# Correlated interaction effects in three-dimensional semi-Dirac semimetal

Jing-Rong Wang  High Magnetic Field Laboratory of Anhui Province, Anhui
Province Key Laboratory of Condensed Matter Physics at Extreme Conditions,
Chinese Academy of Sciences, Hefei 230031, China  Wei Li  Corresponding
author<EMAIL_ADDRESS>Key Laboratory of Materials Physics,
Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031,
China  Chang-Jin Zhang  Corresponding author<EMAIL_ADDRESS>High
Magnetic Field Laboratory of Anhui Province, Anhui Province Key Laboratory of
Condensed Matter Physics at Extreme Conditions, Chinese Academy of Sciences,
Hefei 230031, China  Institute of Physical Science and Information Technology,
Anhui University, Hefei 230601, China

###### Abstract

Understanding the correlation effects in unconventional topological materials,
in which the fermion excitations take unusual dispersion, is an important
topic in recent condensed matter physics. We study the influence of short-
range four-fermion interactions on three-dimensional semi-Dirac semimetal with
an unusual fermion dispersion, that is linear along two directions but
quadratic along the third one. Based on renormalization group theory, we find
all of 11 unstable fixed points including 5 quantum critical points, 5
bicritical points, and one tricritical point. The physical essences of the
quantum critical points are determined by analyzing the susceptibility
exponents for all of the source terms in particle-hole and particle-particle
channels. We also verify phase diagram of the system in the parameter space
carefully through numerically studying the flows of the four-fermion coupling
parameters and behaviors of the susceptibility exponents. These results are
helpful for us to understand the physical properties of candidate materials
for three-dimensional semi-Dirac semimetal such as ZrTe5.

##  I Introduction

The past 15 years have witnessed that study about topological materials
becomes one of the most important fields in condensed matter physics Vafek14 ;
Wehling14 ; Yan17 ; Hasan17 ; Armitage18 ; Kruthoff17 ; Tang19Wan ;
Zhang19FangChen ; VergnioryWangZ19 . Topological materials have wide potential
implications as electronic devices due to their some fascinating physical
properties. In some topological materials, such as Dirac semimetal (DSM)
including Cd3As2 and Na3Bi, and Weyl semimetal (WSM) including TaAs, TaP,
NbAs, and NbP, the low-energy fermion excitations are Dirac fermions or Weyl
fermions which resemble the elementary particles in high energy physics. Thus,
these materials provide a platform to verify some important concepts in high
energy physics.

Besides Dirac and Weyl fermions, there could be unconventional fermions with
unusual dispersion in topological materials. In double- (triple-) WSM, the
fermion dispersion is quadratic (cubic) along two directions but linear along
the third one Xu11 ; Fang12 . At the topological quantum critical point (QCP)
between DSM or WSM and band insulator, the fermion dispersion is mixture of
linear and quadratic ones Dietl08 ; Yang14 . Higher spin fermions with
multiband crossing have also attracted a lot of interest recently Bradyln16 ;
Tang17 ; Zhang18 . Spin-1 chiral fermions characterized by combination of a
Dirac-like band and a flat band with three-bands crossing, and spin-3/2 chiral
fermions displaying a birefringent spectrum with two distinct Fermi velocities
have been observed recently Takane19 ; Rao19 ; Sanchez19 ; Schroter19 ;
Schroter20 .

The correlation effects in Dirac and Weyl fermion systems are extensively
studied, and are well understood relatively Kotov12 ; WangLiu12 ; Hofmann14 ;
Goswami11 ; Hosur12 ; Tang18 ; Leaw19 ; Herbut06 ; Herbut09 ; Maciejko14 ;
Roy16 ; Szabo21 . The influence of many-body interaction on unconventional
fermion systems also attracted much interest and is an important topic. There
have been studies about influence of long-range Coulomb interaction
YangNatPhys14 ; Abrikosov72 ; Isobe16 ; Cho16 ; WangLiuZhang17A ; Lai15 ;
Jian15 ; WangLiuZhang17B ; ZhangShiXin17 ; WangLiuZhang18 ; WangLiuZhang19 ;
Han19 ; ZhangSX18 ; Roy18Birefringent ; Kotov20 , short-range four-fermion
interaction Roy17B ; Roy18A ; WangJing18 ; Roy18B ; Boettcher20 , and quantum
fluctuation of order parameter Savary14 ; Uryszek19 ; Sur19 ; Uryszek20  on
some unconventional fermion systems. These studies reveled many novel
behaviors, such as various quantum phase transitions, non-Fermi liquid
behaviors, anisotropic screening effect _etc_. These studies also showed that
the correlation effects in unconventional fermion systems depend on the
fermion dispersion subtly.

