled through their mass operators. One possibility is to consider
scalar field theories with a single finite group $G$. One can choose
$\vec{\phi}_{1}$ and $\vec{\phi}_{2}$ to be two irreducible representations of
$G$, and then search for solutions to (1.6), such that $H_{ij}$ has no
negative eigenvalues, while $M_{ij}$ has at least one. Preferably, the order
of the finite group should be also small. This is a well defined group
theoretical question. However, to work in this vast space of theories requires
a more systematic study. The GAP system gapgap  for computational discrete
algebra and in particular its small group library will be useful in searching
for potential finite groups and their representations. For an application of
GAP to study SUSY fixed points, see Liendo:2021wpo .

The beta functions we have used are one loop perturbative beta functions in
$4-\epsilon$. It will be interesting to continue calculating higher loop
corrections to shed light on the (non)existence of 2+1D CFTs whose SSB phase
persists at all temperatures ($\epsilon=1$). Our models are closely related to
the models with non-trivial large N limit discussed in PhysRevB.11.239 , as
the Cubic(N) CFT at large N can be treated as N copies of the Ising model
coupled together. The critical exponents of the Cubic(N) CFT can then be
calculated by the diagrammatic method developed in Binder:2021vep  using the
conformal data of the three dimensional Ising model from the numerical
bootstrap el2012solving ; el2014solving2 ; kos2014bootstrapping ;
kos2016precision ; simmons2017lightcone . The Cubic(M)$\times$Cubic(N) CFTs
which we discussed in Section 2.6 may be treated in a similar manner when $M$
and $N$ are both large (with $M/N$ fixed). Such an analysis could give
evidence for the existence of 2+1D CFTs with an all temperature SSB phase,
beyond the one-loop results of this paper. We leave this for future work.

One can also try to engineer a two dimensional lattice Hamiltonian by coupling
$M+N$ copies of the quantum Ising model together. To start, let us consider
the transverse field Ising model sachdev2011quantum  on a square lattice,

$H_{\text{Ising}}=-J\sum_{\langle ij\rangle}Z_{i}Z_{j}-h\sum_{i}X_{i}\,.$
(3.1)

Here $Z_{i}$ and $X_{i}$ are the Pauli matrices. The symbol $\langle
ij\rangle$ means that the sites $i$ and $j$ are nearest neighbors. It is well-
known that as one tunes the parameter $h/J$, such a model goes through a
second order phase transition, which is simply the 2+1 dimensional Ising CFT.
Coupling M-copies of Ising models together, we get the quantum Cubic model

$H_{\text{cubic}}=-J\sum_{a=1}^{M}\sum_{\langle
ij\rangle}Z^{(a)}_{i}Z^{(a)}_{j}-J_{2}\sum_{a\neq b}\sum_{\langle
ij\rangle}Z^{(a)}_{i}Z^{(a)}_{j}Z^{(b)}_{i}Z^{(b)}_{j}-h\sum_{i,a}X^{(a)}_{i}\,.$
(3.2)

Here the symbol $a$ enumerates the M copies of the Ising spin, that live on
the same site. The size of the Hilbert space on each site is therefore
$2^{M}$. One can also introduce other coupling terms, such as

$-h_{2}\sum_{i,a\neq b}X^{(a)}_{i}X^{(b)}_{i}\,,$

as long as it preserves the Cubic symmetry. This model can be viewed as a 2+1
dimensional quantum version of the N-color grest1981n  Ashin-Teller model
PhysRev.64.178 . See solyom1981duality ; o2015symmetry  for the study of the
two-color quantum Ashin-Teller model in 1+1 dimensions. In the two dimensional
parameter space, $(J/h,J_{2}/h)$, there should exist a line of critical points
corresponding to the Cubic(M) CFT. We can now try to couple two Cubic
Hamiltonians together,

