# Critical structure and emergent symmetry of Dirac fermion systems

Jiang Zhou<EMAIL_ADDRESS>Department of Physics, Guizhou University,
Guiyang 550025, PR China

###### Abstract

Emergent symmetry in Dirac system means that the system acquires an
enlargement of two basic symmetries at some special critical point. The
continuous quantum criticality between the two symmetry broken phases can be
described within the framework of Gross-Neveu-Yukawa (GNY) model. Using the
first-order $\epsilon$ expansion in $4-\epsilon$ dimensions, we study the
critical structure and emergent symmetry of the chiral GNY model with $N_{f}$
flavors of four-component Dirac fermions coupled strongly to an $O(N)$ scalar
field under a small $O(N)$-symmetry breaking perturbation. After determining
the stable fixed point, we calculate the inverse correlation length exponent
and the anomalous dimensions (bosonic and fermionic) for general $N$ and
$N_{f}$. Further, we discuss the emergent-symmetry and the emergent
supersymmetric critical point for $N\geq 4$ on the basis of $O(N)$-GNY model.
It turns out that the chiral emergent-$O(N)$ universality class is physically
meaningful if and only if $N<2N_{f}+4$. On this premise, the small
$O(N)$-symmetry breaking perturbation is always irrelevant in the chiral
emergent-$O(N)$ universality class. Our studies show that the emergent
symmetry in Dirac systems has an upper boundary $O(2N_{f}+3)$, depending on
the flavor numbers $N_{f}$. As a result, the emergent-$O(4)$ and $O(5)$
symmetries are possible to be found in the systems with fermion flavor
$N_{f}=1$, and the emergent-$O(4)$, $O(5)$, $O(6)$ and $O(7)$ symmetries are
expected to be found in the systems with fermion flavor $N_{f}=2$. Our result
also suggests some rich transitions with emergent-$Z_{2}\times O(2)\times
O(3)$ symmetry and so on. Interestingly, in the emergent-$O(4)$ universality
class, there is a supersymmetric critical point which is expected to be found
in the systems with fermion flavor $N_{f}=1$.

##  I Introduction

In quantum field theory, the Gross-Neveu model is a fundamental model of half
spin fermions and contains a quartic self-interaction of fermion fieldsgross ;
gracey1 ; handfour . It has some interesting properties which are commonly
shared by the quantum chromodynamicsgross . For example, the phenomena of
asymptotic freedom and dynamical symmetry breaking. Due to its close
connections with the phase transitions in graphene, the general Gross-Neveu
theory is also of interest in condensed matter theory. For instance, it has
been suggested that the transition from the semimetal to the Mott-insulating
phase in graphene can be described by what is termed chiral Gross-Neveu
modelgracey2 ; sorella1 ; herbut1 ; assaad1 ; herbut2 ; janssen1 ; chan1 ;
knorr1 . Based on the specific interaction and underlying symmetry, the chiral
Gross-Neveu model is captured by different universality classes, referring to
chiral Ising, XY and $O(3)$ universality classeszerf1 . More recently, it is
intriguing that the Gross-Neveu model has been found to be related to the
AdS/CFT theoriesgiombi1 .

In the case of chiral Ising universality class, the Gross-Neveu model is only
renormalizable in $1+1$ spacetime dimensions. Another more widely studied
model in this class is the GNY model herbut1 ; assaad1 ; herbut2 ; janssen1 ;
chan1 ; knorr1 ; zerf1 ; otsuka1 ; rosen1 ; mihai1 ; gracey3 ; lang1 ; herbut3
; roy1 ; scherer1 ; torres1 ; janssen2 ; torres2 ; ihrig1 , which is
renormalizable in $3+1$ spacetime dimensions. In contrast to the purely
fermionic Gross-Neveu model, the GNY model can be rewritten by introducing an
auxiliary scalar field so that the Lagrangian is quadratic in fermion
fieldhandfour . But now, the scalar field has a canonical kinetic term and the
scalar field interacts quarticallyzerf1 . For its wide applications in many
areas of physics ranging from high-energy physics to quantum criticality, the
GNY model has been studied by a broad range of different methods, including
functional renormalization groupjanssen1 ; knorr1 ; gies1 , conformal
bootstrapiliesiu1 ; iliesiu2 ; bobev1 ; bash1  and $1/N$ expansiongracey3 ;
mana1 ; gracey4 ; gracey5 . The sign-free quantum Monte Carlo has been broadly
used to study the fermionic quantum criticality of interacting Dirac fermions
on latticeotsuka1 ; mchuff1 ; lzx1 ; chan1 ; lzx2 ; otsuka2 ; wangl1 .
Finally, the $4-\epsilon$ expansion is a suitable algorithms to facilitate
higher-order calculations for fermionic criticalityrosen1 . By now, the most
accurate calculation has been carried up to four loops zerf1 ; ihrig1 .

In particular, the fermionic criticality in Dirac systems such as graphene has
been extensively discussedherbut3 ; otsuka1 . On graphene’s honeycomb lattice,
the low-energy gapless Dirac fermions with relativistic dispersion emerge at
the two inequivalent Dirac points in the Brillouin zone. A typical quantum
criticality of the interacting electrons on graphene’s lattice is the
transition from a semimetal to an ordered charge-density-wave which breaks
sublattice symmetryherbut1 ; chan1 , triggered by sufficient large nearest-
neighbor repulsion interaction. Another typical example on this lattice is the
semimetal-antiferromagnetic transition favored by an onsite Hubbard
repulsionotsuka1 ; assaad1 ; janssen1 . Other examples can be constructed
according to the ordered partner on the specific lattice otsuka3 ; raghu1 ;
ryu1 ; hou1 ; roy2 . The nature of the quantum phase transition for
interacting Dirac fermions has been under debate for a long timesore1 ;
herbut1 . However, recent developments on the fermion-driven quantum critical
point (FIQCP) suggest a second-order continuous phase transition between the
semimetallic and the gapped phase scherer1 ; torres1 ; torres2 ; janssen2 ;
lzx2 ; classen1 ; jsk1 ; jsk2 . The gapless Dirac fermions emerge as a new
degrees of freedom, consequently, their quantum criticality can effectively be
described by a chiral transitions appearing in different $2+1$ dimensional
Gross-Neveu model, giving rise to the chiral universality classeszerf1 . Even
in those system which supports cubic terms of order parameter, the strong
fluctuations of gapless fermions would render the putatively first-order
transition continuous, this is right the central idea of FIQCPclassen1 ; lzx2
; yin1 ; yin2 . Based on the honeycomb lattice, the evidence for FIQCP have
been proposed near the semimetal-Kekule valence-bond-solid transitionlzx2 ;
classen1 , similar scenario was also proposed in three-dimensional double-Weyl
semimetals for semimetal-nodal-nematic order transitionjsk2 . The FIQCP can be
traced back to the fluctuations of gapless Dirac degrees of freedom at
criticality.

Aside from the close connections with the quantum criticality from a Dirac
semimetal to a symmetry broken phase, the GNY model also has connections with
both supersymmetry sslee1 ; sslee2 ; grover1 ; jsk3 ; lzx3 ; witc1 ; rahm1
and emergent symmetry at criticality sato1 ; senthil1 ; grover2 ; tanka1 ;
nahum1 ; lzx5 ; sree1 ; roy3 . It is intriguing that the critical point
possess some symmetries which absent in the original model at the microscopic
level. For instance, right at the fixed point of the special GNY model in
which the velocities for both massless Dirac fermions and relativistic bosons
are not equal to each other, the Lorentz symmetry get restored at criticality,
leading to the notation of an emergent Lorentz symmetryroy4 . The emergent
symmetry at criticality can be attributed to the presence of fluctuations of
new degrees of freedom (gapless Dirac fermions).

