††institutetext:  The Racah Institute of Physics, The Hebrew University of
Jerusalem
Jerusalem, 91904, Israel

# The bi-conical vector model at $1/N$

Noam Chai , Eliezer Rabinovici , Ritam Sinha  and Michael Smolkin
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study finite $N$ aspects of the $O(m)\times O(N-m)$ vector model with
quartic interactions in general $2\leq d\leq 6$ spacetime dimensions. This
model has recently been shown Chai:2020zgq ; Chai:2020onq to display the
phenomenon of persistent symmetry breaking at a perturbative Wilson-Fisher-
like fixed point in $d=4-\epsilon$ dimensions. The large rank limit of the bi-
conical model displays a conformal manifold and a moduli space of vacua. We
find a set of three double trace scalar operators that are respectively
irrelevant, relevant and marginal deformations of the conformal manifold in
general $d$. We calculate the anomalous dimensions of the single and multi-
trace scalar operators to the first sub-leading order in the large rank
expansion. The anomalous dimension of the marginal operator does not vanish in
general, indicating that the conformal manifold is lifted at finite $N$. In
the case of equal ranks we are able to derive explicitly the scaling
dimensions of various operators as functions of only $d$.

##  1 Introduction

Conformal field theories (CFTs) have been extensively studied over the decades
due to their applications in condensed matter physics, statistical mechanics
and string theory. The conformal bootstrap Parisi:1972zm ; Polyakov:1974gs ;
Ferrara:1973yt ; ElShowk:2012ht ; Simmons-Duffin:2016gjk , see also
Mack:2019akh and references therein, is one of the approaches to unravel the
structure of a CFT. Rather than using a specific Lagrangian, it works with
general building blocks, such as the scaling dimensions of local operators and
their operator product expansion coefficients. In this paper we adopt the
diagrammatic $1/N$ conformal perturbation theory Vasiliev:1981yc ;
Vasiliev:1981dg to explore a CFT data of the new class of critical $O(m)\times
O(N-m)$ vector models recently constructed in Chai:2020zgq ; Chai:2020onq .

The large-$N$ (rank) limit of these models, which go by the name of bi-conical
models, displays a conformal manifold, a moduli space of vacua, and a deformed
moduli spaces of vacua at finite temperature. Moreover, it turns out that some
of the internal symmetries of the bi-conical model are broken at arbitrary
finite temperature. That said, studies in the vicinity111There is also a non-
supersymmetric example in $3+1$ dimensions Chaudhuri:2020xxb . of $3+1$
dimensions showed that while the conclusion regarding persistent symmetry
breaking at finite temperature withstands the $1/N$ corrections, the conformal
manifold and the moduli space of vacua are lifted at finite $N$ Chai:2020zgq ;
Chai:2020onq .

This is reminiscent of what happens with $\phi^{6}$ vector model in $2+1$
dimensions Bardeen:1983rv ; Rabinovici:1987tf . In the large rank limit it
also exhibits a conformal manifold and a moduli space of vacua, which are
lifted when 1/N corrections are considered. However, there is a substantial
difference between the two models: while the interacting bi-conical model is
stable below $d=4$, the non-trivial fixed point Pisarski:1982vz of the finite
rank $\phi^{6}$ theory has no ground state Bardeen:1983rv . Although
perturbatively in $1/N$ there are no signs of pathologies, e.g., the scaling
dimensions of various operators satisfy unitarity bounds Goykhman:2020ffn
,222This is similar to the higher dimensional critical vector models which
also exhibit a non-perturbative instability in $1/N$, see e.g., Fei:2014yja ;
Giombi:2019upv . the very presence of instability furnishes an obstacle to
address the question of persistent symmetry breaking in the interacting
$\phi^{6}$ model.

The focus of the current paper is to explore the effect of finite rank
corrections on the conformal manifold of the bi-conical model in general
$2<d<6$ dimensions. To this end, we use a unique hallmark of the conformal
manifolds - existence of exactly marginal operators in the theory. In
particular, we identify such an operator in the large rank limit, and
calculate its anomalous dimension assuming that $1/N$ is the smallest
parameter in the problem. In addition, we evaluate the anomalous dimensions of
the single and multi-trace scalar operators of the bi-conical model to order
$1/N$.

Our calculations suggest that there is a new strongly interacting CFT in $2+1$
dimensions. It does not exhibit a persistent symmetry breaking due to the
Mermin-Wagner-Hohenberg-Coleman no-go theorem Mermin:1966fe ; Hohenberg:1967zz
; Coleman:1973ci , but it is tractable within $1/N$ expansion and may serve as
a theoretical lab to study the structure of strongly coupled CFTs in $2+1$
dimensions as well as duality in the context of Vasiliev’s higher-spin gravity
Klebanov:2002ja ; Elitzur:2005kz ; Elitzur:2007zz .

We find that quite generally the anomalous dimension of the marginal operator
does not vanish, implying that the conformal manifold is lifted at $1/N$
order, and only a discrete number of fixed points is retained. Yet, there
exists one value of $d$ ($2<d<3$) for which the anomalous dimension vanishes.
The higher order corrections in $1/N$ only slightly modify this value, since
the rank of the group is large by assumption. Hence, in principle, the line of
fixed points might survive in this fractional number of dimensions. However,
it is unclear whether there is an integer rank such that $d$ is an integer
too.

In section 2 we review the definition of the bi-conical model, present its
perturbative RG flow, as well as provide an evidence for the existence of
conformal manifold in general dimension and infinite $N$. Then in section 3 we
calculate the anomalous dimensions of the single and multi-trace scalar
operators of the model at the next-to-leading order in $1/N$. In section 4 we
conduct a comprehensive discussion of our results and outline potential future
directions. In Appendix A we calculate perturbative beta functions of the
model in the context of Wilsonian RG flow. In Appendix B we summarize various
identities of the conformal perturbation theory, and use them to evaluate
different Feynman graphs used throughout the text.

##  2 Setup

In this paper, we study the critical $O(m)\times O(N-m)$ vector model in
$2\leq d\leq 6$ dimensions, described by the following Euclidean action333In
Rychkov:2018vya ; Osborn:2020cnf ; Hogervorst:2020gtc a general class of
closely related critical models was analyzed at lowest order in the $\epsilon$
expansion. Earlier works on this topic are reviewed in Pelissetto:2000ek ;
AAharony:2003 .

$\displaystyle
S=\int\,d^{d}y\,\Big{(}\frac{1}{2}(\partial\phi_{i})^{2}+\frac{g_{ij}}{N}\phi_{i}^{2}\phi_{j}^{2}\Big{)}$
(1)

where the flavour indices $i,j=1,2$ are summed over, $\phi_{1}$ and $\phi_{2}$
are real-valued vectors of $O(m)$ and $O(N-m)$ respectively, and the vector
indices are suppressed for brevity. For convenience let us define the
following notations,

$m=Nx=Nx_{1},\hskip 5.69046ptN-m=N(1-x)=Nx_{2}$ (2)

In the large-$N$ limit this model exhibits a line of fixed points Chai:2020onq
; Chai:2020zgq . One can illustrate it explicitly in the vicinity of $d=4$.
Indeed, the perturbative beta function in $d=4-\epsilon$ dimensions is given
by (see Appendix A for details),

$\displaystyle\mu{d\alpha_{11}\over
d\mu}=\epsilon\Big{(}-\alpha_{11}+x\,\alpha_{11}^{2}+(1-x)\alpha_{12}^{2}+\frac{8\,\alpha_{11}^{2}}{N}\Big{)}\leavevmode\nobreak\
,$ $\displaystyle\mu{d\alpha_{22}\over
d\mu}=\epsilon\Big{(}-\alpha_{22}+(1-x)\alpha_{22}^{2}+x\,\alpha_{12}^{2}+\frac{8\,\alpha_{22}^{2}}{N}\Big{)}\leavevmode\nobreak\
,$ (3) $\displaystyle\mu{d\alpha_{12}\over
d\mu}=\epsilon\,\alpha_{12}\Big{(}-1+x\alpha_{11}+(1-x)\alpha_{22}+\frac{2}{N}\big{(}\alpha_{11}+\alpha_{22}+2\alpha_{12}\big{)}\Big{)}\leavevmode\nobreak\
,$

where we introduced a set of dimensionless couplings, $\alpha_{ij}$, defined
by $g_{ij}=2\pi^{2}\epsilon\,\alpha_{ij}\mu^{\epsilon}$ with $\mu$ being a
floating cut off scale. The $1/N$ terms can be dropped in the large $N$ limit.
As a result, the fixed points of the flow shape a conformal manifold
parametrized by $\alpha_{12}$,

$\displaystyle\alpha_{11}^{*}=\frac{1\pm\sqrt{1-4x(1-x)\alpha_{12}^{*2}}}{2x}\leavevmode\nobreak\
,\quad\alpha_{22}^{*}=\frac{1\mp\sqrt{1-4x(1-x)\alpha_{12}^{*2}}}{2(1-x)}\leavevmode\nobreak\
,$
$\displaystyle|\alpha_{12}^{*}|\leq\frac{1}{2\sqrt{x(1-x)}}\leavevmode\nobreak\
.$ (4)

In fact, the two branches are connected at the end points, so together they
form a closed curve. Moreover, the matrix of critical couplings is degenerate,
i.e., $\text{det}(\alpha_{ij}^{*})=0$. The sub-leading $1/N$ terms lift the
conformal manifold (4), and only isolated fixed points remain at finite $N$
Chai:2020onq ; Chai:2020zgq , e.g., the $\mathcal{O}(N)$ invariant fixed point
corresponding to $\alpha_{11}^{*}=\alpha_{22}^{*}=\alpha_{12}^{*}$ and a CFT
with $\alpha_{12}^{*}=0$. The latter fixed point represents a pair of
decoupled critical $\phi^{4}$ vector models. There is a vast literature on
these CFTs, and we do not aim to further explore them in what follows.
However, there is an additional fixed point with $\alpha_{12}^{*}<0$ and
$\text{det}(\alpha_{ij}^{*})=0+\mathcal{O}(1/N)$ which has been far less
studied. This CFT, save the case $x=1/2$, exhibits a symmetry breaking at all
temperatures Chai:2020onq ; Chai:2020zgq , and we explore it in the current
work.

The standard lore in RG flow provides a simple recipe for calculating the
scaling dimensions of various local operators which appear in the effective
action of a CFT. Starting from linearizing the RG flow equations around a
given fixed point, one looks for the eigenvalues, $\omega_{i}$, of the matrix,
$\omega_{ij}$, which represents first-order derivatives of the beta functions
with respect to the couplings. These eigenvalues are simply related to the
scaling dimensions, $\Delta_{i}$, of the operators defined by the
corresponding eigenvectors,

$\Delta_{i}=d+\omega_{i}\leavevmode\nobreak\ .$ (5)

Using (3), and dropping the $1/N$ terms, yields

$\displaystyle\omega_{ij}=-\epsilon\leavevmode\nobreak\ \mathbb{I}_{3\times
3}+\epsilon\left(\begin{array}[]{ccc}2x\alpha_{11}^{*}&0&2(1-x)\alpha_{12}^{*}\\\
0&2(1-x)\alpha^{*}_{22}&2x\alpha^{*}_{12}\\\
x\alpha^{*}_{12}&\quad(1-x)\alpha^{*}_{12}&\quad
x\alpha^{*}_{11}+(1-x)\alpha^{*}_{22}\end{array}\right)\leavevmode\nobreak\ .$
(9)

The three eigenvalues of this matrix at any point on the conformal manifold
(4) are $(-\epsilon,+\epsilon,0)$. For $\epsilon>0$ they represent a relevant,
an irrelevant and a marginal deformation of a fixed point, respectively.

The above analysis is valid in the neighbourhood of $d=4$. It does not rely on
$N$ being large or small, instead it makes use of the weak couplings only. In
what follows we use the large $N$ techniques to identify various deformations
of the fixed points in general $d$ and express the corresponding operators in
terms of the fundamental fields.