$H=H_{1}+H_{2}+H_{3}\,,\\\ $ (3.3)

with

$\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle-J\sum_{a=1}^{M}\sum_{\langle
ij\rangle}Z^{(a)}_{i}Z^{(a)}_{j}-J_{2}\sum_{a\neq b}\sum_{\langle
ij\rangle}Z^{(a)}_{i}Z^{(a)}_{j}Z^{(b)}_{i}Z^{(b)}_{j}-h\sum_{i,a}X^{(a)}_{i}\,,$
$\displaystyle H_{2}$ $\displaystyle=$
$\displaystyle-J\sum_{\rho=1}^{N}\sum_{\langle
ij\rangle}Z^{\prime(\rho)}_{i}Z^{\prime(\rho)}_{j}-J^{\prime}_{2}\sum_{\rho\neq\sigma}\sum_{\langle
ij\rangle}Z^{\prime(\rho)}_{i}Z^{\prime(\rho)}_{j}Z^{\prime(\sigma)}_{i}Z^{\prime(\sigma)}_{j}-h^{\prime}\sum_{i,\rho}X^{\prime(\rho)}_{i},$
$\displaystyle H_{3}$ $\displaystyle=$
$\displaystyle-K\sum_{a=1}^{M}\sum_{\rho=1}^{N}\sum_{\langle
ij\rangle}Z^{\prime(\rho)}_{i}Z^{\prime(\rho)}_{j}Z^{(a)}_{i}Z^{(a)}_{j}\,.$

Other terms such as

$-H\sum_{i}(\sum_{a}X^{(a)}_{i})(\sum_{\rho}X^{\prime(\rho)}_{i}),$ (3.4)

are also allowed as long as they preserve the Cubic(M)$\times$Cubic(N)
symmetry. Based on the field theory calculation, we conjecture that in the
five dimensional parameter space
$(J_{2}/J,h/J,J_{2}^{\prime}/J,h^{\prime}/J,K/J)$, there exist a two
dimensional critical surface, corresponding to the Cubic(M)$\times$Cubic(N)
fixed points discussed in Section 2.6. In other words, one needs to fine tune
three parameters to reach this critical surface. Changing the other two
parameters corresponds to turning on irrelevant operators of the CFT, which
will not drive the RG flow away from the critical surface. Given that we
choose the size of the group properly, once we turn on temperature at this
critical surface, the SSB phase persists at all temperatures. Since the size
of our cubic groups are large, testing this conjecture might be challenging.

The SSB phases at all temperature are interesting for black hole physics
Buchel:2009ge ; Buchel:2020jfs ; Buchel:2020thm ; Buchel:2021ead ;
Buchel:2022zxl  through the AdS/CFT correspondence Maldacena:1997re ;
Gubser:1998bc ; Witten:1998qj , if the corresponding CFT has a dual AdS theory
with the Einstein Hilbert action as the gravitational term. Our models, like
the models in Chai:2020zgq , are vector models whose large N limit will not
have standard AdS duals. The single trace spectrum contains higher spin
operators, but not conserved Rong:2017cow . If the dual theory indeed exist,
it should be a Vasiliev’s higher spin theory VASILIEV1990378 ; Vasiliev:1999ba
with deformations that breaks the higher spin symmetry.

###### Acknowledgements.

We thank Noam Chai, Anatoly Dymarsky and Eliezer Rabinovici for insightful
discussion about thermal order in conformal field theories and thank Zheng Yan
for valuable discussion about tetracritical phenomena. JR is supported by the
Huawei Young Talents Program at IHES. HZ is grateful to his advisor Chris
Pope’s support. HZ is supported in part by DOE grant DE-FG02-13ER42020 and the
Mitchell Institue. PL acknowledges support from the DFG through the Emmy
Noether research group “The Conformal Bootstrap Program” project number
400570283, and through the German-Israeli Project Cooperation (DIP) grant
“Holography and the Swampland”.