In addition, the enlargement of symmetry was put forward to explain the
continuous phase transition between two phases with different broken
symmetries sato1 ; senthil1 ; grover2 ; tanka1 ; nahum1 ; lzx5 ; sree1 ; roy5
; sandvik1 ; sandvik2 ; zhang1 ; roy3 ; pujari1 . It is argued that the
deconfined quantum critical point, which separates the Neel and valence-bond-
solid orders of spin-1/2 Heisenberg quantum antiferromagnet on square lattice,
possesses an enlarged emergent symmetry senthil2 ; senthil3 ; mzy1 ; wangc1 .
Very recently, the similar enlarged emergent symmetry has been stressed in the
Dirac system with competing orderszhou1 ; ghae1 ; janssen2 ; roy3 . As pointed
above, the critical properties of the transition from a semimetal to a gapped
broken phase can be captured by GNY modelxxy2 ; liu3 ; liu4 ; sei1 , therefore
it is believed that the GNY model is accessible to the emergent symmetry with
an appropriate wayclassen6 ; roy6 . Indeed, on the basis of GNY model, it is
stressed that the emergent symmetry is responsible for various deconfined
transitions in Dirac systemsgrover1 ; lzx5 ; roy3 ; lzh1 . On the other hand,
the GNY model also connect closely with the supersymmetry sslee1 ; sslee2 ;
grover1 ; jsk3 ; lzx3 . For example, the GNY model with $1/4$ flavor of four-
component fermions in the chiral Ising universality class relates to the
$\mathcal{N}=1$ supersymmetryzerf1 , and the GNY model with $1/2$ flavor of
four-component fermions in the chiral XY universality class relates to the
$\mathcal{N}=2$ supersymmetrylzx3 .

Although the quantum criticality from a semimetal to an $O(N)$ symmetry broken
phase for $N\leq 3$ has achieved satisfactory understood, the impact of
fermion degrees of freedom on the stability of the quantum critical point with
emergent-$O(N)$ symmetry for $N\geq 4$ is under debatejanssen2 . For the
fermionic criticality in Dirac systems, a variety of critical points with
emergent symmetry have been put forwardryu1 ; roy3 ; pujari1 ; roy5 , by now,
however, what kinds of emergent symmetry is allowed is still unclear. Besides,
whether the supersymmetric quantum critical point can emerge in the chiral
emergent-$O(N)$ universality class remains unknown so far. The stability of
the critical point with emergent-$O(N)$ symmetry, as far as we are aware, is
partly answered in Refs.[janssen2, ] and [roy3, ]. Motivated by these issues,
in this paper, without asking the specific lattice model as well as the
specific forms of symmetry broken orders , we study the critical structure and
the emergent symmetry of the chiral GNY model with $N_{f}$ flavors four-
component fermions coupled strongly to an $O(N)$ scalar field. The present
study has three purposes. The first is to determine the meaningful fixed point
that controls the critical properties in the emergent-$O(N)$ universality
class. The second is to confirm the reasonable emergent-$O(N)$ symmetry in
Dirac systems. Our final purpose is to find the possible supersymmetric
critical point in the chiral emergent-$O(N)$ universality class. To
investigate the impact of gapless Dirac fermion degrees of freedom, we also
introduce a small $O(N)$-anisotropy that breaks $O(N)$ symmetry in the chiral
GNY model.

The rest of the paper is organized as follows. We define the chiral GNY model
in Sec.II. In Sec.III, we review the basic renormalization group (RG)
procedure and give our results for beta functions and anomalous dimensions. In
Sec.IV, we discuss the stability and the emergent symmetry for the
emergent-$O(N)$ fixed point, the supersymmetric quantum critical point in the
chiral emergent-$O(N)$ universality class is also discussed in this section.
Finally, our conclusions and some comments are provided in Sec.V. More details
for the determination of renormalization constants are presented in Appendix
A.

##  II The Gross-Neveu-Yukawa model

We first define the chiral GNY model under investigation. As pointed in the
introduction, for various interacting relativistic fermions systems in $2+1$
dimensions, the quantum phase transition towards a symmetry-broken phase can
be captured by the chiral GNY model in which the fermions couple strongly to a
multicomponent boson fields via Yukawa coupling. Formally, the general chiral
$O(N)$-GNY model in $D$-dimensions Euclidean spacetime can be described by the
effective action

$S=\int d^{D}x(\mathcal{L_{\psi}}+\mathcal{L_{\phi}}+\mathcal{L_{\psi\phi}}).$
(1)

The Lagrangian for fermions is simply abbreviated as

$\mathcal{L}_{\psi}=\bar{\psi}_{i}i\not{\partial}\psi_{i},$ (2)

where the notation $\not{\partial}=\gamma_{\mu}\partial_{\mu}$ is the Feynman
slash, and the $4\times 4$ gamma matrices $\gamma_{\mu}$ form a four-
dimensional representation of the Clifford algebra, i.e.,
$\\{\gamma_{\mu},\gamma_{\nu}\\}=2\delta_{\mu\nu}1_{4}$, with the indices
$\mu$, $\nu=0,1,...,D-1$ and $1_{4}$ denotes the identity matrix. The Dirac
spinor $\psi_{i}$ is a four-component fermion spinor and its conjugate is
defined as $\bar{\psi}_{i}={\psi}_{i}^{{\dagger}}\gamma_{0}$. For generality,
we have introduced $N_{f}$ flavors of four-component spinor such that the
Dirac spinor carries a flavor index $i$, $i=1,...,N_{f}$. The summation
convention over repeated indices is also assumed here and in the following.

The second term in (1) describes the purely bosonic part. Explicitly, it takes
the form as

$\mathcal{L}_{\phi}=\frac{1}{2}(\partial_{\mu}\phi_{a})^{2}+\frac{m^{2}}{2}\phi_{a}\phi_{a}+\frac{\lambda_{1}}{4!}(\phi_{a}\phi_{a})^{2}+\frac{\lambda_{2}}{4!}(\phi_{a})^{4}.$
(3)

Here, the index $a$ in $\phi_{a}$ takes the value range from $1$ to $N$. In
addition to the kinetic term and quartic interactions with strength
$\lambda_{1}$, a small anisotropy (with strength $\lambda_{2}$) that breaks
$O(N)$ symmetry is added to investigate the impact of gapless Dirac fermion
degrees of freedom. The mass-square $m^{2}$ plays the role of tuning parameter
for phase transition, $m^{2}>0$ corresponds to symmetric phase with
$\langle\phi_{a}\rangle=0$, $m^{2}<0$ corresponds to symmetry broken phase and
$m^{2}=0$ at criticality. In the symmetry broken ground state, the scale field
$\phi_{a}$ acquires a nonzero vacuum expectation, then the fermion mass is
generated dynamically. For the pure $O(N)$ scalar model, although the symmetry
is broken by the small anisotropy presented in the last term, it is shown that
the symmetry is restored at criticality for $N<3$kleinert1 ; varn1 . The
$O(N)$ fixed point is unstable under anisotropy for $N>3$. In this paper, we
will demonstrate the impact of gapless Dirac fermion degrees of freedom on the
stability of the $O(N)$ fixed point.

Finally, $\mathcal{L}_{\psi\phi}$ defines the Yukawa coupling between gapless
Dirac fermions and $O(N)$-symmetric scalar fields with strength $g$:

$\mathcal{L}_{\psi\phi}=g\bar{\psi}_{i}\left[(\Sigma_{a})_{4N_{f}\times
4N_{f}}\cdot\phi_{a}\right]\psi_{i}.$ (4)

Each $\Sigma_{a}$ signals the broken pattern of various gapped phases, their
dimension depend on the specific lattice model. More precisely, the dimension
for these sigma matrices coincides with the components of Dirac spinor. For
example, the spinless fermions on honeycomb lattice define a four-component
Grassmann spinor, the dimension for the sigma matrix is four. On the other
hand, the spinful fermions on honeycomb lattice define an eight-component
Dirac spinor, now the dimension for the sigma matrix is eight.

To continue the following calculation, commutating rules between these sigma
matrices and Dirac gamma matrices are required. In the chiral Ising
universality class for $N=1$, the GNY model Eq.(1) includes an one-component
real scalar field, $\Sigma_{1}$ is a trivial identity matrix in this case, so
we have $[\gamma_{\mu},\Sigma_{1}]=0$ for chiral Ising-GNY model. In the
chiral XY universality class for $N=2$, the GNY model includes a complex order
parameter, now the Yukawa term can be generally written as
$\mathcal{L}_{\psi\phi}=g\bar{\psi}_{i}(\phi_{1}+i\gamma^{5}\phi_{2})\psi_{i}$,
where $\\{\gamma^{5},\gamma_{\mu}\\}=0$ and the explicit choice of
$\gamma^{5}$ depends on the specific modelroy1 ; zerf3 ; janssen4 . Note that
$\gamma^{5}S_{\psi}(p)=-S_{\psi}(p)\gamma^{5}$, where $S_{\psi}(p)$ is the
fermion propagator. Since in any non-vanished Feynman loop, each fermion
propagator is attached by two $i\gamma^{5}$ factors at both sides
respectively, then the minus generated from the interchange of $\gamma^{5}$
and $S_{\psi}(p)$ is countered by $i^{2}=-1$. Therefore, the vanished
commutator $[\vec{\Sigma}_{\text{XY}},\gamma_{\mu}]=0$ works in the practical
calculations for chiral XY-GNY model. Finally, in the chiral $O(3)$
universality class for $N=3$, the three-component order parameters break spin-
rotational symmetry spontaneously. An explicit choice of the Yukawa
interactions is given in Ref.[janssen1, ], and the commutator
$[\vec{\Sigma}_{O(3)},\gamma_{\mu}]=0$ is satisfied.