Unlike the traditional perturbative approach, the large-$N$ expansion is
particularly useful sufficiently far away from $d=4$, where the theory becomes
strongly coupled, and the corresponding CFT cannot be studied perturbatively.
In particular, it provides a method to reveal the structure of a strongly
coupled CFT. In our case, we use it to explore the fate of conformal manifold
in $d$ dimensions. To this end we evaluate, for instance, the anomalous
dimension of the operator which happens to be exactly marginal in the
large-$N$ limit. This operator shifts between the CFTs along the line of fixed
points.

###  2.1 Conformal manifold in general dimension

In the previous subsection we reviewed a perturbative analysis which reveals
the existence of a conformal manifold in the vicinity of $d=4$. Here we build
on the large-$N$ expansion to provide an evidence that conformal manifold
exists in general $2<d<6$, save the case $d=4$, where the model is free.

We start from rewriting (1) in terms of auxiliary Hubbard-Stratonovich (HS)
fields, $\sigma_{i}$, for a comprehensive review see Moshe:2003xn ,

$\displaystyle S[\sigma_{i},\phi_{i}]$
$\displaystyle=\int\,d^{d}y\left(\frac{1}{2}(\partial_{\mu}\phi_{i})^{2}-\frac{h^{ij}}{4}\sigma_{i}\sigma_{j}+\frac{1}{\sqrt{Nx_{i}}}\sigma_{i}\phi_{i}^{2}\right)\leavevmode\nobreak\
,$ (10)

where $h^{ij}$ denotes the inverse of rescaled matrix of couplings
$h_{ij}=\sqrt{x_{i}x_{j}}\leavevmode\nobreak\ g_{ij}$. Integrating out the
auxiliary fields gives back the original action of the model.

The large-$N$ propagators of $\phi_{i}$’s are

$\langle\phi_{i}(x)\phi_{j}(y)\rangle=\frac{C_{\phi}}{|x-y|^{2\Delta_{\phi}}}\delta_{ij}\leavevmode\nobreak\
,\quad\Delta_{\phi}=\frac{d}{2}-1\leavevmode\nobreak\ ,\quad
C_{\phi}={\Gamma\left({d-2\over 2}\right)\over 4\pi^{d/2}}\leavevmode\nobreak\
,$ (11)

where the vector indices are suppressed for brevity, and we used a Fourier
transform relation,

$\int\frac{d^{d}p}{(2\pi)^{d}}\,e^{ip\cdot
y}\frac{1}{(p^{2})^{\frac{d}{2}-\Delta}}=\frac{2^{2\Delta-d}}{\pi^{\frac{d}{2}}}\frac{\Gamma(\Delta)}{\Gamma\left(\frac{d}{2}-\Delta\right)}\frac{1}{|y|^{2\Delta}}\,.$
(12)

The $\phi_{i}$ propagators will be represented by a solid line in the Feynman
graphs, see Fig.1 (a).

In fact, the path integral over $\phi_{i}$ can be done in a closed form,
because (10) is gaussian in these fields, and we get

$\displaystyle
S_{\mbox{\footnotesize{eff}}}[\sigma_{i}]=-\frac{h^{ij}}{4}\int\,d^{d}y\leavevmode\nobreak\
\sigma_{i}\sigma_{j}+\frac{N}{2}\sum_{i}x_{i}\leavevmode\nobreak\
\text{tr}\log\Big{(}-\partial^{2}+\frac{2}{\sqrt{Nx_{i}}}\sigma_{i}\Big{)}\leavevmode\nobreak\
.$ (13)

Expanding the trace log term around the conformal vacuum, yields

$\displaystyle S_{\mbox{\footnotesize{eff}}}[\sigma_{i}]$
$\displaystyle=-\int\,d^{d}y_{1}d^{d}y_{2}\leavevmode\nobreak\
\sigma_{i}(y_{1})\Big{(}{h^{ij}\over
4}\delta(y_{1}-y_{2})+\delta^{ij}B(y_{1}-y_{2})\Big{)}\sigma_{j}(y_{2})+\ldots$
(14)

where we explicitly keep quadratic terms only, whereas ellipsis encode the
rest, and444$B(x)$ denotes the product of two free massless scalar field
propagators.

$\displaystyle B(y)={C_{\phi}^{2}\over|y|^{2(d-2)}}\leavevmode\nobreak\ .$
(15)

The large-$N$ effective action is gaussian, but in general it includes an
infinite tower of higher order effective vertices suppressed by powers of
$1/N$. For our needs, however, the infinite series of terms can be truncated
at the fourth order in $\sigma_{i}$, because higher order vertices do not
contribute to the leading order anomalous dimensions.

As usual, inverting the quadratic part of (14) yields the matrix of
propagators for the HS fields. In momentum space, it takes the form

$\displaystyle\langle\sigma_{i}(p)\sigma_{j}(-p)\rangle$
$\displaystyle=-\frac{2}{[1+4B(p)\text{Tr}(\textbf{h})+16B^{2}(p)\text{det}(\textbf{h})]}\big{(}h_{ij}+4B(p)\text{det}(\textbf{h})\delta_{ij}\big{)}\leavevmode\nobreak\
,$ (16)

To identify the scaling dimensions of propagating fields at the fixed point,
it is useful to diagonalize the matrix of couplings, h. Let us denote by
$\lambda_{\pm}$ and $U^{i}_{\pm}$ its eigenvalues and normalized eigenvectors
respectively. Projecting (16) onto $U^{i}_{\pm}$, yields the following
propagators

$\displaystyle\langle\sigma_{+}(p)\sigma_{+}(-p)\rangle$ $\displaystyle=$
$\displaystyle-{2\lambda_{+}\over 1+4\lambda_{+}B(p)}\leavevmode\nobreak\
,\quad\langle\sigma_{+}(p)\sigma_{-}(-p)\rangle=0\leavevmode\nobreak\ ,$
$\displaystyle\langle\sigma_{-}(p)\sigma_{-}(-p)\rangle$ $\displaystyle=$
$\displaystyle-{2\lambda_{-}\over 1+4\lambda_{-}B(p)}\leavevmode\nobreak\ ,$
(17)

where $\sigma_{\pm}=U^{i}_{\pm}\sigma_{i}$.

As we pointed out in the previous subsection,
$\text{det}(\textbf{h}^{*})\sim\mathcal{O}(1/N)$ at any point of the conformal
manifold (4). This degeneracy is true to all orders in $\epsilon$-expansion.
Therefore one of the eigenvalues is suppressed relative to the other at the
fixed points that we are interested in, i.e.,
$\lambda_{-}\sim\mathcal{O}(1/N)$, whereas $\lambda_{+}\sim\mathcal{O}(1)$,
because
$\lambda_{+}\lambda_{-}=\text{det}(\textbf{h}^{*})\sim\mathcal{O}(1/N)$.
Hence, in the deep IR (or UV) regime, defined by $N\gg B(p)\lambda_{+}\gg 1$,
in $2\leq d\leq 4$ (or $4<d\leq 6$) dimensions, the propagators simplify

$\displaystyle\langle\sigma_{+}(p)\sigma_{+}(-p)\rangle$ $\displaystyle=$
$\displaystyle-{1\over 2B(p)}\leavevmode\nobreak\
,\quad\langle\sigma_{+}(p)\sigma_{-}(-p)\rangle=0\leavevmode\nobreak\ ,$
$\displaystyle\langle\sigma_{-}(p)\sigma_{-}(-p)\rangle$ $\displaystyle=$
$\displaystyle-2\lambda_{-}\Big{(}1-4\lambda_{-}B(p)\Big{)}\leavevmode\nobreak\
,$ (18)

Note that we keep the next-to-leading order term in the propagator of
$\sigma_{-}$, because to leading order in the large-$N$ expansion it boils
down to a contact term in position space.

Using the Fourier transform formula (12) one can get the large-$N$ scaling
dimension of the fields based on their momentum space behaviour. Since
$B(p)\sim p^{d-4}$, we find that in the large $N$ limit, $\sigma_{+}$ scales
as $\Delta_{+}=2$. As for the $\sigma_{-}$ propagator, the first term within
the parentheses in (18) corresponds to a contact term, and therefore the
scaling dimension of $\sigma_{-}$ is determined solely by the last term, i.e.,
$\Delta_{-}=d-2$.

Having $\sigma_{\pm}$ at hand one can construct new primary fields. Thus, for
instance, in the large-$N$ limit the scaling dimensions of primary operators
$\sigma_{+}^{2}$, $\sigma_{-}^{2}$ and $\sigma_{+}\sigma_{-}$ are equal to
$4,2d-4$ and $d$ respectively. Indeed, the model is gaussian in this limit,
and therefore the scaling dimensions are additive under composition of
operators. In particular, the above scaling dimensions match what we have
found in the previous subsection using a perturbative beta function in the
vicinity of $d=4$. Moreover, it can be explicitly verified that
$\sigma_{+}^{2}$, $\sigma_{-}^{2}$ and $\sigma_{+}\sigma_{-}$ correspond to
the eigenvectors of $\omega$. This completes our construction of the primary
operators which represent admissible nearly marginal deformations of the
critical model in the vicinity of $d=4$.

Remarkably, we showed that the large-$N$ scaling dimension of
$\sigma_{+}\sigma_{-}$ equals $d$ in any number of dimensions, implying that
to leading order in the $1/N$ expansion this operator is marginal. This
provides an evidence for the existence of conformal manifold in general $d$.
Our primary goal is to calculate the leading order anomalous dimensions of
$\sigma_{+}^{2}$, $\sigma_{-}^{2}$ and $\sigma_{+}\sigma_{-}$. If the scaling
dimension of the marginal operator $\sigma_{+}\sigma_{-}$ acquires a non-
trivial correction, then the line of fixed points is lifted by the finite rank
corrections in any number of dimensions.

(a)

(b)

(c)

Figure 1: Feynman rules for various propagators in the model.

###  2.2 Feynman rules and useful identities

We finish this section by introducing a number of Feynman rules used
throughout the rest of our work. First we notice that in the large-$N$ limit
the propagators of the Hubbard-Stratonovich fields in a non-diagonal basis is
dominated by $\langle\sigma_{+}\sigma_{+}\rangle$. Indeed, by definition

$\langle\sigma_{i}(y)\sigma_{j}(0)\rangle=U_{+}^{i}U_{+}^{j}\langle\sigma_{+}(x)\sigma_{+}(0)\rangle+U_{-}^{i}U_{-}^{j}\langle\sigma_{-}(x)\sigma_{-}(0)\rangle\leavevmode\nobreak\
,$ (19)

where according to (18) the contribution of
$\langle\sigma_{-}\sigma_{-}\rangle$ is $1/N$ suppressed.

In position space we have

$\displaystyle\langle\sigma_{+}(y)\sigma_{+}(0)\rangle$ $\displaystyle=$
$\displaystyle\frac{C_{+}}{|y|^{2\Delta_{+}}}\leavevmode\nobreak\ ,\quad
C_{+}=\frac{2^{d}\Gamma\left(\frac{d-1}{2}\right)\sin\left(\frac{\pi
d}{2}\right)}{\pi^{3/2}\Gamma\left(\frac{d}{2}-2\right)}\leavevmode\nobreak\
,$ $\displaystyle\langle\sigma_{-}(y)\sigma_{-}(0)\rangle$ $\displaystyle=$
$\displaystyle-2\lambda_{-}\Big{(}\delta(y)-4\lambda_{-}B(y)\Big{)}\leavevmode\nobreak\
,$ (20)

where we used (12).

The $\langle\sigma_{+}\sigma_{+}\rangle$ propagator and the leading order
contact term in $\langle\sigma_{-}\sigma_{-}\rangle$ will be represented by
the wavy and dashed lines respectively, see Fig.1 (b),(c). Expanding (13)
around $\sigma_{i}=0$ results in an infinite series of non-local effective
vertices. Two of them, namely cubic and quartic interactions, are essential
for our needs. They are given by

$V_{3}={4\over
3}\sum_{i=1}^{2}{C_{\phi}^{3}\over\sqrt{Nx_{i}}}\int{\prod_{j=1}^{3}d^{d}y_{j}\leavevmode\nobreak\
\sigma_{i}(y_{j})\over\big{(}|y_{12}||y_{23}||y_{31}|\big{)}^{d-2}}\leavevmode\nobreak\
,\leavevmode\nobreak\ V_{4}=-2\sum_{i=1}^{2}{C_{\phi}^{4}\over
Nx_{i}}\int{\prod_{j=1}^{4}d^{d}y_{j}\leavevmode\nobreak\
\sigma_{i}(y_{j})\over\big{(}|y_{12}||y_{23}||y_{34}||y_{41}|\big{)}^{d-2}}\leavevmode\nobreak\
,$ (21)

Diagrammatically, they can be represented as in Fig. 2.