However, there are still a lot of open questions about the correlation effects
in unconventional fermion systems. In this article, we pay attention to the
influence of short-range four-fermion interactions on three-dimensional (3D)
semi-DSM, in which the fermion dispersion is linear along two directions but
quadratic along the third one as shown in Fig. 1. 3D semi-DSM state could be
realized at the topological QCP between 3D DSM and band insulator Yang14 .

The rest of paper is structured as follows. The model is defined in Sec. II.
In Sec. III, we show the results based on renormalization group (RG) theory.
The results are summarized and some discussion is given in Sec. IV. The
detailed derivation of the RG equations and susceptibility exponents for the
source terms and careful numerical analysis are presented in Appendices.

Figure 1: Energy dispersion of fermions in 3D semi-DSM.

##  II Model

The free action for 3D semi-DSM can be written as

$\displaystyle
S_{0}=\int\frac{d\omega}{2\pi}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\bar{\Psi}(\omega,\mathbf{k})\left[i\omega\gamma_{0}+\mathcal{H}(\mathbf{k})\right]\Psi(\omega,\mathbf{k}),$
(1)

where the Hamiltonian density $\mathcal{H}(\mathbf{k})$ is given by

$\displaystyle\mathcal{H}(\mathbf{k})=iv\gamma_{1}k_{1}+ivk_{2}\gamma_{2}+iAk_{3}^{2}\gamma_{3},$
(2)

with $v$ and $A$ being model parameters. $\Psi$ is four component spinor, and
$\bar{\Psi}=\Psi^{{\dagger}}\gamma_{0}$. The gamma matrices are defined as
$\gamma_{0}=\tau_{2}\otimes\sigma_{0}$,
$\gamma_{1}=\tau_{3}\otimes\sigma_{1}$,
$\gamma_{2}=\tau_{3}\otimes\sigma_{2}$,
$\gamma_{3}=\tau_{3}\otimes\sigma_{3}$, and
$\gamma_{5}=\tau_{1}\otimes\sigma_{0}$, where $\tau_{1,2,3}$ and
$\sigma_{1,2,3}$ are Pauli matrices. The gamma matrices satisfy the
anticommutation relation
$\left\\{\gamma_{\mu},\gamma_{\nu}\right\\}=2\delta_{\mu\nu}$ for
$\mu,\nu=0,1,2,3,5$. The energy dispersion of fermions takes the form
$E(\mathbf{k})=\pm\sqrt{v^{2}k_{\bot}^{2}+A^{2}k_{3}^{4}}$ where
$k_{\bot}^{2}=k_{1}^{2}+k_{2}^{2}$. Density of states (DOS) is given by
$\rho(\omega)\propto\omega^{3/2}/(v\sqrt{A})$, which vanishes at the Fermi
level.

Due to the vanishing DOS, the weak four-fermion interaction is irrelevant in
3D semi-DSM. However, if the four-fermion interaction is strong enough, the
system could be driven to a new phase. As shown in the Appendices, there are
formally 12 kinds of four-fermion interactions. Whereas, due to the constraint
by Fierz identity, five of them are linearly independent. Here, we consider
the interacting Lagrangian as following

$\displaystyle\mathcal{L}_{int}$ $\displaystyle=$ $\displaystyle
g_{1}\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}+g_{2}\left(\bar{\Psi}\Psi\right)^{2}+g_{4}\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}$
(3)
$\displaystyle+g_{5}\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}+g_{3z}\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}.$

In this article, we study the influence of four-fermion interactions on 3D
semi-DSM through the RG method Shankar94 .

##  III RG results

As shown in the Appendices, we firstly calculate all of the corrections from
the one-loop Feynman diagrams, by employing a momentum shell
$b\Lambda<\sqrt{v^{2}k_{\bot}^{2}+A^{2}k_{3}^{4}}<\Lambda$, where
$b=e^{-\ell}$ with $\ell$ being the RG running parameter. Then, we consider
these corrections, and perform RG transformations to restore the original form
of the actions. Accordingly, we obtain the RG equations for $g_{a}$, which can
be formally written as

$\displaystyle\frac{dg_{a}}{d\ell}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}g_{a}+F_{a}\left(g_{1},g_{2},g_{4},g_{5},g_{3z}\right),$
(4)

where $a=1,2,4,5,3z$, The concrete expressions of $F_{a}$ can be found in the
Appendices.