Since the emergent $O(N)$ symmetry for $N\geq 4$ can be builded on the
competing orders that break these three basic symmetriesgrover2 ; roy3 ; ryu1
, it is reasonable to assume the commutating rules

$[\Sigma_{a},\gamma_{\mu}]=0,\quad\forall a,\mu,$ (5)

in chiral $O(N)$-GNY model. This vanished commutator is essential for the
derivation of fixed points and critical exponents in the chiral
emergent-$O(N)$ universality class. In the practical calculation, all beta and
eta functions are independent from the explicit matrix representation of the
Dirac gamma matrix, and only the Clifford algebra and the dimensions of the
representation matrix are required in the renormalization group calculations.
At the tree level, the scaling dimensions for the field variables and coupling
constants can be identified from Eq.(1),

$\displaystyle[\psi]=\frac{D-1}{2},\quad[\phi]=\frac{D-2}{2},$ (6)
$\displaystyle[\lambda]=4-D,\quad[g]=\frac{4-D}{2}.$ (7)

The Yukawa coupling $g$, quartic $O(N)$-interaction $\lambda_{1}$, as well as
the strength for small $O(N)$-symmetry-breaking perturbations $\lambda_{2}$,
are all marginal at the upper critical dimension $D_{uc}=4$, implying that the
critical properties is accessible via standard epsilon expansion in
$D=4-\epsilon$ spacetime dimensions.

##  III Field theory and Renormalization group

This section present the renormalization group (RG) analysis of the chiral
$O(N)$-GNY model under a small $O(N)$-symmetry-breaking perturbation, see
Eq.(1). To perform standard RG analysis in $4-\epsilon$ dimensions, we employ
dimensional regularization and modified minimal subtraction (MS) scheme. The
bare Lagrangian is defined by replacing the field variables and different
couplings with their bare counterparts, i.e.,

$\psi\rightarrow\psi_{0},\phi\rightarrow\phi_{0},\lambda\rightarrow\lambda_{0},g\rightarrow
g_{0},m^{2}\rightarrow m^{2}_{0}.$ (8)

The renormalized Lagrangian is then written as

$\displaystyle\mathcal{L}$
$\displaystyle=Z_{\psi}\bar{\psi}_{i}i\not{\partial}\psi_{i}+\frac{1}{2}Z_{\phi}(\partial_{\mu}\phi_{a})^{2}+\frac{1}{2}Z_{\phi}Z_{m^{2}}m^{2}\phi_{a}^{2}$
$\displaystyle+\frac{\lambda_{1}}{4!}\mu^{\epsilon}Z_{\lambda_{1}}Z_{\phi}^{2}\left(\phi_{a}\phi_{a}\right)^{2}+\frac{\lambda_{2}}{4!}\mu^{\epsilon}Z_{\lambda_{2}}Z_{\phi}^{2}\left(\phi_{a}\right)^{4}$
$\displaystyle+g\mu^{\epsilon/2}Z_{g}Z_{\psi}\sqrt{Z_{\phi}}\bar{\psi}_{i}\left(\Sigma_{a}\phi_{a}\right)\psi_{i}.$
(9)

where $\mu$ is the energy-scale parameterizing the RG flow of the coupling
constants. These different $Z$-factors are named renormalization constants,
which are used to absorbed the divergence in the loop corrections. The wave-
function renormalization constants $Z_{\psi}$, and $Z_{\phi}$, relate the bare
and renormalized field variables upon the field rescalings
$\psi_{0}=\sqrt{Z_{\psi}}\psi$, $\phi_{0}=\sqrt{Z_{\phi}}\phi$. Accordingly,
the bare mass-square, quartic couplings, and the Yukawa coupling are related
to their dimensionless partner as following,

$\displaystyle m^{2}_{0}=\mu^{2}m^{2}Z_{m^{2}},$ (10)
$\displaystyle\lambda_{10}=\mu^{\epsilon}\lambda_{1}Z_{\lambda_{1}},$ (11)
$\displaystyle\lambda_{20}=\mu^{\epsilon}\lambda_{2}Z_{\lambda_{2}},$ (12)
$\displaystyle g_{0}=\mu^{\epsilon/2}gZ_{g}.$ (13)

We have also introduced the rescaling couplings

$g\rightarrow
g\mu^{\epsilon/2},\lambda_{1}\rightarrow\lambda_{1}\mu^{\epsilon},\lambda_{2}\rightarrow\lambda_{2}\mu^{\epsilon},$
(14)

such that different couplings in the renormalized Lagrangian are
dimensionless, this lead to explicit energy-scale dependencies Lagrangian.

The RG beta functions are defined as the logarithmic derivatives with respect
to $\mu$, $\beta(X)=dX/d\ln\mu$, where $X=m^{2},\lambda_{1},\lambda_{2},g$.
These beta functions can be derived using the fact that the bare value are
independent of $\mu$. Defining the mass-square anomalous dimensions
$\gamma_{m^{2}}=d\ln Z_{m^{2}}/d\ln\mu$, we find

$\beta(m^{2})=-(2+\gamma_{m^{2}})m^{2}.$ (15)

The $\ln\mu$ derivative of Eqs. (11)-(13) give the beta function of
dimensionless couplings:

$\displaystyle\beta(\lambda_{1})=-\epsilon\lambda_{1}-\frac{1}{Z_{\lambda_{1}}}\frac{\partial
Z_{\lambda_{1}}}{\partial\ln\mu}\lambda_{1},$ (16)
$\displaystyle\beta(\lambda_{2})=-\epsilon\lambda_{2}-\frac{1}{Z_{\lambda_{2}}}\frac{\partial
Z_{\lambda_{2}}}{\partial\ln\mu}\lambda_{2},$ (17)
$\displaystyle\beta(g^{2})=-\epsilon g^{2}-\frac{2}{Z_{g}}\frac{\partial
Z_{g}}{\partial\ln\mu}g^{2}.$ (18)

Here, we have introduced the squared Yukawa coupling $g^{2}$ for notational
simplicity. In terms of these RG beta functions, the inverse correlation
length exponent $\nu^{-1}$ is determined by mihai1 ; zerf1

$\nu^{-1}=-\frac{d\beta(m^{2})}{m^{2}}=2+\gamma_{m^{2}}.$ (19)

Furthermore, the fermion anomalous dimensions $\gamma_{\psi}$, and boson
anomalous dimensions $\gamma_{\phi}$ are obtained from

$\gamma_{\phi}=\frac{1}{Z_{\phi}}\frac{\partial
Z_{\phi}}{\partial\ln\mu},\quad\gamma_{\psi}=\frac{1}{Z_{\psi}}\frac{\partial
Z_{\psi}}{\partial\ln\mu}.$ (20)

Both the beta functions and anomalous dimensions can be determined from the
renormalization constants in the context of the standard perturbative RG. In
the next subsection, we will determine the renormalization constants at the
leading order.

###  III.1 Renormalization constants

Figure 1: One-loop 1PI diagrams. (a), (b) and (c) for boson self-energy, (d)
for fermion self-energy, (e) for Yukawa vertex, (f), (g) and (h) for
$O(N)$-symmetry quartic vertex, (i) and (j) for $O(N)$-anisotropy quartic
vertex.