Figure 2: Effective cubic and quartic vertices obtained by expanding (13)
around a conformal vacuum. Note that we implicitly sum over the vector indices
of the fields $\phi_{i}$ which are running in the loop.

Furthermore, using the completeness relations, we get a useful set of
identities,

$\displaystyle h^{*}_{ij}$ $\displaystyle=$
$\displaystyle\lambda_{+}U_{+}^{i}U_{+}^{j}+\lambda_{-}U_{-}^{i}U_{-}^{j}\quad\Rightarrow\quad
U_{+}^{i}U_{+}^{j}={h^{*}_{ij}\over\text{Tr}(\textbf{h}^{*})}+\mathcal{O}(1/N)\leavevmode\nobreak\
,$ $\displaystyle\delta_{ij}$ $\displaystyle=$ $\displaystyle
U_{+}^{i}U_{+}^{j}+U_{-}^{i}U_{-}^{j}\quad\Rightarrow\quad
U_{-}^{i}U_{-}^{j}=\delta_{ij}-{h^{*}_{ij}\over\text{Tr}(\textbf{h}^{*})}+\mathcal{O}(1/N)\leavevmode\nobreak\
.$ (22)

##  3 Anomalous dimensions

In this section we calculate the anomalous dimensions of various operators. We
use the diagrammatic $1/N$ technique, e.g., Vasiliev:1975mq ; Vasiliev:1981yc
; Vasiliev:1981dg ; Derkachov:1997ch ; Gracey:2018ame ; Goykhman:2019kcj . To
simplify our notation we suppress asterisk in the superscript of critical
couplings. This should not result in confusion, since we never consider a non-
critical model.

Our final results passed numerous checks. For instance, in the vicinity of
$d=4$, one can use the perturbative beta function (59) to evaluate the scaling
dimensions of the double trace operators up to $1/N$ order through the use of
(5). We carried out such a comparison and found that the results match.

###  3.1 Single trace scalars

The full conformal two point functions of $\sigma_{\pm}$ have the following
form

$\langle\sigma_{\pm}(y)\sigma_{\pm}(0)\rangle={C_{\pm}(1+A_{\pm})\over|y|^{2(\Delta_{\pm}+\gamma_{\pm})}}\mu^{-2\gamma_{\pm}}\leavevmode\nobreak\
,$ (23)

where $\mu$ represents an arbitrary floating cut off scale,
$A_{\pm}\sim\mathcal{O}(1/N)$ are associated with the $1/N$ corrections to the
leading order amplitudes $C_{+}$ and $C_{-}=8\lambda_{-}^{2}C_{\phi}^{2}$, as
defined in (20), and $\gamma_{\pm}\sim\mathcal{O}(1/N)$ are the anomalous
dimensions.

(a)

(b)

(c)

Figure 3: Feynman diagrams representing a $1/N$ correction to the effective
action (14).

To derive $\gamma_{\pm}$ we consider $1/N$ correction, $V_{2}$, to the
quadratic effective action (14). It is determined by the sum of three diagrams
shown in Fig. 3,

$-V_{2}=\text{Fig.}\ref{Vsigsig}\text{(a)}+\text{Fig.}\ref{Vsigsig}\text{(b)}+\text{Fig.}\ref{Vsigsig}\text{(c)}\leavevmode\nobreak\
.$ (24)

The two-loop Feynman graphs in Fig. 3 (a),(b) are built off the quartic term
in the power series expansion of (13) around the conformal vacuum, whereas the
three-loop diagram in Fig. 3 (c) is obtained by contracting two cubic terms in
the expansion of (13). These effective vertices are shown in Fig. 2. Thus, for
instance,

$\text{Fig.}\ref{Vsigsig}\text{(a)}=-{4(d-2)\pi^{d/2}\over
N\Gamma\left({d+2\over 2}\right)}\sum_{i}{U_{+}^{i}U_{+}^{i}\over
x_{i}}\,C_{\phi}^{4}C_{+}\int
d^{d}y_{1}d^{d}y_{2}{\sigma_{i}(y_{1})\sigma_{i}(y_{2})\over|y_{12}|^{d-2}}\int{d^{d}y_{3}\over|y_{13}|^{d-2}|y_{23}|^{d}}\leavevmode\nobreak\
,$ (25)

where we used the following identities to integrate over $y_{4}$,

${2d\over|y|^{d+2}}={\partial}_{\mu}{\partial}^{\mu}{1\over|y|^{d}}\quad,\quad\quad{\partial}_{\mu}{\partial}^{\mu}{1\over|y|^{d-2}}=-{2(d-2)\pi^{d/2}\over\Gamma\left({d\over
2}\right)}\leavevmode\nobreak\ \delta(y)\leavevmode\nobreak\ .$ (26)

The integral over $y_{3}$ diverges in the vicinity of $y_{2}$. Introducing a
floating spherical sharp cut off $\mu$, yields555Specifically,  $\int
d^{d}y_{3}{1\over|y_{13}|^{d-2}|y_{23}|^{d}}={1\over|y_{12}|^{d-2}}\int_{1/\mu}{d^{d}y_{3}\over|y_{3}|^{d}}+\ldots={1\over|y_{12}|^{d-2}}{2\pi^{d/2}\over\Gamma\left({d\over
2}\right)}\log\mu+\ldots\leavevmode\nobreak\ .$ (27)

$\text{Fig.}\ref{Vsigsig}\text{(a)}=-{16(d-2)\pi^{d}\over
Nd\,\Gamma^{2}\left({d\over 2}\right)}\sum_{i}{U_{+}^{i}U_{+}^{i}\over
x_{i}}\,C_{\phi}^{4}C_{+}\int
d^{d}y_{1}d^{d}y_{2}{\sigma_{i}(y_{1})\sigma_{i}(y_{2})\over|y_{12}|^{2(d-2)}}\log\mu+\ldots\leavevmode\nobreak\
,$ (28)

where ellipsis encode $\mu$-independent terms. Similarly,

Fig.3(b) $\displaystyle=$ $\displaystyle{4\over N}U\Big{(}{d-2\over
2},{d-2\over 2},2\Big{)}\sum_{i}{U_{+}^{i}U_{+}^{i}\over
x_{i}}\,C_{\phi}^{4}C_{+}\int
d^{d}y_{1}d^{d}y_{2}{\sigma_{i}(y_{1})\sigma_{i}(y_{2})\over|y_{12}|^{d-4}}$
(29) $\displaystyle\times\int
d^{d}y_{3}{1\over|y_{13}|^{d}|y_{23}|^{d}}\leavevmode\nobreak\ ,$

where we used the star-triangle identity (65) to integrate over $y_{4}$. This
time the integral over $y_{3}$ diverges in the vicinity of both $y_{1}$ and
$y_{2}$. As before, we regulate it by introducing a spherical sharp cut off
$\mu$ and retain the divergent part only,

$\text{Fig.}\ref{Vsigsig}\text{(b)}={16\pi^{d/2}\over N\Gamma\left({d\over
2}\right)}U\Big{(}{d-2\over 2},{d-2\over
2},2\Big{)}\sum_{i}{U_{+}^{i}U_{+}^{i}\over x_{i}}\,C_{\phi}^{4}C_{+}\int
d^{d}y_{1}d^{d}y_{2}{\sigma_{i}(y_{1})\sigma_{i}(y_{2})\over|y_{12}|^{2(d-2)}}\log\mu\leavevmode\nobreak\
.$ (30)

Finally,

Fig.3(c) $\displaystyle=$ $\displaystyle{64\pi^{d/2}\over N\Gamma\left({d\over
2}\right)}U^{2}\Big{(}{d-2\over 2},{d-2\over
2},2\Big{)}U\Big{(}1,2,d-3\Big{)}\sum_{i,j}\left(U_{+}^{i}U_{+}^{j}\right)^{2}{C_{\phi}^{6}C_{+}^{2}\over\sqrt{x_{i}x_{j}}}$
(31) $\displaystyle\times\int
d^{d}y_{1}d^{d}y_{2}{\sigma_{i}(y_{1})\sigma_{j}(y_{2})\over|y_{12}|^{2(d-2)}}\log\mu\leavevmode\nobreak\
,$

where the star-triangle identity (65) was used to integrate over $y_{6}$ and
$y_{5}$ first, and then over $y_{4}$. The remaining divergent integral over
$y_{3}$ was regulated through the use of spherical sharp cut off $\mu$.

Combining all together and using (22) to express everything in terms of
critical couplings, yields

$-V_{2}={4\Gamma(d-2)\sin\left({\pi d\over 2}\right)\over
Nd(d-2)\pi^{d+1}}\sum_{i,j}\Big{(}{h_{ii}\,\delta_{ij}\over
x_{i}\text{Tr}(\textbf{h})}+{d(d-3)\over
2\sqrt{x_{i}x_{j}}}{h^{2}_{ij}\over\text{Tr}^{2}(\textbf{h})}\Big{)}\int\int{\sigma_{i}(y_{1})\sigma_{j}(y_{2})\over|y_{12}|^{2(d-2)}}\log\mu\leavevmode\nobreak\
.$ (32)

This correction to the effective action determines the leading order anomalous
dimensions, $\gamma_{\pm}$, which can be read off after substituting
$\sigma_{i}=U^{i}_{+}\sigma_{+}+U^{i}_{-}\sigma_{-}$ and factoring out a term
proportional to $B(x)$ in the leading order quadratic action (14),

$\gamma_{\pm}=\pm{2^{d}\Gamma\left({d-1\over 2}\right)\sin\left({\pi d\over
2}\right)\over N\pi^{3/2}\Gamma\left({d+2\over
2}\right)}\sum_{i,j}U_{\pm}^{i}U_{\pm}^{j}\Big{(}{h_{ii}\,\delta_{ij}\over
x_{i}\text{Tr}(\textbf{h})}+{d(d-3)\over
2\sqrt{x_{i}x_{j}}}{h^{2}_{ij}\over\text{Tr}^{2}(\textbf{h})}\Big{)}\leavevmode\nobreak\
.$ (33)

Or equivalently, using the completeness relations (22), yields

$\displaystyle\gamma_{+}$ $\displaystyle=$
$\displaystyle{2^{d}\Gamma\left({d-1\over 2}\right)\sin\left({\pi d\over
2}\right)\over N\pi^{3/2}\Gamma\left({d+2\over
2}\right)}\sum_{i,j}\Big{(}{1\over
x_{i}}{h^{2}_{ii}\delta_{ij}\over\text{Tr}^{2}(\textbf{h})}+{d(d-3)\over
2\sqrt{x_{i}x_{j}}}{h^{2}_{ij}h_{ij}\over\text{Tr}^{3}(\textbf{h})}\Big{)}\leavevmode\nobreak\
,$ $\displaystyle\gamma_{-}$ $\displaystyle=$
$\displaystyle-{2^{d}\Gamma\left({d-1\over 2}\right)\sin\left({\pi d\over
2}\right)\over N\pi^{3/2}\Gamma\left({d+2\over
2}\right)}\sum_{i}\Big{(}{h_{ii}\over x_{i}\text{Tr}(\textbf{h})}+{d(d-3)\over
2}{h^{2}_{ii}\over
x_{i}\text{Tr}^{2}(\textbf{h})}\Big{)}+\gamma_{+}\leavevmode\nobreak\ .$ (34)

It is hard to further simplify this expression without knowing the critical
values of the coupling constants in general $d$. However, in the special case
of equal ranks, $x_{1}=x_{2}=1/2$, the anomalous dimensions can be written in
terms of dimension $d$ only. Indeed, in this case the fixed points (4), which
survive the $1/N$ corrections, respect the $\mathbb{Z}_{2}$ symmetry
$\phi_{1}\leftrightarrow\phi_{2}$, i.e., they satisfy $h_{11}=h_{22}$, as can
be explicitly checked within $\epsilon$-expansion. Recall that to leading
order in $1/N$ these couplings are also equal to $|h_{12}|$, because
$\text{det}(\textbf{h})\sim\mathcal{O}(1/N)$. Hence,

$\gamma_{\pm}\Big{|}_{m={N\over 2}}=\pm{2^{d}\Gamma\left({d-1\over
2}\right)\sin\left({\pi d\over 2}\right)\over N\pi^{3/2}\Gamma\left({d+2\over
2}\right)}\Big{(}1+{d(d-3)\over
4}\big{(}1\pm\text{sign}(h_{12})\big{)}\Big{)}\leavevmode\nobreak\ .$ (35)

###  3.2 Double trace scalars

In this subsection, we proceed to calculate the leading order anomalous
dimensions of the double trace scalars $\sigma_{+}^{2},\sigma_{-}^{2}$ and
$\sigma_{+}\sigma_{-}$. As shown in the previous section, the scaling
dimensions of these operators in the large $N$ limit are equal to $4,2d-4$ and
$d$ respectively. Hence, the anomalous dimensions of $\sigma_{+}^{2}$ and
$\sigma_{-}^{2}$ cannot change their RG behaviour in $d\neq 4$, i.e., they
stay relevant or irrelevant provided that $N$ is sufficiently large. However,
a non-zero sub-leading correction to the scaling dimension of the marginal
operator, $\sigma_{+}\sigma_{-}$, determines whether it becomes relevant or
irrelevant, depending on the sign of correction. In particular, if the leading
order anomalous dimension vanishes, then it implies that the line of fixed
points survives up to $\mathcal{O}(1/N)$ order. In either case, the results
would carry important information regarding the phase structure of the theory.