##  Appendix A large N analysis

###  1.1 Cubic(M) $\times$ Cubic(N)

We can now perform a large N analysis of the fixed points. More precisely, we
take the $M\rightarrow\infty$ limit the ratio $N/M$ fixed. From the numerical
results notice the coupling constants have the following large M behavior
$\lambda_{2}\sim\lambda_{4}\sim\frac{1}{3}+O(\frac{1}{N}),\lambda_{1}\sim\lambda_{3}\sim\lambda_{5}\sim
O(\frac{1}{N})$. Inspired by the above observation, we take the following
ansatz

$\displaystyle
N=xM,\quad\lambda_{1}=\frac{a_{1}}{M}+\frac{a_{11}}{M^{2}}+\cdots,\lambda_{2}=\frac{1}{3}+\frac{a_{2}}{M}+\frac{a_{22}}{M^{2}}+\cdots,\lambda_{3}=\frac{a_{3}}{M}+\frac{a_{33}}{M^{2}}+\cdots,$
(A.1)
$\displaystyle\lambda_{4}=\frac{1}{3}+\frac{a_{4}}{M}+\frac{a_{44}}{M^{2}}+\cdots,\lambda_{5}=\frac{a_{5}}{M}+\frac{a_{55}}{M^{2}}+\cdots$
(A.2)

and do $\frac{1}{M}$ expansion for beta function, leading order equations are

$\displaystyle\beta_{1}=\frac{2a_{5}a_{55}x+8a_{1}^{2}+2\left(a_{2}+a_{11}\right)a_{1}-\frac{a_{11}}{3}}{M^{2}}+\frac{a_{5}^{2}x+a_{1}^{2}-\frac{a_{1}}{3}}{M}+\cdots$
(A.3)
$\displaystyle\beta_{2}=\frac{3a_{2}^{2}+12a_{1}a_{2}+4a_{11}+a_{22}}{M^{2}}+\frac{4a_{1}+a_{2}}{M}+\cdots$
(A.4)
$\displaystyle\beta_{3}=\frac{2a_{3}\left(a_{33}x+a_{4}\right)+8a_{3}^{2}+2a_{5}a_{55}-\frac{a_{33}}{3}}{M^{2}}+\frac{a_{3}^{2}x-\frac{a_{3}}{3}+a_{5}^{2}}{M}+\cdots$
(A.5)
$\displaystyle\beta_{4}=\frac{3a_{4}^{2}+12a_{3}a_{4}+4a_{33}+a_{44}}{M^{2}}+\frac{4a_{3}+a_{4}}{M}+\cdots$
(A.6)
$\displaystyle\beta_{5}=\frac{a_{5}\left(a_{33}x+2a_{1}+a_{2}+2a_{3}+a_{4}+4a_{5}+a_{11}\right)+a_{55}\left(a_{3}x+a_{1}-\frac{1}{3}\right)}{M^{2}}+$
(A.7)
$\displaystyle\quad\quad\quad+\frac{a_{5}\left(a_{3}x+a_{1}-\frac{1}{3}\right)}{M}+\cdots$
(A.8)

Solving the leading order equation, we can find

$a_{1}=\frac{1}{3}\left(1-3a_{3}x\right),\quad
a_{2}=\frac{4}{3}\left(3a_{3}x-1\right),\quad a_{4}=-4a_{3},\quad
a_{5}=-\frac{\sqrt{a_{3}-3a_{3}^{2}x}}{\sqrt{3}},$ (A.9)

with $a_{3}$ left free, indicating the existence of a large-N perturbative
conformal manifold. If we include the sub-leading corrections, we find

$\displaystyle a_{1}=\frac{1}{6},\quad a_{2}=-\frac{2}{3},\quad
a_{3}=\frac{1}{6x},a_{4}=-\frac{2}{3x},\quad a_{5}=-\frac{1}{6\sqrt{x}},\quad
a_{11}=\frac{-3a_{33}x^{2}+x+2\sqrt{x}+1}{3x},$ (A.10) $\displaystyle\quad
a_{22}=\frac{4\left(3a_{33}x^{2}-x-2\sqrt{x}-1\right)}{3x},\quad
a_{44}=-4a_{33},\quad a_{55}=0.$ (A.11)

Plugging them back to the mass matrix, we find the two eigenvalues

$m_{1}=\frac{1}{6}\left(3-\sqrt{x}\right)+\cdots,\quad
m_{2}=\frac{1}{6}(3-\frac{1}{\sqrt{x}})+\cdots.$ (A.12)

Apparently, we need either $x>9$ or $x<1/9$ so that one of the eigenvalues
will be negative.