In order to access the properties of critical point, one must calculate all
the renormalization constants. Using dimensional regularization and MS scheme,
they depend only on the dimensionless couplings and can be expanded into the
following formal Laurent seriespeskin1 ,

$Z_{X}(1/\epsilon,\lambda_{1},\lambda_{2},g)=1+\sum_{k=1}^{\infty}Z^{(k)}_{X}(\lambda_{1},\lambda_{2},g)\frac{1}{\epsilon^{k}},$
(21)

which can be determined order by order in pertubative RG. To the leading
order, we expand $Z_{X}=1+\delta_{X}$ and demand these $\delta_{X}$s cancel
the ultraviolet divergences of the one-loop corrections. The one-particle
irreducible (1PI) diagrams for two-point boson self-energy, two-point fermion
self-energy, three-point Yukawa vertex, $O(N)$-symmetry quartic vertex, and
$O(N)$-anisotropy quartic vertex are indicated in Fig.1. To calculate these
one-loop 1PI divergent diagrams, the fermion and boson propagators are

$\displaystyle S^{\psi}_{ab}(p)=-i\delta_{ab}\frac{\not{p}}{p^{2}},$ (22)
$\displaystyle D^{\phi}_{ij}(p)=\frac{\delta_{ij}}{p^{2}+m^{2}}.$ (23)

And the complete Feynman rules are illustrated in Fig.2. We present a more
detailed calculations of these one-loop 1PI diagrams in Appendix A, here we
only quote the final results:

$\displaystyle Z_{\psi}=1-Ng^{2}\frac{K}{\epsilon},$ (24) $\displaystyle
Z_{\phi}=1-4N_{f}g^{2}\frac{K}{\epsilon},$ (25) $\displaystyle
Z_{m^{2}}=1+(\frac{N+2}{3}\lambda_{1}+\lambda_{2}+4N_{f}g^{2})\frac{K}{\epsilon},$
(26) $\displaystyle
Z_{\lambda_{1}}=1+(\frac{N+8}{3}\lambda_{1}+2\lambda_{2}-48N_{f}g^{4}/\lambda_{1}+8N_{f}g^{2})\frac{K}{\epsilon},$
(27) $\displaystyle
Z_{\lambda_{2}}=1+(3\lambda_{2}+4\lambda_{1}+8N_{f}g^{2})\frac{K}{\epsilon},$
(28) $\displaystyle Z_{g}=1+(2N_{f}+4-N)g^{2}\frac{K}{\epsilon}.$ (29)

It is seen from above that all renormalization constants have a simple pole at
$\epsilon$. Here and in the following, the constant $K=1/(4\pi)^{2}$. To
derive these renormalization $Z$-factors, we have made use of the sigma matrix
with dimensions four but without asking its explicit representation. The
dimensions-dependence on the sigma matrix is generalized by introducing
$N_{f}$ flavors of four-component fermions.

Figure 2: Feynman rules for the fermion and boson propagators, Yukawa vertex,
$O(N)$-symmetry and $O(N)$-anisotropy quartic vertex.

###  III.2 Beta functions and anomalous dimensions

To derive the exact expressions of the beta functions, we first use the chain
rule, to write

$\frac{\partial Z_{X}}{\partial\ln\mu}=\sum_{Y}\frac{\partial Z_{X}}{\partial
Y}\beta(Y),$ (30)

where $X,Y\in\\{m^{2},\lambda_{1},\lambda_{2},g\\}$. Inserting the
renormalization $Z$-factors into Eqs.(16)-(18) then produces the beta
functions for $O(N)$ scalar coupling $\lambda_{1}$, $O(N)$-anisotropy strength
$\lambda_{2}$, and squared Yukawa coupling $g^{2}$, respectively. We find

$\displaystyle\beta(\lambda_{1})$
$\displaystyle=-\epsilon\lambda_{1}+(N+8)K\lambda^{2}_{1}/3+2K\lambda_{1}\lambda_{2}$
$\displaystyle+8KN_{f}g^{2}\lambda_{1}-48KN_{f}g^{4},$ (31)
$\displaystyle\beta(\lambda_{2})=-\epsilon\lambda_{2}+3K\lambda^{2}_{2}+4K\lambda_{1}\lambda_{2}+8KN_{f}g^{2}\lambda_{2},$
(32) $\displaystyle\beta(g^{2})=-\epsilon g^{2}+(4N_{f}+8-2N)Kg^{4}.$ (33)

Our beta function can be verified in some specific limits. For example,
setting $g=0$, we recover the one-loop beta function for the $O(N)$ scalar
model under a cubic-anisotropyamit1 . Setting $g=0$ and $\lambda_{2}=0$,
$\beta(\lambda_{1})$ agrees exactly with the results of $O(N)$ scalar model.
In the chiral GNY limit upon setting $\lambda_{2}=0$, rescaling the coupling
$\alpha/8\pi^{2}\mapsto\alpha$ for $\alpha=g^{2}$, $\lambda_{1}$,
$\lambda_{2}$, and replacing the boson self-interactions according to
$\lambda/4!\mapsto\lambda$, our beta function agrees fully with the
corresponding expressions derived in the chiral GNY model zerf1 .

In terms of Eq.(30), the inverse correlation length exponent and the anomalous
dimensions [see Eqs.(19)-(20)] can be obtained, to the one-loop order, as

$\displaystyle\nu^{-1}$
$\displaystyle=2-(\frac{N+2}{3}\lambda_{1}+\lambda_{2}+4N_{f}g^{2})K.$ (34)
$\displaystyle\gamma_{\phi}$
$\displaystyle=4KN_{f}g^{2},\quad\gamma_{\psi}=KNg^{2}.$ (35)

The couplings dependence of these exponents is meant to be evaluated at the
stable fixed point that controls the critical behaviors.

##  IV RG analysis

We began the RG analysis by searching for the fixed points of the beta
function. The beta function for the Yukawa coupling exhibits two fixed points:
zero and nonzero. It is easy to see that only the fixed point with finite
Yukawa value is stable, since at which the slope
$\partial\beta(g^{2})/\partial g^{2}$ is positive. In the case of $g^{2}=0$,
the beta functions in Eqs.(III.2)-(32) admit four well understood fixed
pointsvarn1 ; amit1 : the trivial Gaussian fixed point $(0,0)$, Ising fixed
point $[\epsilon/(3K),0]$, anisotropic fixed point
$[\epsilon/(NK),(N-4)\epsilon/(3NK)]$, and isotropic Wilson-Fisher fixed point
$[3\epsilon/((N+8)K),0]$. The most intriguing fixed points are the Wilson-
Fisher and the anisotropic fixed points as they exchange their stability at
the critical dimensionality $N_{c}\approx 3$, see for example in Ref.[varn1,
].

At the fixed point with finite Yukawa coupling, solving the common zero of
Eqs.(III.2) and (32), we find four different fixed points: two Yukawa-Wilson-
Fisher (YWF) fixed points and two fixed points with small $O(N)$-symmetry-
breaking anisotropy which we will term anisotropic fixed point (AFP). Among
all those fixed points, we are interesting in the ones with positive
$\lambda_{1}$, so-called YWF2 fixed point and AFP2, as their stability depend
on the value of $N$ and fermion flavors $N_{f}$. The YWF2 fixed point locates
at

$\displaystyle\lambda_{1*}$
$\displaystyle=\frac{3}{2}\frac{F+N^{2}-8N-4N_{f}^{2}+16}{(N+8)(2N_{f}-N+4)^{2}}\frac{\epsilon}{K},$
(36) $\displaystyle\lambda_{2*}$ $\displaystyle=0,$ (37) $\displaystyle
g^{2}_{*}$ $\displaystyle=\frac{1}{2(2N_{f}-N+4)}\frac{\epsilon}{K},$ (38)

defining

$F\equiv\sqrt{[(N-4)^{2}+4N_{f}^{2}+20N_{f}N+112N_{f}]F_{0}},$ (39)

with $F_{0}=(2N_{f}-N+4)^{2}$, and the AFP2 locates at

$\displaystyle\lambda_{1*}$
$\displaystyle=\frac{W+2N_{f}+N-4}{2N(N-2N_{f}-4)}\frac{\epsilon}{K},$ (40)
$\displaystyle\lambda_{2*}$
$\displaystyle=\frac{-2W+(N+2N_{f}-6)N-4N_{f}+8}{3N(N-2N_{f}-4)}\frac{\epsilon}{K},$
(41) $\displaystyle g^{2}_{*}$
$\displaystyle=\frac{1}{2(2N_{f}-N+4)}\frac{\epsilon}{K},$ (42)

where $W=[4N_{f}^{2}+4N_{f}(37N-4)+(N-4)^{2}]^{1/2}$. The stable fixed point
occurs at the nonzero Yukawa coupling, thus the chiral $O(N)$ universality
class has strongly boson-fermion coupled critical fluctuations. We will
analysis the interplay between the boson-fermion coupled critical fluctuations
and the small $O(N)$-symmetry-breaking anisotropy.