The diagrams which appear at the first sub-leading order in the $1/N$
expansion have the same topologies for all the three operators. One of them is
simple. It corresponds to the sub-leading corrections of the single trace
operators $\sigma_{\pm}$, and we therefore present the remaining topologies
only. They are shown in Fig.4. In this section, we evaluate these diagrams for
each of the double trace operators separately, by plugging in their individual
propagators and combinatorial factors.

(a)

(b)

(c)

Figure 4: Three different topologies contributing to the anomalous dimensions
of the double trace scalars at the next-to-leading order in $1/N$. These
diagrams are built off effective vertices in Fig.2. The cross-caps correspond
to the insertion points of the double trace operators. The solid bold lines
denote propagators of the single trace operators.

#### The $\sigma_{+}^{2}$ operator

The full correlator of $\sigma^{2}_{+}$ is given by

$\langle\sigma^{2}_{+}(y)\sigma^{2}_{+}(0)\rangle=\frac{2C_{+}^{2}(1+A_{++})}{|y|^{2(\Delta_{++}+\gamma_{++})}}\mu^{-2\gamma_{++}}\leavevmode\nobreak\
,\quad\Delta_{++}=4\leavevmode\nobreak\ .$ (36)

where $A_{++}\sim\mathcal{O}(1/N)$ is a sub-leading correction to the
amplitude, and $\gamma_{++}\sim\mathcal{O}(1/N)$ is the anomalous dimension.
The large $N$ limit of this correlator is entirely fixed by a single diagram
in Fig.5(a). In contrast, the $1/N$ correction is determined by the four
diagrams in Fig.5(b)-(e).

(a)

(b)

(c)

(d)

(e)

Figure 5: Feynman diagrams contributing to the
$\langle\sigma^{2}_{+}\sigma^{2}_{+}\rangle$ correlator up to $1/N$ order: (a)
leading order diagram (b)-(e) next-to-leading order graphs. A wavy propagator
with a grey blob in diagram (b) represents the full
$\langle\sigma_{+}\sigma_{+}\rangle$ propagator.

The contribution of Fig.5(b) to $\gamma_{++}$ is simply $2\gamma_{+}$, and we
proceed to the rest of the diagrams. The graph in Fig.5(c) is built off two
effective cubic vertices in Fig.2 with $U^{i}_{+}\sigma_{+}$ substituted for
$\sigma_{i}$ and contracted with two external operators $\sigma_{+}^{2}$. The
diagram itself can be calculated with the help of repeated use of the star-
triangle identity (65). We relegate the details of calculation to Appendix B
(see (66) there),

$\displaystyle\text{Fig.}\ref{sigma_dt-1}(c)$ $\displaystyle=$
$\displaystyle{1\over N}{4(8\pi^{d})^{3}(d-2)\Gamma^{2}\left({2-d\over
2}\right)\over(d-4)^{3}\,\Gamma^{2}(d-3)}\,{\log\mu\over|y|^{8}}\,\sum_{i,j}{(U_{+}^{i}U_{+}^{j})^{3}\over\sqrt{x_{i}x_{j}}}C_{\phi}^{6}C_{+}^{5}$
(37) $\displaystyle=$ $\displaystyle{32\over
N}{(d-3)^{2}\Gamma(d-1)\sin\left({\pi d\over
2}\right)\over\pi\Gamma^{2}\left({d\over
2}\right)}\,C_{+}^{2}\,{\log\mu\over|y|^{8}}\,\sum_{i,j}{h_{ij}^{3}\over\sqrt{x_{i}x_{j}}\,\text{Tr}^{3}(\textbf{h})}\leavevmode\nobreak\
,$

where in the second equality we used (22).

Similarly, using the effective quartic vertex in Fig.2, with
$U^{i}_{+}\sigma_{+}$ substituted for $\sigma_{i}$, and contracting it with
the external operators $\sigma_{+}^{2}$, yields

$\displaystyle\text{Fig.}\ref{sigma_dt-1}(d)$ $\displaystyle=$
$\displaystyle-{128\over N}{\pi^{2d}(d-2)\Gamma\left({2-d\over
2}\right)\over(d-4)\,\Gamma(d-3)\Gamma(d/2)}\,{\log\mu\over|y|^{8}}\,\sum_{i}{(U_{+}^{i})^{4}\over
x_{i}}C_{\phi}^{4}C_{+}^{4}$ (38) $\displaystyle=$ $\displaystyle-{8\over
N}{(d-3)(d-4)\Gamma(d-1)\sin\left({\pi d\over
2}\right)\over\pi\Gamma^{2}\left({d\over
2}\right)}\,C_{+}^{2}\,{\log\mu\over|y|^{8}}\,\sum_{i}{h_{ii}^{2}\over
x_{i}\text{Tr}^{2}(\textbf{h})}\leavevmode\nobreak\ ,$

where in the first equality we repeatedly used the star-triangle relation (65)
to carry out the loop integrals, whereas in the second equality we substituted
(22). Finally, based on (67), we obtain

$\displaystyle\text{Fig.}\ref{sigma_dt-1}(e)$ $\displaystyle=$
$\displaystyle{(32\pi^{d})^{2}\Gamma\left({8-d\over 2}\right)\over
N(d-4)^{3}\,\Gamma(d-3)\Gamma(d/2)}\,{\log\mu\over|y|^{8}}\,\sum_{i}{(U_{+}^{i})^{4}\over
x_{i}}C_{\phi}^{4}C_{+}^{4}$ (39) $\displaystyle=$ $\displaystyle-{8\over
N}{(d-3)(d-6)\Gamma(d-1)\sin\left({\pi d\over
2}\right)\over\pi\Gamma^{2}\left({d\over
2}\right)}\,C_{+}^{2}\,{\log\mu\over|y|^{8}}\,\sum_{i}{h_{ii}^{2}\over
x_{i}\text{Tr}^{2}(\textbf{h})}\leavevmode\nobreak\ ,$

Combining all together, yields

$\gamma_{++}=2\gamma_{+}+{4\over N}{(d-3)\Gamma(d-1)\sin\left({\pi d\over
2}\right)\over\pi\Gamma^{2}\left({d\over
2}\right)}\,\sum_{i,j}\left({(d-5)h_{ii}^{2}\delta_{ij}\over
x_{i}\text{Tr}^{2}(\textbf{h})}-{2(d-3)h_{ij}^{3}\over\sqrt{x_{i}x_{j}}\text{Tr}^{3}(\textbf{h})}\right)\leavevmode\nobreak\
.$ (40)

The dependence on the critical values of the couplings drops out for the equal
rank case, $x_{1}=x_{2}=1/2$, because the $\mathbb{Z}_{2}$ symmetry, which
swaps between $\phi_{1}$ and $\phi_{2}$, is respected by the fixed points of
interest, i.e., in this case $h_{11}=h_{22}=|h_{12}|$.666It can be explicitly
verified using the standard $\epsilon$-expansion around $d=4$. Hence,

$\displaystyle\gamma_{++}\Big{|}_{m={N\over 2}}$ $\displaystyle=$
$\displaystyle{2^{d-1}(d-4)(d^{2}-4d-1)\Gamma\left({d-1\over
2}\right)\sin\left({\pi d\over 2}\right)\over N\pi^{3\over
2}\Gamma\left({d+2\over 2}\right)}\quad\text{for}\quad
h_{12}<0\leavevmode\nobreak\ ,$ $\displaystyle\gamma_{++}\Big{|}_{m={N\over
2}}$ $\displaystyle=$ $\displaystyle-{2^{d}(d-4)(d-1)\Gamma\left({d+1\over
2}\right)\sin\left({\pi d\over 2}\right)\over N\pi^{3\over
2}\Gamma\left({d+2\over 2}\right)}\quad\text{for}\quad
h_{12}>0\leavevmode\nobreak\ .$ (41)

The plot of $\gamma_{++}$ for negative $h_{12}$ is shown in Fig. 6.

As expected, (41) for $h_{12}>0$ matches the anomalous dimension of the
Hubbard-Stratonovich field in the critical $O(N)$ model Ma:1974qh ;
Lang:1993ct . Indeed, as stressed in section 2, the symmetry of the bi-conical
model is enhanced to $O(N)$ provided that ranks are equal and $h_{12}>0$. Now
the single trace scalar, $s\sim\phi_{1}^{2}+\phi_{2}^{2}$, of the enhanced
symmetry corresponds to the Hubbard-Stratonovich field of the critical $O(N)$
model. In terms of auxiliary fields of the equal rank bi-conical model, it is
given by $s\sim\sigma_{1}+\sigma_{2}\sim\sigma_{+}$, and therefore its two
point function is entirely determined by the
$\langle\sigma_{+}\sigma_{+}\rangle$ correlator.

Figure 6: The anomalous dimension $\gamma_{++}$ as a function of $d$ for
$h_{12}<0$. For simplicity we set $N=1$.

#### The $\sigma_{-}^{2}$ operator

As in the case of $\sigma_{+}^{2}$, the anomalous dimension of
$\sigma^{2}_{-}$ is determined by matching the diagrammatic expansion of
$\langle\sigma^{2}_{-}\sigma^{2}_{-}\rangle$ with the general form of the
exact correlation function

$\langle\sigma^{2}_{-}(y)\sigma^{2}_{-}(0)\rangle=\frac{2C_{-}^{2}(1+A_{--})}{|x|^{2(\Delta_{--}+\gamma_{--})}}\mu^{-2\gamma_{--}}\leavevmode\nobreak\
,\quad\Delta_{--}=2(d-2)\leavevmode\nobreak\ ,\quad
C_{-}=8\lambda_{-}^{2}C_{\phi}^{2}\leavevmode\nobreak\ ,$ (42)

where $A_{--}\sim\mathcal{O}(1/N)$ is associated with a sub-leading correction
to the leading order amplitude, and $\gamma_{--}\sim\mathcal{O}(1/N)$ is the
anomalous dimension.

The leading order behaviour of this correlator is entirely fixed by a single
diagram in Fig.7(a). Note that the correlation function of the
$\sigma_{-}^{2}$ operator, likewise that of $\sigma_{-}$, are suppressed by
powers of $1/N$ relative to the analogous correlators for $\sigma_{+}$. This
is because the dashed propagator is proportional to
$\lambda_{-}\sim\mathcal{O}(1/N)$.

The $1/N$ correction to the leading order diagram is represented in terms of
four diagrams in Fig.7(b)-(e).