###  1.2 Cubic(M)$\times$Tetrahedral(N)

For the Cubic(M)$\times$Tetrahedral(N) fixed points, we find a similar large N
behavio in this
case$\lambda_{2}\sim\frac{1}{9}+O(\frac{1}{N}),\lambda_{4}\sim\frac{1}{3}+O(\frac{1}{N}),\lambda_{1}\sim\lambda_{3}\sim\lambda_{5}\sim
O(\frac{1}{N})$. We take our ansatz to be

$N=xM,\lambda_{1}=\frac{a_{1}}{M},\lambda_{2}=\frac{1}{9}+\frac{a_{2}}{M},\lambda_{3}=\frac{a_{3}}{M},\lambda_{4}=\frac{1}{3}+\frac{a_{4}}{M},\lambda_{5}=\frac{a_{5}}{M}.$
(A.13)

The leading order beta function becomes

$\displaystyle\beta_{1}=\frac{1}{M}(a_{1}^{2}x-\frac{5a_{1}}{9}+a_{5}^{2}+\frac{4}{81x})$
(A.14)
$\displaystyle\beta_{2}=\frac{1}{M}(\frac{4a_{1}}{3}+a_{2}-\frac{1}{27x})+\cdots$
(A.15)
$\displaystyle\beta_{3}=\frac{1}{M}(a_{5}^{2}x+a_{3}^{2}-\frac{a_{3}}{3})+\cdots$
(A.16) $\displaystyle\beta_{4}=\frac{1}{M}(4a_{3}+a_{4})+\cdots$ (A.17)
$\displaystyle\beta_{5}=\frac{a_{5}}{M}\left(a_{1}x+a_{3}-\frac{4}{9}\right)+\cdots$
(A.18)

Solving them gives

$\quad a_{2}=\frac{1-36a_{1}x}{27x}\quad
a_{3}=\frac{1}{9}\left(4-9a_{1}x\right)\quad
a_{4}=\frac{4}{9}\left(9a_{1}x-4\right)\quad
a_{5}=-\frac{\sqrt{-81a_{1}^{2}x^{2}+45a_{1}x-4}}{9\sqrt{x}},$ (A.19)

with $a_{1}$ left free, indicating the existence of a large-N perturbative
conformal manifold. Plugging them back to mass matrix, we find

$m_{1}=-\frac{\sqrt{-81a_{1}^{2}x^{2}+45a_{1}x-4}}{9\sqrt{x}}+a_{1}x+\frac{2}{9},m_{2}=-\frac{1}{9}\sqrt{x}\sqrt{-81a_{1}^{2}x^{2}+45a_{1}x-4}+\frac{1}{9}\left(4-9a_{1}x\right)+\frac{1}{3}$
(A.20)

Again, do the same sub-leading analysis, we find $x,a_{1}$ must satisfy the
constraint.

$\displaystyle-5832a_{1}^{3}x^{7/2}-324a_{1}^{2}x^{2}\left(x\sqrt{-81a_{1}^{2}x^{2}+45a_{1}x-4}-\sqrt{-81a_{1}^{2}x^{2}+45a_{1}x-4}-15\sqrt{x}\right)$
(A.21)
$\displaystyle-40x\sqrt{-81a_{1}^{2}x^{2}+45a_{1}x-4}+18a_{1}x\left(13x\sqrt{-81a_{1}^{2}x^{2}+45a_{1}x-4}-7\sqrt{-81a_{1}^{2}x^{2}+45a_{1}x-4}-66\sqrt{x}\right)$
(A.22) $\displaystyle+28\sqrt{-81a_{1}^{2}x^{2}+45a_{1}x-4}+80\sqrt{x}=0$
(A.23)