###  IV.1 Stability analysis

Figure 3: Stability exponents as a function of $N$ in the non-Yukawa limit.
$\omega^{\text{(WF)}}$ (blue) and $\omega^{\text{(AFP)}}$ (red) denote the
stability exponents evaluated at the isotropic Wilson-Fisher and the
anisotropic fixed point, respectively.

The stability of the fixed points is determined by a matrix $M_{ij}$, which is
termed stability matrix and defined as the first derivatives of the beta
functions with respect to the couplings

$\displaystyle M_{ij}=\frac{\partial\beta(\lambda_{i})}{\partial\lambda_{j}},$
(43)

where $i,j=1,2,3$ and $\lambda_{3}=g^{2}$. The reliable conclusion about the
stability of fixed point can be given by calculating the eigenvalues of the
stability matrix taken at the fixed point. If the real part of the eigenvalues
are all positive, the fixed point is stable and corresponds to a sink. On the
other hand, if the real part of the eigenvalues have opposite sign, the fixed
point is of a saddle-point. A saddle-point type fixed point acquires at least
an unstable direction on the surface spanned by the coupling constants. An
important property of the stability matrix is that its $i$-th eigenvalue
(denoted by $\omega_{\lambda_{i}}$) controls the RG flow approaching the fixed
point along $\lambda_{i}$-direction. In turn, these eigenvalues are called
stability exponents.

For the non-Yukawa limit with $g^{2}=0$, the stability matrix reads

$\displaystyle
M_{ij}=\left(\begin{array}[]{cc}-\epsilon+\frac{N+8}{3}2K\lambda_{1}+2K\lambda_{2}&2K\lambda_{1}\\\
4K\lambda_{2}&-\epsilon+6K\lambda_{2}+4K\lambda_{1}\end{array}\right).$

To obtain the stability exponents of the fixed point, we calculate the
eigenvalues of the stability matrix. The stability exponents at the isotropic
Wilson-Fisher (or named Heisenberg) fixed point and the anisotropic fixed
point are plotted in Fig.3, respectively. It is seen from Fig.3 that the
critical value $N_{c}=4$ separates two distinct regimes of the stable fixed
point. For $N<N_{c}$, the isotropic Wilson-Fisher fixed point is stable. While
for $N>N_{c}$, the anisotropic fixed point is stable as the stability
exponents $\omega^{(\text{AFP})}_{1}>0$ and $\omega^{(\text{AFP})}_{2}>0$ in
this regime. Therefore, both fixed points merge into a single point and
exchange their stability at $N_{c}$. For the most accurate value of $N_{c}$,
early four and five-loop approximations suggest it lies bellow $3$kleinert1 ;
varn1 .

Let us now concentrate our attentions on the YWF2 fixed point. In the presence
of finite Yukawa coupling, the stability matrix can be derived from
Eqs.(III.2)-(33). Diagonalization of the stability matrix shows that the
stability of the YWF2 fixed point depends on $N$. In Fig.4, we plot the
stability exponents of the YWF2 fixed point for $N_{f}=1$ and $N_{f}=2$. In
the case of $N_{f}=1$, the first exponents are constantly positive for
different $N$, while the second exponent changes its sign at $N\approx 11$,
see the blue-square in Fig.4. This implies that the YWF2 fixed point is stable
and governs the critical behaviors for $N<11$. In the case of $N_{f}=2$, the
stability exponents are illustrated by red line in Fig.4, the results are
qualitatively similar as that for $N_{f}=1$ but the stability is separated by
$N=12$ (see red triangle). Further, we also determine the stability of YWF2
fixed point as the function of flavors of fermions, and the results are shown
in Fig.5. From Fig.5, we see that the exponents for different $N$ do not
change their sign as $N_{f}$ increases. In particular, the YWF2 fixed point in
chiral emergent-$O(4)$ and $O(5)$ universality class is stable, this implies
fermion-induced symmetry enhancement in interacting Dirac fermion
systemsjanssen2 ; roy3 .

The stable YWF2 fixed point means that the $O(N)$-anisotropy is irrelevant.
Indeed, for $N=1$, the initial model possesses an exchange symmetry
$\lambda_{1}\Leftrightarrow\lambda_{2}$. As a result, the beta functions
should obey the relation
$\beta(\lambda_{1}+\lambda_{2})=\beta(\lambda_{1})+\beta(\lambda_{2})$, the
anisotropic fixed point and the isotropic fixed point merge to form a new
Yukawa Wilson-Fisher (stable) fixed point. Then the initial model reduce to
the chiral Ising GNY model with an effective coupling
$\lambda_{e}=\lambda_{1}+\lambda_{2}$ in this special case.

Figure 4: Numerical results for the first two stability exponents at the YWF2
fixed point as a function of $N$. dashed blue line is plotted for $N_{f}=1$
and red line is plotted for $N_{f}=2$. Figure 5: Stability exponents of the
YWF2 fixed point for different $N$ as a function of $N_{f}$.

###  IV.2 RG flows

Aside from the stability analysis for the fixed point, RG flows provide us
with an alternative way to investigate the properties of the fixed point.
Since the beta function for Yukawa coupling is independent of $\lambda_{1}$
and $\lambda_{2}$ [see Eq.(33)], the stable fixed point shares the common
Yukawa value. Therefore, it is sufficient only to plot the projected RG flows
in the $\lambda_{1}$-$\lambda_{2}$ plane at the finite Yukawa coupling.

We plot the RG flows for $N_{f}=1$ in Fig.6(a)-(c). For the chiral $O(4)$-GNY
model [Fig.6(a)], the YWF2 fixed point is a stable point, while the AFP2 is a
saddle point. With the increase in $N$, AFP2 moves gradually toward YWF2 fixed
point, then they merge into a single point M at about $N=11$, as shown in
Fig.6(b). Right at the point M, the beta functions have a marginal component
in $\lambda_{2}$-direction. With further increase of $N$, the YWF2 fixed point
acquires one unstable direction, and the AFP2 turns into a stable one in
Fig.6(c). Further, the RG flows for $N_{f}=2$ are illustrated in Fig.6(d)-(f).
As in the case of $N_{f}=1$, the YWF2 fixed point is stable and controls the
critical behavior in the chiral $O(4)$ universality class. With the increase
in $N$, the YWF2 fixed point and AFP2 merge into a single point M at $N=12$.
The YWF2 fixed point is stable for $N<12$, while the AFP2 is stable for
$N>12$. These RG flows are in consistent with the stability analysis.

Although the RG flows suggest another stable fixed point (say AFP2) for
sufficiently large $N$, what has to be emphasized is that such a new fixed
point is not physically meaningful and cannot be reached in Dirac systems
since the Yukawa coupling now is immeasurable for sufficiently large $N$[see
Eq.(42)]. The non-negativity of the squared Yukawa coupling implies that $N$
has an upper boundary. As a consequence, the emergent symmetry has a maximum
value in different Dirac fermion systems.

Figure 6: Projected RG flows in the $\lambda_{1}$-$\lambda_{2}$ plane. The
first row (a)-(c) display the flows for $N_{f}=1$. Panel (a) is plotted for
$N=4$, the YWF2 fixed point is stable and AFP2 is unstable. With the increase
in $N$, two fixed points merge into a single point M at $N\approx 11$ in (b).
Panel (c) is plotted for $N=14$, the YWF2 fixed point is unstable and AFP2 is
stable in (c). The second row (d)-(f) display the flows for $N_{f}=2$. Panel
(d) is plotted for $N=4$, the YWF2 fixed point is stable and AFP2 is unstable.
With the increase in $N$, two fixed points merge into a single point M at
$N=12$ in (e). Panel (f) is plotted for $N=14$, the YWF2 fixed point is
unstable and AFP2 is stable in (f).

###  IV.3 Critical exponents and emergent symmetries

When the system is tuned to criticality, all couplings flow to the infrared
stable fixed point at which the system exhibits scale invariance. Close enough
to the scale invariant point, the correlation length and two-point correlation
functions have the form of simple power-law. These power exponents define the
critical exponents. Here, we calculate the inverse correlation length
exponent, bosonic anomalous dimensions, and the fermionic anomalous dimensions
for the chiral $O(N)$-GNY model at the physically reasonable fixed point.
Furthermore, we also discuss the supersymmetric quantum critical point in the
chiral emergent-$O(N)$ universality class.