(a)

(b)

(c)

(d)

(e)

Figure 7: Feynman diagrams contributing to the
$\langle\sigma^{2}_{-}\sigma^{2}_{-}\rangle$ correlator up to $1/N$ order: (a)
The leading order diagram built off a sub-leading term proportional to $B(y)$
in (20) (b)-(e) Next-to-leading order graphs. A dashed propagator with a grey
blob in diagram (b) represents the full $\langle\sigma_{-}\sigma_{-}\rangle$
propagator.

The diagram in Fig.7(b) contributes $2\gamma_{-}$ to the anomalous dimension
of $\sigma_{-}^{2}$, whereas the graph of Fig.7(d) vanishes, because delta
function propagators of the field $\sigma_{-}$ result in a term proportional
to $\langle\phi_{i}^{2}\rangle^{2}$, which vanishes in a CFT. Similarly, the
diagram in Fig.7(e) is finite. It contributes to $A_{--}$ only, and we ignore
it in what follows.

The upshot of this discussion is that we have only to calculate the graph in
Fig.7(c). It is built off two effective cubic vertices in Fig.2 with
$U^{i}_{+}\sigma_{+}$ substituted for the end points of the internal
propagator of $\sigma_{i}$ (wavy line), and $U^{i}_{-}\sigma_{-}$ replacing
$\sigma_{i}$’s of the cubic vertices which are contracted with two external
operators $\sigma_{-}^{2}$. The diagram itself can be calculated with the help
of repeated use of the star-triangle identity (65) (see (68) in Appendix B for
details),

$\displaystyle\text{Fig.}\ref{sigma_dt-2}(c)$ $\displaystyle=$
$\displaystyle{512\over N}{(d-2)\pi^{d}\over(d-4)\Gamma^{2}\left({d\over
2}\right)}\,{\log\mu\over|y|^{4(d-2)}}\,\sum_{i,j}{(U_{-}^{i}U_{-}^{j})^{2}U_{+}^{i}U_{+}^{j}\over\sqrt{x_{i}x_{j}}}C_{\phi}^{6}C_{+}(-2\lambda_{-})^{4}$
$\displaystyle=$ $\displaystyle{32\Gamma^{2}\left({d-2\over
2}\right)\Gamma(d-2)\sin\left({\pi d\over
2}\right)\over\pi^{2d+1}(d-2)}\,{\lambda_{-}^{4}\over
N}\,{\log\mu\over|y|^{4(d-2)}}\,\sum_{i,j}{h_{ij}\over\sqrt{x_{i}x_{j}}\,\text{Tr}(\textbf{h})}\left(\delta_{ij}-{h_{ij}\over\text{Tr}(\textbf{h})}\right)^{2}\,,$

where in the second equality we used (22). Combining all together, yields

$\gamma_{--}=2\gamma_{-}-{8\,\Gamma(d-1)\sin\left({\pi d\over 2}\right)\over
N\pi\Gamma^{2}\left({d\over
2}\right)}\sum_{i,j}{h_{ij}\over\sqrt{x_{i}x_{j}}\,\text{Tr}(\textbf{h})}\left(\delta_{ij}-{h_{ij}\over\text{Tr}(\textbf{h})}\right)^{2}\leavevmode\nobreak\
.$ (44)

In the case of equal rank $x_{1}=x_{2}=1/2$, this expression simplifies

$\displaystyle\gamma_{--}\Big{|}_{m={N\over 2}}$ $\displaystyle=$
$\displaystyle-{2^{d+1}(d-2)\Gamma\left({d+1\over 2}\right)\sin\left({\pi
d\over 2}\right)\over N\pi^{3\over 2}\Gamma\left({d+2\over
2}\right)}\quad\text{for}\quad h_{12}<0\leavevmode\nobreak\ ,$
$\displaystyle\gamma_{--}\Big{|}_{m={N\over 2}}$ $\displaystyle=$
$\displaystyle-{2^{d+1}(d+2)\Gamma\left({d+1\over 2}\right)\sin\left({\pi
d\over 2}\right)\over N(d-1)\pi^{3\over 2}\Gamma\left({d+2\over
2}\right)}\quad\text{for}\quad h_{12}>0\leavevmode\nobreak\ .$ (45)

The plot of $\gamma_{--}$ for negative $h_{12}$ is shown in Fig. 8.

Figure 8: The anomalous dimension $\gamma_{--}$ as a function of $d$ for
$h_{12}<0$. For simplicity we set $N=1$.

#### The $\sigma_{+}\sigma_{-}$ operator

The operator $\sigma_{+}\sigma_{-}$ is marginal to leading order in the
large-$N$ expansion. Hence, in this limit there exists a conformal manifold in
general $d$. Calculating the anomalous dimension at the next-to-leading order
in $1/N$ will tell us whether the model exhibits a conformal manifold at
finite $N$. If the anomalous dimension does not vanish, it means that this
manifold is lifted by the $1/N$ corrections.

The full conformal correlator takes the form

$\langle\sigma_{+}\sigma_{-}(y)\sigma_{+}\sigma_{-}(0)\rangle=\frac{C_{+}C_{-}(1+A_{+-})}{|x|^{2(\Delta_{+-}+\gamma_{+-})}}\mu^{-2\gamma_{+-}}\leavevmode\nobreak\
,\quad\Delta_{+-}=d\leavevmode\nobreak\ ,$ (46)

where $A_{+-},\gamma_{+-}\sim\mathcal{O}(1/N)$ are the sub-leading correction
to the amplitude and anomalous dimension respectively. Fig.9(a) represents the
leading order behaviour of this correlator. It scales as
$\mathcal{O}(1/N^{2})$, because the dashed propagator is proportional to
$\lambda_{-}\sim\mathcal{O}(1/N)$. The $1/N$ correction is represented in
terms of six diagrams displayed in Fig.9(b)-(g).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 9: Feynman diagrams contributing to (46) up to $1/N$ order: (a) The
leading order diagram built off a sub-leading term proportional to $B(y)$ in
(20) (b)-(g) Next-to-leading order graphs. The propagators with a grey blob in
diagram (b) represent the full propagators of $\sigma_{\pm}$.

The contribution of Fig.9(b) to the anomalous dimension of
$\sigma_{-}\sigma_{+}$ boils down to $\gamma_{-}+\gamma_{+}$. Diagrams in
Fig.9(c),(d) are very similar. They are built off two effective cubic vertices
in Fig.2 with $U^{i}_{\pm}\sigma_{\pm}$ substituted for $\sigma_{i}$ in an
appropriate way. Repeated use of the star-triangle identity (65) eventually
yields (see (69)),

Fig.9(c) $\displaystyle=$
$\displaystyle{-(32\pi^{d})^{2}\,\Gamma\left(2-{d\over 2}\right)\over
N(d-4)^{2}\Gamma(d-3)\Gamma\left({d\over
2}\right)}{\log\mu\over|y|^{2d}}\sum_{i,j}{(U_{+}^{i})^{3}U_{+}^{j}(U_{-}^{j})^{2}\over\sqrt{x_{i}x_{j}}}C_{\phi}^{6}C_{+}^{3}(-2\lambda_{-})^{2}$
$\displaystyle=$ $\displaystyle{16(d-3)\Gamma(d-1)\sin\left({\pi d\over
2}\right)\over\pi\Gamma^{2}\left({d\over
2}\right)}C_{+}C_{-}{\log\mu\over|y|^{2d}}\sum_{i,j}{h_{ii}h_{ij}\over\sqrt{x_{i}x_{j}}\,\text{Tr}^{2}(\textbf{h})}\left(1-{h_{jj}\over\text{Tr}(\textbf{h})}\right)\leavevmode\nobreak\
,$ Fig.9(d) $\displaystyle=$ $\displaystyle{16(d-3)\Gamma(d-1)\sin\left({\pi
d\over 2}\right)\over\pi\Gamma^{2}\left({d\over
2}\right)}C_{+}C_{-}{\log\mu\over|y|^{2d}}\sum_{i,j}{h_{ii}h_{jj}\over\sqrt{x_{i}x_{j}}\,\text{Tr}^{2}(\textbf{h})}\left(\delta_{ij}-{h_{ij}\over\text{Tr}(\textbf{h})}\right)\leavevmode\nobreak\
.$

where in the second equality we used (22).

Figure 10: The anomalous dimension $\gamma_{+-}$ as a function of $d$ for
$h_{12}<0$. For simplicity we set $N=1$. The sign change happens around
$d=2.57437$.

Next using the identities (26) one can verify that the diagram in Fig.9(f) is
finite, and therefore it only contributes to $A_{+-}$. In contrast,

Fig.9(e) $\displaystyle=$ $\displaystyle-{32(d-2)\pi^{d}\over
N\,d\,\Gamma^{2}\left({d\over
2}\right)}{\log\mu\over|y|^{2d}}\sum_{i}{(U_{+}^{i}U_{-}^{i})^{2}\over
x_{i}}C_{\phi}^{4}C_{+}^{2}(-2\lambda_{-})^{2}$ $\displaystyle=$
$\displaystyle-{8\Gamma(d-2)\sin\left({\pi d\over
2}\right)\over\pi\Gamma\left({d-4\over 2}\right)\Gamma\left({d+2\over
2}\right)}C_{+}C_{-}{\log\mu\over|y|^{2d}}\sum_{i}{h_{ii}\over
x_{i}\text{Tr}(\textbf{h})}\left(1-{h_{ii}\over\text{Tr}(\textbf{h})}\right),$
Fig.9(g) $\displaystyle=$ $\displaystyle{64(d-2)\pi^{d}\over
N\,(d-4)\,\Gamma^{2}\left({d\over
2}\right)}{\log\mu\over|y|^{2d}}\sum_{i}{(U_{+}^{i}U_{-}^{i})^{2}\over
x_{i}}C_{\phi}^{4}C_{+}^{2}(-2\lambda_{-})^{2}$ (48) $\displaystyle=$
$\displaystyle{2^{d+1}\,\Gamma\left({d-1\over 2}\right)\sin\left({\pi d\over
2}\right)\over\pi^{3\over 2}\Gamma\left({d\over
2}\right)}C_{+}C_{-}{\log\mu\over|y|^{2d}}\sum_{i}{h_{ii}\over
x_{i}\text{Tr}(\textbf{h})}\left(1-{h_{ii}\over\text{Tr}(\textbf{h})}\right),$

where we used (22) and (70) to get the expression for Fig.9(g), whereas the
expression for Fig.9(e) is derived based on (26). As a result, we get

$\displaystyle\gamma_{+-}$ $\displaystyle=$
$\displaystyle{4(d-3)\Gamma(d-1)\sin\left({\pi d\over 2}\right)\over
N\pi\Gamma^{2}\left({d\over
2}\right)}\Bigg{(}\,\text{sign}(h_{12})-2\sum_{i,j}{h_{ii}h_{ij}\over\sqrt{x_{i}x_{j}}\,\text{Tr}^{2}(\textbf{h})}\left(1-2{h_{jj}\over\text{Tr}(\textbf{h})}\right)$
(49) $\displaystyle-$ $\displaystyle 2\sum_{i}{h_{ii}^{2}\over
x_{i}\,\text{Tr}^{2}(\textbf{h})}-{(d+4)\over 2d(d-3)}\sum_{i}{h_{ii}\over
x_{i}\text{Tr}(\textbf{h})}\left(1-{h_{ii}\over\text{Tr}(\textbf{h})}\right)\Bigg{)}\leavevmode\nobreak\
.$

where the first term on the right hand side represents $\gamma_{+}+\gamma_{-}$
given by (35). For equal rank, we have

$\displaystyle\gamma_{+-}\Big{|}_{m={N\over 2}}$ $\displaystyle=$
$\displaystyle-{2(6d^{2}-17d+4)\Gamma(d-1)\sin\left({\pi d\over 2}\right)\over
N\pi d\,\Gamma^{2}\left({d\over 2}\right)}\quad\text{for}\quad
h_{12}<0\leavevmode\nobreak\ ,$ $\displaystyle\gamma_{+-}\Big{|}_{m={N\over
2}}$ $\displaystyle=$ $\displaystyle-{2(2d^{2}-5d+4)\Gamma(d-1)\sin\left({\pi
d\over 2}\right)\over N\pi d\,\Gamma^{2}\left({d\over
2}\right)}\quad\text{for}\quad h_{12}>0\leavevmode\nobreak\ .$ (50)

The plot of $\gamma_{+-}$ for negative $h_{12}$ is shown in Fig. 10. In
general, $\gamma_{+-}\neq 0$ which implies that conformal manifold is lifted.
However, there is a sign flip around $d_{*}=2.57437$, and therefore the
$\sigma_{+}\sigma_{-}$ operator remains marginal in $d_{*}$ dimensions. In
this special case, the line of fixed points might survive up to $1/N$ order.
Furthermore, the higher order corrections in $1/N$ only slightly modify the
value of $d_{*}$, because $N$ is large by assumption. Note that if the value
of $N$ is decreased, there could be a critical, $N_{*}$, such that the
corresponding $d_{*}$ becomes integer. It is, however, unclear whether an
integer $N_{*}$ of this kind exists.