Combining with (A.20), we can numerically find the bound $x>9.800000\cdots$

## References

  * (1) N. Chai, S. Chaudhuri, C. Choi, Z. Komargodski, E. Rabinovici and M. Smolkin, _Symmetry breaking at all temperatures_ , _Phys. Rev. Lett._ 125 (2020) 131603.
  * (2) N. Chai, S. Chaudhuri, C. Choi, Z. Komargodski, E. Rabinovici and M. Smolkin, _Thermal Order in Conformal Theories_ , _Phys. Rev. D_ 102 (2020) 065014 [2005.03676].
  * (3) S. Weinberg, _Gauge and global symmetries at high temperature_ , _Physical Review D_ 9 (1974) 3357.
  * (4) F. Jona and G. Shirane, _Ferroelectric crystals, international series of monographs on solid state physics_ , Pergamon press Oxford, UK: (1962).
  * (5) P.C. Hohenberg, _Existence of long-range order in one and two dimensions_ , _Physical Review_ 158 (1967) 383.
  * (6) S. Coleman, _There are no goldstone bosons in two dimensions_ , _Communications in Mathematical Physics_ 31 (1973) 259.
  * (7) N.D. Mermin and H. Wagner, _Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic heisenberg models_ , _Physical Review Letters_ 17 (1966) 1133.
  * (8) N. Chai, A. Dymarsky and M. Smolkin, _Model of Persistent Breaking of Discrete Symmetry_ , _Phys. Rev. Lett._ 128 (2022) 011601 [2106.09723].
  * (9) N. Chai, A. Dymarsky, M. Goykhman, R. Sinha and M. Smolkin, _A model of persistent breaking of continuous symmetry_ , 2111.02474.
  * (10) S. Chaudhuri and E. Rabinovici, _Symmetry breaking at high temperatures in large N gauge theories_ , _JHEP_ 08 (2021) 148 [2106.11323].
  * (11) S. Chaudhuri, C. Choi and E. Rabinovici, _Thermal order in large N conformal gauge theories_ , _JHEP_ 04 (2021) 203 [2011.13981].
  * (12) S. Rychkov and A. Stergiou, _General Properties of Multiscalar RG Flows in $d=4-\varepsilon$_, _SciPost Phys._ 6 (2019) 008 [1810.10541].
  * (13) A. Aharony and M.E. Fisher, _Critical behavior of magnets with dipolar interactions. i. renormalization group near four dimensions_ , _Phys. Rev. B_ 8 (1973) 3323.
  * (14) A. Aharony, _Critical behavior of anisotropic cubic systems_ , _Phys. Rev. B_ 8 (1973) 4270.
  * (15) D.J. Wallace, _Critical behaviour of anisotropic cubic systems_ , _Journal of Physics C: Solid State Physics_ 6 (1973) 1390.
  * (16) R.K.P. Zia and D.J. Wallace, _Critical behaviour of the continuous n-component potts model_ , _Journal of Physics A: Mathematical and General_ 8 (1975) 1495.
  * (17) A. Pelissetto and E. Vicari, _Critical phenomena and renormalization group theory_ , _Phys. Rep._ 368 (2002) 549.
  * (18) P. Francesco, P. Mathieu and D. Sénéchal, _Conformal field theory_ , Springer Science & Business Media (2012).
  * (19) J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert, _Duality and defects in rational conformal field theory_ , _Nucl. Phys. B_ 763 (2007) 354 [hep-th/0607247].
  * (20) C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, _Topological Defect Lines and Renormalization Group Flows in Two Dimensions_ , _JHEP_ 01 (2019) 026 [1802.04445].
  * (21) S.A. Kivelson, G. Aeppli and V.J. Emery, _Thermodynamics of the interplay between magnetism and high-temperature superconductivity_ , _Proceedings of the National Academy of Sciences_ 98 (2001) 11903.
  * (22) X. Hu, _Bicritical and tetracritical phenomena and scaling properties of the so (5) theory_ , _Physical Review Letters_ 87 (2001) 057004\.