Near the stable YWF2 fixed point, the anomalous dimensions for bosons as well
as fermions, to the leading order in $\epsilon=4-D$, read

$\displaystyle(\eta_{\phi},\eta_{\psi})=\left[\frac{2N_{f}}{(2N_{f}-N+4)},\frac{N}{2(2N_{f}-N+4)}\right]\epsilon,$
(44)

which control the scaling of two-point boson and fermi correlation functions.
Inserting the YWF2 fixed point into Eq.(34), we find for the inverse
correlation length exponent

$\displaystyle\frac{1}{\nu_{y}}=2-$
$\displaystyle\frac{(N+2)(F+N^{2}-8N-4N_{f}^{2}+16)}{2(N+8)(2N_{f}-N+4)^{2}}\epsilon$
$\displaystyle-$ $\displaystyle\frac{2N_{f}}{2N_{f}-N+4}\epsilon.$ (45)

For $N_{f}=2$, the numerical evaluation of the expression provides the
following series in $\epsilon$ : $\nu_{y}^{-1}=2-0.9524\epsilon$ $(N=1)$ and
$\nu_{y}^{-1}=2-1.2\epsilon$ $(N=2)$, in full agreement with the previous
study in the chiral Ising and chiral XY universality classzerf1 . When
extrapolated to $\epsilon=1$, the anomalous dimensions [see Eq.(44)] at the
linear order in $\epsilon$ must be non-negative. Then, the non-negativity of
the anomalous dimensions imposes a constraint on $N$, or on the emergent
$O(N)$ symmetry, reading

$\displaystyle N<2N_{f}+4.$ (46)

Meanwhile, we note that this constraint is also a natural result of the
measurability of Yukawa coupling. Due to the constraint, the AFP2 is therefore
a non-physical fixed point for $N_{f}=1$ and $N_{f}=2$.

The numerical series for the inverse correlation length exponent
$\nu_{y}^{-1}$ have been provided in Table 1. In general, the chiral $O(N)$
universality class are divided into two classes. The first class is the
conventional chiral $O(N)$ universality class in which the ordered phase
breaks some basic symmetries. Another class is the chiral emergent-$O(N)$
universality class in which the critical point possesses an enlargement of the
basic symmetries. For instance, the chiral Ising-GNY model is relevant to the
quantum criticality of semimetal-CDW transition on graphene’s honeycomb
lattice, where the ordered state breaks $Z_{2}$ sublattice symmetry
spontaneously. The chiral XY-GNY model has connections with superconducting or
Kekulé valence-bond-solid transition in graphene, where the ordered state
breaks $U(1)\simeq O(2)$ symmetry. The chiral $O(3)$-GNY model describes the
transition towards an antiferromagnetic order that breaks $SU(2)$ symmetry in
related materials. These are three typical examples in the conventional chiral
$O(N)$ universality class. For the chiral emergent-$O(N)$ universality class,
however, it cannot be achieved by breaking a single conventional symemtry. In
fact, the emergent symmetry can be realized in some special cases when two
ordered phases with different broken symmetry meet at a common critical
pointjanssen2 ; roy3 .

Limited by the constraint on $N$, the emergent-$O(N)$ symmetry depends on the
number of fermion flavor and has an upper boundary $O(2N_{f}+3)$.
Specifically, in the Dirac system with single flavor ($N_{f}=1$) of four-
component fermions, the emergent-$O(4)$ and $O(5)$ symmetries at criticality
are possible. And in the system with two flavors ($N_{f}=2$) of four-component
fermions, the emergent-$O(N)$ symmetry for $N=4,5,6,7$ are possible to be
found. More importantly, since the constraint on $N$ are obtained from the
leading-order anomalous dimensions which are not affected by higher-order
corrections, thus the constraint holds in any higher-order computations. What
we want to emphasize in particular is that close enough to the critical point
the emergent symmetry-$O(2)$ ($Z_{2}\times Z_{2}$) and $O(3)$ ($Z_{2}\times
O(2)$roy3 ) are also compatible with the constraint.

Finally, let us briefly discuss the emergent supersymmetric critical point.
For $N_{f}=1/4$, the GNY model in the chiral Ising universality class exhibits
an emergent supersymmetry, the YWF2 fixed point now becomes a supersymmetric
fixed point at which $\eta_{\phi}=\eta_{\psi}=\epsilon/7$. Another interesting
case is the $N_{f}=1/2$ GNY model in the chiral XY universality classlzx3 ,
this version of model has been argued that the supersymmetry might emerges at
the critical point, with the anomalous dimensions are given by
$\eta_{\phi}=\eta_{\psi}=\epsilon/3$. In general, we note from Eq.(44) that
the supersymmetry is ensured by $N=4N_{f}$. As a result, the supersymmetry is
expected to be emerged from the quantum critical point characterized by an
emergent-$O(4)$ symmetry, such supersymmetric critical point is expected to be
found in the chiral GNY model with single flavor ($N_{f}=1$) of four-component
fermions.

Table 1:  Numerical series of the inverse correlation length exponent
$\nu_{y}^{-1}$ at the stable YWF2 fixed point.

Basic $O(N)$ symmetry | $N_{f}=1$ | $N_{f}=2$
---|---|---
$Z_{2}$ ($N$=1) | $2-0.8347\epsilon$ | $2-0.9524\epsilon$
$O(2)$ | $2-1.1325\epsilon$ | $2-1.2\epsilon$
$O(3)$ | $2-1.5988\epsilon$ | $2-1.5273\epsilon$
Emergent $O(N)$ symmetry | $N_{f}=1$ | $N_{f}=2$
$O(4)$ | $2-2.5\epsilon$ | $2-2.0\epsilon$
$O(5)$ | $2-5.1583\epsilon$ | $2-2.7692\epsilon$
$O(6)$ | not exist | $2-4.2857\epsilon$
$O(7)$ | not exist | $2-8.8\epsilon$

##  V Conclusions and comments

Within the first-order $\epsilon$ expansion, we have studied the critical
structure and the emergent symmetry of chiral GNY model in the presence of a
small $O(N)$-anisotropy. This model includes a Yukawa term with $N_{f}$
flavors of four-component Dirac fermions strongly coupled to an $O(N)$ scalar
field. We have determined the stability of the fixed points and computed the
critical exponents by means of perturbative renormalization in $4-\epsilon$
dimensions. On the basis of $O(N)$-GNY model, we have discussed the physically
reasonable emergent symmetry in Dirac systems. Further, the supersymmetric
quantum critical point in the emergent-$O(4)$ universality class have also
been discussed briefly. The main three conclusions of our findings can be
summarized as follows.

(i) The GNY model in the chiral Ising, chiral XY, or chiral $O(3)$
universality class has an unique infrared-stable fixed point, the so called
Wilson-Fisher-Yukawa fixed point. For the GNY model in the chiral
emergent-$O(N)$ universality class with $N\geq 4$, in order to meet the
requirements of measurability for Yukawa coupling, the chiral emergent-$O(N)$
universality class is physically meaningful if and only if $N$ is less than
$2N_{f}+4$, where $N_{f}$ is the number of flavors of four-component Dirac
fermions. On the premise that the emergent-$O(N)$ universality class is
meaningful, the GNY model in the chiral emergent-$O(N)$ universality class is
also dominated by the Wilson-Fisher-Yukawa fixed point. This result holds at
least in the cases with $N_{f}=1$ and $N_{f}=2$. As a result, the small
$O(N)$-anisotropy that breaks $O(N)$ symmetry is irrelevant in the chiral
emergent-$O(N)$ universality class.

(ii) The non-negativity of the anomalous dimensions and the measurability of
the Yukawa coupling impose the constraint $N<2N_{f}+4$ on the emergent-$O(N)$
symmetry. As a result, the emergent symmetry has an upper boundary
$O(2N_{f}+3)$ in Dirac systems. The enlarged emergent-$O(4)$ and $O(5)$
symmetries are possible to be found in the system with single flavor
($N_{f}=1$) of four-component fermions, and the enlarged emergent-$O(4)$,
$O(5)$, $O(6)$ and $O(7)$ symmetries are expected to be found in the systems
with two flavors ($N_{f}=2$) of four-component fermions. These results hold in
any higher-loops calculations. Moreover, the emergent-$O(2)$ ($Z_{2}\times
Z_{2}$) and $O(3)$ ($Z_{2}\times O(2)$roy3 ) symmetries are also compatible
with the constraint.