###  3.3 Multi-trace scalars

(a)

(b)

(c)

Figure 11: Feynman diagrams contributing to
$\langle\mathcal{O}_{k,\ell}\leavevmode\nobreak\ \mathcal{O}_{k,\ell}\rangle$
up to the next-to-leading order in $1/N$. Cross-caps represent insertion
points of $\mathcal{O}_{k,\ell}$. (a) Leading order correlator (b) $1/N$
correction associated with the single trace correlator (c) $1/N$ correction
associated with the double trace correlator.

The information about the single trace operators was crucial for the
calculation of the anomalous dimensions of the double trace operators.
However, this information was not sufficient. There were additional
contributions, that needed to be accounted for while calculating the full
answer. In contrast, the anomalous dimensions, $\gamma_{k,\ell}$, of the
multi-trace scalar operators of the form
$\mathcal{O}_{k,\ell}\equiv\sigma_{+}^{k}\sigma_{-}^{\ell}$, to leading order
in $1/N$, are entirely determined by the diagrams that we evaluated already.
That is to say, the complete answer can be written in terms of the anomalous
dimensions of the single and double trace scalars. The latter can be readily
understood based on the effective action analysis.

Indeed, $\gamma_{k,\ell}$ is defined by

$\langle\mathcal{O}_{k,\ell}(y)\leavevmode\nobreak\
\mathcal{O}_{k,\ell}(0)\rangle={k!\,\ell!\,C_{+}^{k}\,C_{-}^{\ell}\big{(}1+A_{k,\ell}\big{)}\over|y|^{2(2k+\ell(d-2)+\gamma_{k,\ell})}}\leavevmode\nobreak\
,$ (51)

where $A_{k,\ell}\sim 1/N$ represents a sub-leading correction to the
amplitude. Now apart from the diagrams built off the effective cubic and
quartic vertices in Fig.2, there are no other vertices which give rise to the
$1/N$ corrections. In particular, as shown in Fig.11, all possible diagrams,
which contribute to the two point function of $\mathcal{O}_{k,\ell}$ up to
$1/N$ order, factorize into a product of zero order correlators
$\langle\sigma_{\pm}\sigma_{\pm}\rangle$ and previously calculated two point
function of either single or double trace operators. Hence, calculating
$\gamma_{k,\ell}$ boils down to a simple problem in combinatorics, and the
leading order anomalous dimension of $\mathcal{O}_{k,\ell}$ takes the form

$\gamma_{k,\ell}=(2-k-\ell)(k\gamma_{+}+\ell\gamma_{-})+k\ell\gamma_{+-}+{k(k-1)\over
2}\gamma_{++}+{\ell(\ell-1)\over 2}\gamma_{--}\leavevmode\nobreak\ .$ (52)

Recall now that if the ranks of the groups are equal, i.e., $m=N/2$, then the
anomalous dimensions of the single and double trace scalars become simple
functions of only the dimension $d$. Likewise, $\gamma_{k,\ell}$ simplify in
this special case.

##  4 Discussion

In this paper we studied the critical $O(m)\times O(N-m)$ vector model (1) in
general $2\leq d\leq 6$ dimensions. In the large $N$ limit and
$\epsilon=4-d\ll 1$, this model exhibits a line of fixed points, which is
lifted by the $1/N$ corrections provided that $\epsilon$ is the smallest
expansion parameter in the problem Chai:2020onq ; Chai:2020zgq . To reveal
what happens with the conformal manifold in general dimension, we employed the
large-$N$ technique to identify an operator which happens to be exactly
marginal in any $d$ in the large-$N$ limit. Such an operator accompanies the
existence of conformal manifold. We calculated the $1/N$ correction to its
leading order scaling dimension and found that quite generically the anomalous
dimension does not vanish, see (49), (50). The outcome of this calculation is
that only a discrete set of fixed points survive the $1/N$ corrections. An
$O(N)$ invariant critical vector model is among them. This corresponds to the
point on the conformal manifold with $g_{11}=g_{22}=g_{12}$, in both the equal
and unequal rank models. This model has been extensively studied throughout
the literature for several decades, and we are not going to discuss it here.

We will discuss a new CFT with $g_{12}<0$. Note that in the equal rank model,
this corresponds to the point $g_{11}=g_{22}=-g_{12}$ on the conformal
manifold. In the unequal rank model, the critical couplings are a non-linear
function of the ranks, and can only be determined numerically even in the
perturbative regime near $d=4$. As shown in Chai:2020onq ; Chai:2020zgq , it
exhibits a persistent symmetry breaking at finite temperature. This is a rare
example where a continuous global symmetry is broken at all temperatures for
finite $N$. There is, however, a caveat here: this phenomenon occurs only in a
fractional number of dimensions. While for infinite $N$ there is a candidate
for a $3+1$ dimensional CFT with some of its internal symmetries broken at all
temperatures Chaudhuri:2020xxb , a fully fledged example of a finite $N$ CFT
with persistent symmetry breaking in integer $d$ has not yet been found. For
instance, the bi-conical model is free in $d=4$, whereas in $d=3$ it fails due
to the Mermin-Wagner-Hohenberg-Coleman theorem Mermin:1966fe ;
Hohenberg:1967zz ; Coleman:1973ci . Yet, understanding better the structure of
such a CFT may help to either build a finite $N$ candidate with persistent
symmetry breaking in realistic dimensions, e.g., a model with persistent
breaking of a discrete symmetry in $d=3$, or lead to a new no-go theorem
forcing a mechanism for symmetry breaking restoration at finite $N$. To this
end, we calculated a CFT data (52) associated with the scaling dimensions of
the multi-trace scalar operators up to $1/N$ order. The dependence on the
critical couplings is somewhat disguised in the closed form formulas, thereby
allowing for explicit results only in $d=4-\epsilon$, where the coupling
constants are perturbatively known. Away from the perturbative regime, the
model becomes strongly coupled in the IR, and the critical couplings cannot be
evaluated anymore. However, there is a significant simplification in the limit
of equal ranks, i.e., for $m=N/2$. In this case we are able to obtain the
scaling dimensions of the multi-trace scalar operators of the model as
functions of only $d$ ($2\leq d\leq 6$).

New analytical and numerical bootstrap techniques have been developed in
recent times, many of which have been tested successfully in the $O(N)$ vector
model at the Wilson-Fisher fixed point. The bi-conical vector model, for both
equal and unequal ranks, can serve as a brand new testing ground for many of
these bootstrap methods, which in turn could provide new conformal data for
the model.

While the model with equal ranks is a special case of (1), there is no much
difference between the equal and unequal ranks at zero temperature. Yet, the
behaviour of the models is substantially different at finite temperature. In
particular, if the ranks are equal and $N$ is finite, there is no persistent
symmetry breaking in the equal rank case Chai:2020onq ; Chai:2020zgq . Hence,
one might wonder whether a CFT data of the equal rank model may shed light on
the structure of a theory with a persistent symmetry breaking. Hereafter we
argue that the answer to this question is affirmative to a certain extent.

### The slightly unequal rank model

Let us introduce a new parameter $\delta=x-{1\over 2}$. It measures how far
the model deviates from the equal rank case. The critical coupling constants
of the conformal manifold, which survive the finite $N$ corrections, are
functions of $\delta$, and we denote them by $h_{ij}(\delta)$. The equal rank
model satisfies $h_{11}(0)=h_{22}(0)$, because the fixed points in this case
respect the $\mathbb{Z}_{2}$ symmetry $\phi_{1}\leftrightarrow\phi_{2}$, as
can be explicitly checked using $\epsilon$-expansion Chai:2020onq ;
Chai:2020zgq . Therefore we deduce that the following relations between the
critical couplings hold

$h_{11}(\delta)=h_{22}(-\delta)\leavevmode\nobreak\ ,\quad
h_{12}^{2}(\delta)=h_{11}(\delta)h_{11}(-\delta)\leavevmode\nobreak\ ,$ (53)

where the first identity rests on $x_{1}\leftrightarrow x_{2}$ under
$\delta\to-\delta$ (recall that $x_{1}={1\over 2}+\delta$ and $x_{2}={1\over
2}-\delta$), whereas the second one follows from $\text{det}({\bf h})=0$. In
particular, $h_{12}(\delta)$ is an even function of $\delta$, and all of the
anomalous dimensions that we have calculated in this work depend on just one
unknown function $h_{11}(\delta)$. Likewise, using these relations and (34),
(40), (44), (49), it can be readily verified that the leading order anomalous
dimensions are also even functions of $\delta$.

In particular, for a slightly unequal rank characterized by $\delta\ll 1$,
various anomalous dimensions in general dimension $d$ satisfy777Assuming
$g_{12}<0$ and $\epsilon=4-d\ll 1$, one can verify it explicitly within
$\epsilon$-expansion,
$\displaystyle\gamma_{+}={-2\epsilon(1+8\delta^{2})\over N}+\ldots,\hskip
8.5359pt\gamma_{-}={2\epsilon(3-8\delta^{2})\over N}+\ldots$
$\displaystyle\gamma_{++}={-\epsilon^{2}(1-32\delta^{2})\over N}+\ldots,\hskip
5.69046pt\gamma_{--}={12\epsilon-13\epsilon^{2}+16\epsilon(3+2\epsilon)\delta^{2}\over
N}+\ldots$ $\displaystyle\gamma_{+-}={16\epsilon(1-6\delta^{2})\over
N}+\ldots$ (54) where ellipsis encode higher order terms in $1/N,\delta$ and
$\epsilon$.

$\gamma(\delta)=\gamma(0)+\mathcal{O}(\delta^{2}/N,1/N^{2})\leavevmode\nobreak\
,$ (55)

where $\gamma(0)$ collectively represents the anomalous dimensions of the
equal rank model, e.g., it can be either (35), (41), (45) or (50).

In contrast, the order parameter of the symmetry breaking scales linearly with
$\delta\ll 1$ at any temperature Chai:2020onq ; Chai:2020zgq , implying that
to linear order in $\delta$ a CFT data of the unequal rank model matches that
of an equal rank case, albeit for unequal ranks the system already exhibits a
pattern of persistent symmetry breaking. Moreover, in these models, the non-
zero value of the marginal operator to the leading order in $\delta$ indicates
that the conformal manifold is lifted. We cannot, however, determine the fate
of the conformal manifold in the most general unequal rank model due to lack
of knowledge of the critical couplings.
As an example, consider $N=10$ and $\delta=0.1$. This corresponds to an
$O(4)\times O(6)$ model, which can be classified as a slightly unequal rank
model. In this model, the anomalous dimension of the marginal operator will be
given for any $2<d<6$ by (50) $+O(\delta^{2})$ corrections, indicating that
the conformal manifold is lifted. The same cannot be said, however, for
example in an $O(1)\times O(9)$ model with $\delta=0.4$, where the leading
anomalous dimension of the marginal operator itself is dependent on the
critical couplings.

### A non-trivial fixed point in $d=3$

Our calculations provide an additional evidence for the existence of an
interesting strongly interacting CFT in three dimensions. Most illuminating in
this regard is the equal rank model, where we evaluated the numerical values
of the scaling dimensions of various operators.