  * (23) G. Group et al., _Gap–groups, algorithms, and programming, version 4.11. 1, 2021_ , _URL: https://www. gap-system. org_ .
  * (24) P. Liendo and J. Rong, _Seeking SUSY fixed points in the 4 $-$ $\epsilon$ expansion_, _JHEP_ 12 (2021) 033 [2107.14515].
  * (25) V.J. Emery, _Critical properties of many-component systems_ , _Phys. Rev. B_ 11 (1975) 239.
  * (26) D.J. Binder, _The cubic fixed point at large $N$_, _JHEP_ 09 (2021) 071 [2106.03493].
  * (27) S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, _Solving the 3d ising model with the conformal bootstrap_ , _Physical Review D_ 86 (2012) 025022.
  * (28) S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, _Solving the 3d ising model with the conformal bootstrap ii. c-minimization and precise critical exponents_ , _Journal of Statistical Physics_ 157 (2014) 869.
  * (29) F. Kos, D. Poland and D. Simmons-Duffin, _Bootstrapping mixed correlators in the 3d ising model_ , _Journal of High Energy Physics_ 2014 (2014) 1.
  * (30) F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, _Precision islands in the ising and o (n) models_ , _Journal of High Energy Physics_ 2016 (2016) 1.
  * (31) D. Simmons-Duffin, _The lightcone bootstrap and the spectrum of the 3d ising cft_ , _Journal of High Energy Physics_ 2017 (2017) 1\.
  * (32) S. Sachdev, _Quantum phase transitions_ , Cambridge university press (2011).
  * (33) G.S. Grest and M. Widom, _N-color ashkin-teller model_ , _Physical Review B_ 24 (1981) 6508.
  * (34) J. Ashkin and E. Teller, _Statistics of two-dimensional lattices with four components_ , _Phys. Rev._ 64 (1943) 178.
  * (35) J. Sólyom, _Duality of the block transformation and decimation for quantum spin systems_ , _Physical Review B_ 24 (1981) 230.
  * (36) A. O’Brien, S.D. Bartlett, A.C. Doherty and S.T. Flammia, _Symmetry-respecting real-space renormalization for the quantum ashkin-teller model_ , _Physical Review E_ 92 (2015) 042163\.
  * (37) A. Buchel and C. Pagnutti, _Exotic Hairy Black Holes_ , _Nucl. Phys. B_ 824 (2010) 85 [0904.1716].
  * (38) A. Buchel, _Fate of the conformal order_ , _Phys. Rev. D_ 103 (2021) 026008 [2011.11509].
  * (39) A. Buchel, _Thermal order in holographic CFTs and no-hair theorem violation in black branes_ , _Nucl. Phys. B_ 967 (2021) 115425 [2005.07833].
  * (40) A. Buchel, _Compactified holographic conformal order_ , _Nucl. Phys. B_ 973 (2021) 115605 [2107.05086].
  * (41) A. Buchel, _The quest for a conifold conformal order_ , 2205.00612.
  * (42) J.M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Adv. Theor. Math. Phys._ 2 (1998) 231 [hep-th/9711200].
  * (43) S.S. Gubser, I.R. Klebanov and A.M. Polyakov, _Gauge theory correlators from noncritical string theory_ , _Phys. Lett. B_ 428 (1998) 105 [hep-th/9802109].
  * (44) E. Witten, _Anti-de Sitter space and holography_ , _Adv. Theor. Math. Phys._ 2 (1998) 253 [hep-th/9802150].
  * (45) J. Rong and N. Su, _Scalar CFTs and Their Large N Limits_ , _JHEP_ 09 (2018) 103 [1712.00985].
  * (46) M. Vasiliev, _Consistent equations for interacting gauge fields of all spins in 3+1 dimensions_ , _Physics Letters B_ 243 (1990) 378.
  * (47) M.A. Vasiliev, _Higher spin gauge theories: Star product and AdS space_ , hep-th/9910096.