(iii) In the chiral emergent-$O(4)$ universality class, there is a
supersymmetric critical point, with the anomalous dimensions
$\eta_{\phi}=\eta_{\psi}=\epsilon$. The supersymmetry is expected to be found
in the system with fermion flavor $N_{f}=1$.

Our result has close connections with the recent study on the quantum
multicritical point that possesses enlarged emergent-$O(N)$ symmetry in Dirac
systems. Refs.[janssen2, ] and [roy3, ] pointed that the multicritical point
between the distinct $O(s_{1})$ and $O(s_{2})$ symmetry broken phases is
generically characterized by an emergent $O(s_{1}+s_{2})$ symmetry. For the
conventional broken symmetries in Dirac systems, e.g., $Z_{2}$, $O(2)$ and
$O(3)$, our result agrees well with Refs.[janssen2, ] and [roy3, ]. In
addition, our result also suggests some rich emergent symmetries for fermionic
criticality in graphene-like systems, for instance, the deconfined transition
with emergent $Z_{2}\times Z_{2}\times O(2)$, $Z_{2}\times O(2)\times O(2)$
and $O(2)\times O(2)\times O(3)$ symmetries and so on. This conjecture
deserves further investigation in future. Importantly, our result is appliable
to the recently observed emergent-$O(4)$ symmetrysato1 , the deconfined
transition between $SO(3)$-semimetal and $U(1)$-insulatorlzh1 , or the
transition between antiferromagnetism and valence-bond-solid in quantum Monte
Calor simulations of a designed synthetic Dirac systemslzx5 .

Finally, the new emergent-$O(4)$ supersymmetric critical point is interesting
on its own. In the future, we expect such supersymmetric critical point could
be cross-checked by other methods, i.e., conformal bootstrap approach and
quantum Monte Carlo simulationlzx3 .

###### Acknowledgements.

We acknowledge the support from the startup grant under No.20175788 in Guizhou
University.

##  Appendix A Renormalization constants at one-loop order

This appendix devotes to calculate the renormalization constants. At one-loop
order, the renormalization constants are defined as $Z_{X}=1+\delta_{X}$,
these $\delta_{X}$s are known as counterterms which are used to absorb the
divergencies in the 1PI diagram. Therefore, we need to calculate all the
divergence of the 1PI diagrams in Fig. 1.

###  A.1 Vertex tensor product

Before the calculation, let us derive the vertex product contributing to the
effective quartic boson interaction. The general vertex tensor can be
represented as the following symmetrized form:
$\lambda_{ijkl}=\lambda_{1}T^{(1)}_{ijkl}+\lambda_{2}T^{(2)}_{ijkl}$, with

$\displaystyle
T^{(1)}_{ijkl}=\frac{1}{3}(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{kl}+\delta_{il}\delta_{jk}),$
(47) $\displaystyle
T^{(2)}_{ijkl}=\delta_{ijkl}=\left\\{\begin{array}[]{cc}1,&i=j=k=l,\\\
0,&\text{otherwise}.\end{array}\right.$ (50)

Here, $\delta_{ijkl}$ satisfies $\sum_{k}\delta_{ijkk}=\delta_{ij}$,
$\sum_{i}\delta_{ii}=N$. The leading order 1PI diagram for quartic boson
interaction need to calculate the tensor product:
$\lambda_{ijmn}\lambda_{mnkl}$. For $T^{(1)}T^{(1)}$, we have

$T^{(1)}_{ijmn}T^{(1)}_{mnkl}=\frac{1}{9}[(N+4)\delta_{ij}\delta_{kl}+2\delta_{ik}\delta_{jl}+2\delta_{il}\delta_{jk}].$
(51)

After symmetrization, we have the symmetrized result

$\displaystyle\left[T^{(1)}_{ijmn}T^{(1)}_{mnkl}\right]_{s}=\frac{N+8}{9}T^{(1)}_{ijkl}.$
(52)

Accordingly,

$\displaystyle\left[T^{(1)}_{ijmn}T^{(2)}_{mnkl}+T^{(2)}_{ijmn}T^{(1)}_{mnkl}\right]_{s}=\frac{2}{3}T^{(1)}_{ijkl}+\frac{4}{3}T^{(2)}_{ijkl},$
(53)
$\displaystyle\left[T^{(2)}_{ijmn}T^{(2)}_{mnkl}\right]_{s}=T^{(2)}_{ijkl}.$
(54)

Combining the above results together, the symmetrized tensor product is given
by

$\displaystyle\left[\lambda_{ijmn}\lambda_{mnkl}\right]_{s}=$
$\displaystyle\left[\frac{N+8}{9}\lambda^{2}_{1}+\frac{2\lambda_{1}\lambda_{2}}{3}\right]T^{(1)}_{ijkl}$
$\displaystyle+$
$\displaystyle\left[\lambda^{2}_{2}+\frac{4\lambda_{1}\lambda_{2}}{3}\right]T^{(2)}_{ijkl}.$
(55)

Finally, the product $T^{(1)}_{ijmn}\delta_{mn}=(N+2)\delta_{ij}/3$ will be
necessary for boson mass-squared renormalization factor.

###  A.2 Boson two-point function

The 1PI boson two-point function are given in Fig.1(a)-(c). Fig.1(a)
contributes to $Z_{\phi}$, (b) and (c) contribute to the mass-square
renormalization $Z_{m^{2}}$. Fig.1(a) gives

$\displaystyle\mathcal{A}_{1a}(p)$
$\displaystyle=(-1)(-g)^{2}N_{f}\int\frac{d^{d}k}{(2\pi)^{d}}\text{Tr}\left[\Sigma_{a}\frac{1}{i\not{k}}\Sigma_{b}\frac{1}{i(\not{k}+\not{p})}\right]$
$\displaystyle=g^{2}N_{f}D_{\Sigma}\delta_{ab}\int\frac{d^{d}k}{(2\pi)^{d}}\left[\frac{1}{\not{k}}\frac{1}{(\not{k}+\not{p})}\right].$
(56)

where the minus sign $(-1)$ arises from fermion loop. The integral is a
standard Feynman integral, introduce a Feynman parameter and perform the
elementary integral, then we have

$\displaystyle\mathcal{A}_{1a}(p)=-g^{2}N_{f}D_{\Sigma}\frac{K}{\epsilon}\delta_{ab}p^{2},$
(57)

where $D_{\Sigma}$ is the dimensions of $\Sigma_{a}$, $K=1/(4\pi)^{2}$ here
and after in this paper. To reach the final result, the commutating rule
$[\Sigma_{a},\gamma_{\mu}]=0$ has been used [see Eq.(5)]. We only extract the
divergences when $d\rightarrow 4$ since only the divergent part should cancel
out with the $Z$-factors, that is, these $Z$-factors depend only on the
divergences in the dimensional regularization and MS scheme.

For Fig.1(b) and (c), we have

$\displaystyle\mathcal{A}_{1b}+\mathcal{A}_{1c}$
$\displaystyle=\frac{1}{2}(-\lambda_{1}T^{(1)}_{ijkl}-\lambda_{2}T^{(2)}_{ijkl})\int\frac{d^{d}k}{(2\pi)^{d}}\frac{i\delta^{kl}}{k^{2}+m^{2}}$
$\displaystyle=\left[\frac{(N+2)}{3}\lambda_{1}+\lambda_{2}\right]m^{2}\frac{K}{\epsilon}\delta^{ij}.$
(58)

To reach the final result, we have used the useful integral in d-dimensional
Euclidean space:

$\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}_{E}+\Delta)^{n}}=\frac{1}{(4\pi)^{d/2}}\frac{\Gamma(2-d/2)}{\Gamma(n)}\frac{1}{\Delta^{2-d/2}}.$
(59)

In terms of the renormalization condition

$\mathcal{A}_{1a}+\mathcal{A}_{1b}+\mathcal{A}_{1c}-p^{2}\delta_{\phi}-(\delta_{\phi}+\delta_{m^{2}})m^{2}=0,$
(60)

we find

$\displaystyle Z_{\phi}$
$\displaystyle=1-g^{2}N_{f}D_{\Sigma}\frac{K}{\epsilon},$ (61) $\displaystyle
Z_{m^{2}}$
$\displaystyle=1+\left[\frac{(N+2)}{3}\lambda_{1}+\lambda_{2}+g^{2}N_{f}D_{\Sigma}\right]\frac{K}{\epsilon}.$
(62)