Tractable CFTs in three dimensions are interesting from a number of
perspectives. The critical $O(N)$ vector model in $d=3$ presents the most
prominent example in this regard. It has a perturbative IR fixed point in
$d=4-\epsilon$ dimensions, which is the well known Wilson-Fisher fixed point.
In fact, the $\epsilon$-expansion was devised to gain a quantitative
description of a fixed point in $d=3$, which describes the three-dimensional
critical Ising model. The main difficulty encountered during the studies of
the Ising model stems directly from its strongly coupled nature, and not until
very recently (see Simmons-Duffin:2016wlq and Caron-Huot:2020ouj for recent
progress) much was unknown about it due to insufficient non-perturbative
techniques.

Nonetheless, one can study the model in the limit of large $N$ and make much
progress in characterising the low-lying spectrum of the theory. The primary
reason for the effectiveness of the large-$N$ approach is due to the coupling
constant dropping out of the analysis, thereby ridding us off the necessity to
determine its critical value, which is beyond the reach of analytical
approach.

This entire discussion carries over to the equal rank model (and in some
limited way to a slightly unequal rank case). Thus, for instance, a
perturbative analysis of the bi-conical model in the vicinity of $d=4$ results
in the Wilson-Fisher like fixed points. Moreover, due to the $\mathbb{Z}_{2}$
symmetry $\phi_{1}\leftrightarrow\phi_{2}$, the large $N$ analysis of the
equal rank bi-conical model reveals that we can extend the calculations beyond
the perturbative regime in the vicinity of $d=4$.888This is not true for the
unequal rank model where there is no symmetry to simplify the calculations.

The punchline of this analysis is that very much like the case of the critical
$O(N)$ vector model, the values of the couplings are not needed to derive the
large-$N$ CFT data for the equal rank bi-conical model.

Moreover, to leading order in the large-$N$ expansion the bi-conical model
disintegrates into two free scalar fields $\phi_{1}$ and $\phi_{2}$ in the
vector representation of $O(m)$ and $O(N-m)$ respectively, and decoupled
scalar singlets of $O(m)\times O(N-m)$ with scaling dimension $\Delta_{+}=2$
and $\Delta_{-}=d-2$. The sub-leading $1/N$ terms re-instate the interaction
between various fields, thereby giving us a non-trivial interacting model in
$d=3$.

## Acknowledgements

RS dedicates this paper to the loving memory of his father Dipak Kumar Sinha.
We thank Soumyadeep Chaudhuri, Changha Choi, Mikhail Goykhman and Zohar
Komargodski for helpful discussions. Our work is partially supported by the
Israeli Science Foundation Center of Excellence (grant No. 2289/18). The work
of NC, RS and MS is partially supported by the Binational Science Foundation
(grant No. 2016186), and by the Quantum Universe I-CORE program of the Israel
Planning and Budgeting Committee (grant No. 1937/12).

##  Appendix A Beta functions

In this Appendix we consider Wilsonian RG flow of the model (1) to derive the
beta functions of the couplings. The partition function at some arbitrary
scale $\mu$ is given by

$Z_{\mu}=\int\prod_{a=1}^{2}D\phi_{a}\exp\left[-\sum_{i=1}^{2}\int
d^{d}x\leavevmode\nobreak\
\left(\frac{1}{2}\partial_{\nu}\phi_{i}\partial^{\nu}\phi_{i}+\frac{1}{2}g_{i}^{2}\,\mu^{2}\phi^{2}_{i}+\frac{g_{ij}}{N}\mu^{4-d}\,\phi^{2}_{i}\,\phi^{2}_{j}+\ldots\right)\right]\,,$
(56)

where ellipsis encode $\phi^{6}$ and higher order interactions with or without
derivatives. Note that in contrast to the main body of the text, the coupling
constants $g_{ij}$ are dimensionless throughout this Appendix.

The interaction matrix $g_{ij}$ is symmetric, and every principal submatrix of
$g_{ij}$ is positive definite to make sure the theory is stable for large
values of the fields. However, this is not essential for the calculations
carried out in this Appendix.

To find the fixed points, one has to analyze an infinite system of one-loop
exact beta functions. Unfortunately, it cannot be done in full generality.
However, assuming $d=4-\epsilon$ with $|\epsilon|\ll 1$, we can implement the
classic Wilson-Fisher $\epsilon$-expansion. In this case apart from $\phi^{4}$
and $({\partial}\phi)^{2}$, the coupling parameter of any interaction with
$\ell$ factors of the fields will have a power series expansion which begins
with a term of order $\epsilon^{\ell/2}$. Hence, to linear order in $\epsilon$
we can focus on $g_{i}$ and $g_{ij}$ only.

Integrating out field modes with energy between $(\mu,\mu-d\mu)$, yields999No
summation over repeated indices in what follows.

$\displaystyle
g_{i}(\mu-d\mu)(\mu-d\mu)^{2}=g_{i}(\mu)\mu^{2}+4\sum_{k}g_{ik}\mu^{4-d}\left(x_{k}+{2\over
N}\delta_{ik}\right)\int_{\mu-d\mu}^{\mu}{d^{d}p\over(2\pi)^{d}}{1\over
p^{2}+\mu^{2}g_{k}(\mu)}\leavevmode\nobreak\ ,$ $\displaystyle
g_{ij}(\mu-d\mu)(\mu-d\mu)^{4-d}=g_{ij}(\mu)\mu^{4-d}$ (57)
$\displaystyle-4\sum_{k}g_{ik}g_{jk}\mu^{2(4-d)}\left(x_{k}+{2\over
N}(\delta_{ik}+\delta_{jk})\right)\int_{\mu-d\mu}^{\mu}{d^{d}p\over(2\pi)^{d}}{1\over\Big{(}p^{2}+\mu^{2}g_{k}(\mu)\Big{)}^{2}}$
$\displaystyle-{16\over
N}g_{ij}^{2}\mu^{2(4-d)}\int_{\mu-d\mu}^{\mu}{d^{d}p\over(2\pi)^{d}}{1\over\Big{(}p^{2}+\mu^{2}g_{i}(\mu)\Big{)}\Big{(}p^{2}+\mu^{2}g_{j}(\mu)\Big{)}}\leavevmode\nobreak\
.$

The loop integrals are simple since they are done over a thin shell,

$\displaystyle\mu{dg_{i}\over d\mu}$ $\displaystyle=$
$\displaystyle-2g_{i}-{S_{d}\over(2\pi)^{d}}\sum_{k}\Big{(}x_{k}+{2\over
N}\delta_{ik}\Big{)}{4g_{ik}\over 1+g_{k}}\leavevmode\nobreak\ ,$ (58)
$\displaystyle\mu{dg_{ij}\over d\mu}$ $\displaystyle=$
$\displaystyle-\epsilon\,g_{ij}+{4S_{d}\over(2\pi)^{d}}\sum_{k}\Big{(}x_{k}+{2\over
N}(\delta_{ik}+\delta_{jk})\Big{)}{g_{ik}g_{jk}\over(1+g_{k})^{2}}$
$\displaystyle+$ $\displaystyle{16\over
N}{S_{d}\over(2\pi)^{d}}{g_{ij}^{2}\over(1+g_{i})^{2}(1+g_{j})^{2}}+\mathcal{O}(\epsilon^{3})\leavevmode\nobreak\
,$

where $S_{d}={2\pi^{d/2}\over\Gamma(d/2)}$. Since $g_{i}\sim\epsilon$, one can
replace all the denominators in the above equation by 1 without violating the
accuracy. Hence, we finally get

$\displaystyle\mu{dg_{i}\over d\mu}$ $\displaystyle=$
$\displaystyle-2g_{i}-a\sum_{k}x_{k}g_{ik}-{2a\over
N}g_{ii}+\mathcal{O}(\epsilon^{2})\leavevmode\nobreak\ ,$ (59)
$\displaystyle\mu{dg_{4ij}\over d\mu}$ $\displaystyle=$
$\displaystyle-\epsilon\,g_{ij}+a\sum_{k}\Big{(}x_{k}+{2\over
N}(\delta_{ik}+\delta_{jk})\Big{)}g_{ik}g_{jk}+{4a\over
N}g_{ij}^{2}+\mathcal{O}(\epsilon^{3})\leavevmode\nobreak\ .$

where we introduced $a=4S_{d}/(2\pi)^{d}=1/(2\pi^{2})+\mathcal{O}(\epsilon)$.
The fixed points $g_{i}^{*},g_{ij}^{*}$ of the flow satisfy

$\displaystyle g_{i}^{*}$ $\displaystyle=$ $\displaystyle-{a\over
2}(x_{1}\,g_{i1}^{*}+x_{2}\,g_{i2}^{*})-{a\over
N}g_{ii}^{*}+\mathcal{O}(\epsilon^{2})\leavevmode\nobreak\ ,$ (60)
$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\epsilon\,g_{ij}^{*}+a\left(x_{1}\,g_{i1}^{*}g_{j1}^{*}+x_{2}\,g_{i2}^{*}g_{j2}^{*}\right)+{2a\over
N}\left(g_{ii}^{*}+g_{jj}^{*}\right)g_{ij}^{*}+{4a\over
N}g_{ij}^{*2}+\mathcal{O}(\epsilon^{3})\leavevmode\nobreak\ .$

The above algebraic equations are reliable up to linear order in $\epsilon$.
After solving for $g_{ij}^{*}$ one substitutes it into the first equation to
get $g_{i}^{*}$. We ignore equation for $g_{i}^{*}$ since it is not essential
for our needs. For $g_{12}^{*}\neq 0$ the rest reduces to

$\displaystyle\epsilon g_{11}^{*}$ $\displaystyle=$ $\displaystyle
a\left(x_{1}+{8\over
N}\right)g_{11}^{*\,2}+a\,x_{2}\,g_{12}^{*\,2}\leavevmode\nobreak\ ,$
$\displaystyle\epsilon g_{22}^{*}$ $\displaystyle=$ $\displaystyle
a\left(x_{2}+{8\over
N}\right)g_{22}^{*\,2}+a\,x_{1}\,g_{12}^{*\,2}\leavevmode\nobreak\ ,$
$\displaystyle\epsilon$ $\displaystyle=$ $\displaystyle a\left(x_{1}+{2\over
N}\right)g_{11}^{*}+a\left(x_{2}+{2\over N}\right)g_{22}^{*}+{4a\over
N}g_{12}^{*}\leavevmode\nobreak\ .$ (61)

These equations are degenerate in the limit $N\to\infty$, and there is a line
of fixed points

$\epsilon=a\big{(}x_{1}g_{11}^{*}+x_{2}g_{22}^{*}\big{)}\leavevmode\nobreak\
,\quad\text{det}(g_{ij}^{*})=0\leavevmode\nobreak\ .$ (62)

##  Appendix B Conformal perturbation theory

There are simple diagrammatical rules for performing some of the loop
integrals within conformal perturbation theory. In this Appendix we review and
apply these rules to evaluate Feynman diagrams which contribute to the
anomalous dimensions studied in this paper. The calculations are done in
position space. For simplicity, we assume that in any Feynman graph the
interaction vertices and amplitudes of the propagators are normalized to
unity, e.g.,

$2\Delta$

$=\frac{1}{|x|^{2\Delta}}$

In particular, a simple loop diagram satisfies additivity, i.e.,

Furthermore, there is a propagator merging relation of the form

$\int d^{d}x_{2}\leavevmode\nobreak\
{1\over|x_{12}|^{2\Delta_{1}}|x_{23}|^{2\Delta_{2}}}={U(\Delta_{1},\Delta_{2},d-\Delta_{1}-\Delta_{2})\over|x_{13}|^{2(\Delta_{1}+\Delta_{2})-d}}\leavevmode\nobreak\
,$ (63)

where

$U(\Delta_{1},\Delta_{2},\Delta_{3})=\pi^{\frac{d}{2}}\,\frac{\Gamma\left(\frac{d}{2}-\Delta_{1}\right)\Gamma\left(\frac{d}{2}-\Delta_{2}\right)\Gamma\left(\frac{d}{2}-\Delta_{3}\right)}{\Gamma(\Delta_{1})\Gamma(\Delta_{2})\Gamma(\Delta_{3})}\,.$
(64)

This relation can be represented diagrammatically as follows

where we integrate over the insertion point of the bold vertex on the left-
hand side.

Finally, the star-triangle relation for the conformal cubic vertex is given
by,

$\int
d^{d}x_{4}\,\frac{1}{|x_{14}|^{2\Delta_{1}}|x_{24}|^{2\Delta_{2}}|x_{34}|^{2\Delta_{3}}}=\frac{U(\Delta_{1},\Delta_{2},\Delta_{3})}{|x_{12}|^{d-2\Delta_{3}}|x_{13}|^{d-2\Delta_{2}}|x_{23}|^{d-2\Delta_{1}}}\leavevmode\nobreak\
,\quad\Delta_{1}+\Delta_{2}+\Delta_{3}=d\leavevmode\nobreak\ .$ (65)

Diagrammatically it can be represented as follows

Let us illustrate the power of these rules by evaluating the diagrams
contributing to the anomalous dimensions of various operators studied in our
work.

As shown in Fig.12, the star-triangle relation (65) is enough to evaluate the
logarithmically divergent part of the diagram in Fig.5(c).

Figure 12: Calculation of the loop integrals associated with the diagram in
Fig.5(c). In each step the star-triangle relation (65) is applied to integrate
over the insertion point of the bold vertex. Cross-caps denote location of the
external operators.

The remaining integral in Fig.12 diverges. Introducing a spherical sharp cut
off, $\mu$, yields

$\text{Fig}.\ref{sigpp-c}={16\pi^{3d}(d-2)\Gamma^{2}\left({2-d\over
2}\right)\over(d-4)^{3}\,\Gamma^{2}(d-3)}{\log\,\mu\over|y|^{8}}\leavevmode\nobreak\
.$ (66)

Similarly, one can evaluate the diagrams in Fig.5(d,e). For instance, the
necessary steps for the diagram in Fig.5(e) are shown explicitly in Fig.13.

Figure 13: Calculation of the integrals associated with the diagram in
Fig.5(e). In each step the star-triangle relation (65) is applied to integrate
over the insertion point of the bold vertex. Cross-caps denote location of the
external operators.

As before, introducing a spherical cut off $\mu$ to carry out the remaining
integral in Fig.13, yields

$\text{Fig}.\ref{sigpp-e}={(8\pi^{d})^{2}\Gamma\left({8-d\over
2}\right)\over(d-4)^{3}\,\Gamma(d-3)\Gamma\left({d\over
2}\right)}{\log\,\mu\over|y|^{8}}\leavevmode\nobreak\ .$ (67)

Recall now that the non-trivial contribution to $\gamma_{--}$ comes entirely
from Fig.7(b,c). Fig.14 below summarizes all the steps which are necessary to
calculate the diagram in Fig.7(c).

Figure 14: Calculation of the loop integrals associated with the diagram in
Fig.7(c). The dashed lines represent a delta function of the $\sigma_{-}$
propagator, and we integrate over the delta functions first. Next the star-
triangle relation (65) is applied to integrate over the insertion point of the
bold vertex. Cross-caps denote location of the external operators.

In terms of spherical cut off $\mu$, we get

$\text{Fig}.\ref{sigmm-c}={4(d-2)\pi^{d}\over(d-4)\,\Gamma^{2}\left({d\over
2}\right)}{\log\,\mu\over|y|^{4(d-2)}}\leavevmode\nobreak\ .$ (68)

Next let us consider $\gamma_{+-}$. All relevant diagrams for the calculation
of this anomalous dimension are listed in Fig.9. It turns out that the same
calculation should be done to evaluate the loop integrals of the graphs shown
in Fig.9(c,d). It is displayed in Fig.15.

Figure 15: Calculation of the integrals associated with the diagram in
Fig.9(c). The dashed lines represent a delta function of the $\sigma_{-}$
propagator, and we integrate over the delta functions first. The star-triangle
relation (65) is applied to integrate over the insertion point of the bold
vertex for each step thereafter. Cross-caps denote location of the external
operators.

Equivalently,

$\text{Fig}.\ref{sigpm-c}={-16\pi^{2d}\Gamma\left({4-d\over
2}\right)\over(d-4)^{2}\,\Gamma(d-3)\Gamma\left({d\over
2}\right)}{\log\,\mu\over|y|^{2d}}\leavevmode\nobreak\ .$ (69)

Finally, to evaluate Fig.9(g) one should follow the steps in Fig.16 below.

Figure 16: Calculation of the loop integrals associated with the diagram in
Fig.9(g). The dashed lines represent a delta function of the $\sigma_{-}$
propagator, and we integrate over the delta functions first. The star-triangle
relation (65) is applied to integrate over the insertion point of the bold
vertex in the last equality. Cross-caps denote location of the external
operators.

Up to a minor change in the scaling of the figure, the final answer is
identical to (68)

$\text{Fig}.\ref{sigpm_g}={4(d-2)\pi^{d}\over(d-4)\,\Gamma^{2}\left({d\over
2}\right)}{\log\,\mu\over|y|^{2d}}\leavevmode\nobreak\ .$ (70)

## References

  * (1) N. Chai, S. Chaudhuri, C. Choi, Z. Komargodski, E. Rabinovici, and M. Smolkin, “Thermal Order in Conformal Theories,” Phys. Rev. D 102 no. 6, (2020) 065014, arXiv:2005.03676 [hep-th].
  * (2) N. Chai, S. Chaudhuri, C. Choi, Z. Komargodski, E. Rabinovici, and M. Smolkin, “Symmetry Breaking at All Temperatures,” Phys. Rev. Lett. 125 no. 13, (2020) 131603.
  * (3) G. Parisi, “On self-consistency conditions in conformal covariant field theory,” Lett. Nuovo Cim. 4S2 (1972) 777–780.
  * (4) A. Polyakov, “Nonhamiltonian approach to conformal quantum field theory,” Zh. Eksp. Teor. Fiz. 66 (1974) 23–42.
  * (5) S. Ferrara, A. Grillo, and R. Gatto, “Tensor representations of conformal algebra and conformally covariant operator product expansion,” Annals Phys. 76 (1973) 161–188.
  * (6) 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,” Phys. Rev. D 86 (2012) 025022, arXiv:1203.6064 [hep-th].
  * (7) D. Simmons-Duffin, “The Conformal Bootstrap,” in Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, pp. 1–74.  2017\.  arXiv:1602.07982 [hep-th].
  * (8) G. Mack, “Conformal Field Theory in D$>$2 dimensions, representations and harmonic analysis,” arXiv:1902.03812 [hep-th].
  * (9) A. Vasiliev, Y. Pismak, and Y. Khonkonen, “Simple Method of Calculating the Critical Indices in the 1/$N$ Expansion,” Theor. Math. Phys. 46 (1981) 104–113.
  * (10) A. Vasiliev, Y. Pismak, and Y. Khonkonen, “1/$N$ Expansion: Calculation of the Exponents $\eta$ and Nu in the Order 1/$N^{2}$ for Arbitrary Number of Dimensions,” Theor. Math. Phys. 47 (1981) 465–475.
  * (11) S. Chaudhuri, C. Choi, and E. Rabinovici, “Thermal order in large N conformal gauge theories,” arXiv:2011.13981 [hep-th].
  * (12) W. A. Bardeen, M. Moshe, and M. Bander, “Spontaneous Breaking of Scale Invariance and the Ultraviolet Fixed Point in O($N$) Symmetric $\phi^{6}$ Theory,” Phys. Rev. Lett. 52 (1984) 1188.
  * (13) E. Rabinovici, B. Saering, and W. A. Bardeen, “Critical Surfaces and Flat Directions in a Finite Theory,” Phys. Rev. D 36 (1987) 562.
  * (14) R. Pisarski, “Fixed point structure of $\phi^{6}$ in three-dimensions at large $N$,” Phys. Rev. Lett. 48 (1982) 574–576.
  * (15) M. Goykhman, V. Rosenhaus, and M. Smolkin, “The background field method and critical vector models,” arXiv:2009.13137 [hep-th].
  * (16) L. Fei, S. Giombi, and I. R. Klebanov, “Critical $O(N)$ models in $6-\epsilon$ dimensions,” Phys. Rev. D 90 no. 2, (2014) 025018, arXiv:1404.1094 [hep-th].
  * (17) S. Giombi, R. Huang, I. R. Klebanov, S. S. Pufu, and G. Tarnopolsky, “The $O(N)$ Model in ${4<d<6}$ : Instantons and complex CFTs,” Phys. Rev. D 101 no. 4, (2020) 045013, arXiv:1910.02462 [hep-th].
  * (18) N. Mermin and H. Wagner, “Absence of ferromagnetism or antiferromagnetism in one-dimensional or two-dimensional isotropic Heisenberg models,” Phys. Rev. Lett. 17 (1966) 1133–1136.
  * (19) P. Hohenberg, “Existence of Long-Range Order in One and Two Dimensions,” Phys. Rev. 158 (1967) 383–386.
  * (20) S. R. Coleman, “There are no Goldstone bosons in two-dimensions,” Commun. Math. Phys. 31 (1973) 259–264.
  * (21) I. Klebanov and A. Polyakov, “AdS dual of the critical O(N) vector model,” Phys. Lett. B 550 (2002) 213–219, arXiv:hep-th/0210114.
  * (22) S. Elitzur, A. Giveon, M. Porrati, and E. Rabinovici, “Multitrace deformations of vector and adjoint theories and their holographic duals,” JHEP 02 (2006) 006, arXiv:hep-th/0511061.
  * (23) S. Elitzur, A. Giveon, M. Porrati, and E. Rabinovici, “Multitrace deformations of vector and adjoint theories and their holographic duals,” Nucl. Phys. B Proc. Suppl. 171 (2007) 231–242.
  * (24) S. Rychkov and A. Stergiou, “General Properties of Multiscalar RG Flows in $d=4-\varepsilon$,” SciPost Phys. 6 no. 1, (2019) 008, arXiv:1810.10541 [hep-th].
  * (25) H. Osborn and A. Stergiou, “Heavy Handed Quest for Fixed Points in Multiple Coupling Scalar Theories in the $\varepsilon$ Expansion,” arXiv:2010.15915 [hep-th].
  * (26) M. Hogervorst and C. Toldo, “Bounds on multiscalar CFTs in the epsilon expansion,” arXiv:2010.16222 [hep-th].
  * (27) A. Pelissetto and E. Vicari, “Critical phenomena and renormalization group theory,” Phys. Rept. 368 (2002) 549–727, arXiv:cond-mat/0012164.
  * (28) A. Aharony, “Comment on ‘Bicritical and Tetracritical Phenomena and Scaling Properties of the SO (5) Theory’,” Phys. Rev. Lett. 88 (2002) 059703, arXiv:cond-mat/0107585.
  * (29) M. Moshe and J. Zinn-Justin, “Quantum field theory in the large N limit: A Review,” Phys. Rept. 385 (2003) 69–228, arXiv:hep-th/0306133.
  * (30) A. Vasiliev and M. Nalimov, “Analog of Dimensional Regularization for Calculation of the Renormalization Group Functions in the 1/n Expansion for Arbitrary Dimension of Space,” Theor. Math. Phys. 55 (1983) 423–431.
  * (31) S. E. Derkachov and A. Manashov, “The Simple scheme for the calculation of the anomalous dimensions of composite operators in the 1/N expansion,” Nucl. Phys. B 522 (1998) 301–320, arXiv:hep-th/9710015.
  * (32) J. Gracey, “Large $N_{f}$ quantum field theory,” Int. J. Mod. Phys. A 33 no. 35, (2019) 1830032, arXiv:1812.05368 [hep-th].
  * (33) M. Goykhman and M. Smolkin, “Vector model in various dimensions,” Phys. Rev. D 102 no. 2, (2020) 025003, arXiv:1911.08298 [hep-th].
  * (34) S.-k. Ma, “Scaling Variables and Dimensions,” Phys. Rev. A 10 no. 5, (1974) 1818.
  * (35) K. Lang and W. Ruhl, “Critical nonlinear O(N) sigma models at 2 $<$ d $<$ 4: The Degeneracy of quasiprimary fields and it resolution,” Z. Phys. C 61 (1994) 495–510.
  * (36) D. Simmons-Duffin, “The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT,” JHEP 03 (2017) 086, arXiv:1612.08471 [hep-th].
  * (37) S. Caron-Huot, Y. Gobeil, and Z. Zahraee, “The leading trajectory in the 2+1D Ising CFT,” arXiv:2007.11647 [hep-th].