###  A.3 Fermion two-point function

Fig.1(d) shows the 1PI fermion two-point function, which is given by

$\displaystyle\mathcal{A}_{1d}(p)=(-g)^{2}\int\frac{d^{d}k}{(2\pi)^{d}}\Sigma_{a}\frac{-i\not{k}}{k^{2}}\Sigma_{b}\frac{\delta_{ab}}{(p-k)^{2}+m^{2}}.$

Using the feynman parameter integral:

$\frac{1}{k^{2}}\frac{1}{(p-k)^{2}+m^{2}}=\int^{1}_{0}dx\frac{1}{[(k+p(x-1))^{2}+\Delta]^{2}},$

where $\Delta=p^{2}x(1-x)+m^{2}(1-x)$, then shifting the integration variable,
$k\rightarrow k-p(x-1)$, applying the result in Eq.(59), one obtains

$\displaystyle\mathcal{A}_{1d}$
$\displaystyle=g^{2}N(-i)\int^{1}_{0}dx\int\frac{d^{d}k}{(2\pi)^{d}}\frac{-\not{p}(x-1)}{(k^{2}+\Delta)^{2}},$
$\displaystyle=-g^{2}N\frac{K}{\epsilon}i\not{p}.$ (63)

The divergence in Fig.1(d) should cancel out with $\delta_{\psi}$, the
renormalization condition is $-i\not{p}\delta_{\psi}+\mathcal{A}_{1d}(p)=0$,
we thus obtain

$Z_{\psi}=1-g^{2}N\frac{K}{\epsilon}.$ (64)

###  A.4 Yukawa vertex

To the one-loop, there is a single diagram for the 1PI Yukawa vortex, see
Fig.1(e). It can be calculated to give

$\displaystyle\mathcal{A}_{1d}(p,q)$
$\displaystyle=\int\frac{d^{d}k}{(2\pi)^{d}}\frac{\delta_{ac}}{(p-k)^{2}+m^{2}}\times$
$\displaystyle\left[(-g\Sigma_{a})\frac{1}{i(\not{q}+\not{k})}(-g\Sigma_{b})\frac{1}{i\not{k}}(-g\Sigma_{c})\right].$

The divergent part can be obtained by setting the external momentum $p$, $q$
to be zero. Making use of the general Feynman parameters:

$\frac{1}{AB^{n}}=\int_{0}^{1}dxdy\delta(x+y-1)\frac{ny^{n-1}}{(xA+yB)^{n+1}},$
(65)

we have

$\displaystyle\mathcal{A}_{1d}(0,0)$
$\displaystyle=g^{3}\Sigma_{b}(2-N)\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{k^{2}+m^{2}}\frac{1}{k^{2}}$
$\displaystyle=g^{3}(2-N)\Sigma_{b}\int_{0}^{1}dx\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}+xm^{2})^{2}}$
$\displaystyle=g^{3}2(2-N)\Sigma_{b}\frac{K}{\epsilon}.$ (66)

The divergences should cancel out with $Z_{g}Z_{\psi}\sqrt{Z_{\phi}}-1$, so,
it is sufficient to define the renormalization condition

$-g\Sigma_{b}(Z_{g}Z_{\psi}\sqrt{Z_{\phi}}-1)+\mathcal{A}_{1d}(0,0)=0.$ (67)

Inserting $Z_{\psi},Z_{\phi}$ into it leads to

$Z_{g}=1+g^{2}[(4-N+N_{f}D_{\Sigma}/2)]\frac{K}{\epsilon}.$ (68)

###  A.5 $O(N)$-symmetric bosonic quartic-vertex

The relevant diagrams for $O(N)$ bosonic quartic-vertex are given in
Fig.1(f)-(h). Fig.1(f) and (g) contribute partly to the boson self-interaction
$\lambda_{1}$ [see Eq.(A.1)], the result can be evaluated to give

$\displaystyle\mathcal{A}^{(1)}_{1f}+\mathcal{A}^{(1)}_{1g}$
$\displaystyle=\frac{3}{2}[\lambda_{ijmn}\lambda_{mnkl}]^{(1)}_{s}\int\frac{d^{d}k}{(2\pi)^{d}}\left[\frac{1}{k^{2}+m^{2}}\right]^{2}$
$\displaystyle=3\left[\frac{N+8}{9}\lambda^{2}_{1}+\frac{2\lambda_{1}\lambda_{2}}{3}\right]T^{(1)}_{ijkl}\frac{K}{\epsilon},$
(69)

where the number $3$ counts $s$-channel, $t$-channel and $u$-channel in all.
We have set all external momentum to be zero to get the divergences.

For Fig.1(h) with the the external momentum $p=0$, we have

$\displaystyle\mathcal{A}^{(1)}_{1h}(0)$
$\displaystyle=-2N_{f}g^{4}\int\frac{d^{d}k}{(2\pi)^{d}}{\text{Tr}\left[\Sigma_{i}\frac{1}{i\not{k}}\Sigma_{j}\frac{1}{i\not{k}}\Sigma_{k}\frac{1}{i\not{k}}\Sigma_{l}\frac{1}{i\not{k}}\right]}$
$\displaystyle=-2N_{f}g^{4}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{k^{4}}\text{Tr}[\Sigma_{i}\Sigma_{j}\Sigma_{k}\Sigma_{k}],$
(70)

where Tr[…] in the first line denotes the trace over the sigma matrix, the
number $2$ in the first line counts the exchange of $i$ and $j$ (or
equivalently, $k$ and $l$). Using the identity

$\text{Tr}[\Sigma_{i}\Sigma_{j}\Sigma_{k}\Sigma_{k}]=D_{\Sigma}(\delta_{ij}\delta_{kl}-\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jl}),$
(71)

and performing the $k$ integration, we obtain

$\displaystyle\mathcal{A}^{(1)}_{1h}(0)=-12N_{f}D_{\Sigma}g^{4}\frac{K}{\epsilon}T^{(1)}_{ijkl}.$
(72)

To extract the renormalization constant $Z_{\lambda_{1}}$, we define the
renormalization condition

$-\lambda_{1}(Z_{\lambda_{1}}Z^{2}_{\phi}-1)T^{(1)}_{ijkl}+\mathcal{A}^{(1)}_{1f}+\mathcal{A}^{(1)}_{1g}+\mathcal{A}^{(1)}_{1h}(0)=0,$
(73)

Inserting $Z_{\phi}$ into this condition leads to

$\displaystyle Z_{\lambda_{1}}$
$\displaystyle=1+[\frac{N+8}{3}\lambda_{1}+2\lambda_{2}-12D_{\Sigma}N_{f}g^{4}/\lambda_{1}$
$\displaystyle+2N_{f}D_{\Sigma}g^{2}]\frac{K}{\epsilon}.$ (74)

###  A.6 $O(N)$-anisotropy

The diagrams for $O(N)$-anisotropy are given in Fig.1(i) and (j), and the
divergence can be computed to give

$\displaystyle\mathcal{A}^{(2)}_{1i}+\mathcal{A}^{(2)}_{1j}$
$\displaystyle=\frac{3}{2}[\lambda_{ijmn}\lambda_{mnkl}]^{(2)}_{s}\int\frac{d^{d}k}{(2\pi)^{d}}\left[\frac{1}{k^{2}+m^{2}}\right]^{2}$
$\displaystyle=3\left[\lambda^{2}_{2}+\frac{4\lambda_{1}\lambda_{2}}{3}\right]T^{(2)}_{ijkl}\frac{K}{\epsilon}.$
(75)

Here, to reach the final result, we have used the symmetrized tensor product
Eq.(A.1). Defining the renormalization condition

$-\lambda_{2}(Z_{\lambda_{2}}Z^{2}_{\phi}-1)T^{(2)}_{ijkl}+\mathcal{A}^{(2)}_{1i}+\mathcal{A}^{(2)}_{1j}=0,$
(76)

and inserting $Z_{\phi}$ into it, we have

$\displaystyle Z_{\lambda_{2}}$
$\displaystyle=1+[3\lambda_{2}+4\lambda_{1}+2N_{f}D_{\Sigma}g^{2}]\frac{K}{\epsilon}.$
(77)

Making use of the sigma matrix with dimensions four but without asking their
explicit representation, we obtain all the renormalization constants of the
main text.

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