Table 1:  There are 5 QCPs among the 11 non-trivial unstable fixed points. The corresponding order parameters for the five QCPs are shown in second rows.  | FP1 | FP2 | FP3 | FP4 | FP5
---|---|---|---|---|---
Order parameter | $\Delta_{2}$ | $\Delta_{5}$ | $\Delta_{2}/\Delta_{5}$ | $\Delta_{7z}$ | $\Delta_{8z}$

Solving the equations

$\displaystyle\left.\frac{dg_{a}}{d\ell}\right|_{(g_{1},g_{2},g_{4},g_{5},g_{3z})=(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*})}=0,$
(5)

we get 12 fixed points, including the trivial Gauss fixed point
$(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*})=(0,0,0,0,0)$ and 11 non-
trivial fixed points

$\displaystyle\mathrm{FP}i:\quad(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*})=(g_{1,i}^{*},g_{2,i}^{*},g_{4,i}^{*},g_{5,i}^{*},g_{3z,i}^{*}),$
(6)

with $i=1,2,...,11$.

Expanding the RG equations of $g_{a}$ in the vicinity of a fixed point, we
obtain

$\displaystyle\frac{d\delta g_{a}}{d\ell}$ $\displaystyle=$
$\displaystyle\sum_{b}M_{ab}\delta g_{b},$ (7)

where $\delta g_{a}=g_{a}-g_{a}^{*}$. $M$ is five dimension square matrix, and
the matrix elements are expressions of $g_{1}^{*}$, $g_{2}^{*}$, $g_{4}^{*}$,
$g_{5}^{*}$, $g_{3z}^{*}$. From eigenvalues of $M$ at a fixed point
$(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*})$, we can get the
properties of the fixed point. A negative (positive) eigenvalue is
corresponding to a stable (unstable) eigendirection Maciejko14 ; Szabo21 .
There is one unstable direction for QCP, and there are two and three unstable
directions for bicritical point (BCP) and tricritical point (TCP)
respectively. Substituting the values of $g_{a}^{*}$ at each fixed point into
the expression of $M$, we calculate the corresponding eigenvalues of $M$. We
find that FP1, FP2, FP3, FP4, and FP5 are QCPs, FP6, FP7, FP8, FP9, and FP10
are BCPs, and FP11 is a TCP.

For a QCP, the correlation length exponent $\nu$ is determined by the inverse
of the corresponding positive eigenvalue of $M$. For the five QCPs, $\nu$
always satisfies

$\displaystyle\nu^{-1}=1.5.$ (8)

Figure 2: (a)-(e): Flows of $g_{1}$, $g_{2}$, $g_{4}$, $g_{5}$, and $g_{3z}$
with different initial conditions. $g_{1,0}=0$, $g_{4,0}=0$, $g_{5,0}=0$, and
$g_{3z,0}=0$ are taken.

In order to determine the physical essences of the QCPs, we analyze the RG
flows of all the fermion bilinear source terms in particle-hole and particle-
particle channels. The source terms in particle-hole channel can be generally
written as

$\displaystyle
S_{s}=\Delta_{X}\int\frac{d\omega}{2\pi}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\bar{\Psi}(\omega,\mathbf{k})\Gamma_{X}\Psi\left(\omega,\mathbf{k}\right).$
(9)

There are 12 choices for the matrix $\Gamma_{X}$, which corresponds to 12
different order parameters in particle-hole channel. The source terms in
particle-particle channel take the general form

$\displaystyle
S_{s}=\Delta_{Y}\int\frac{d\omega}{2\pi}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\Psi^{{\dagger}}(\omega,\mathbf{k})\Gamma_{Y}\Psi^{*}\left(\omega,\mathbf{k}\right).$
(10)

There are $6$ choices for the matrix $\Gamma_{Y}$, which is corresponding to 6
different superconducting pairings.

Firstly, we calculate the one-loop order corrections to the source terms as
shown in Eqs. (9) and (10) induced by the four-fermion interactions as shown
in Eq. (3). Then, we include these corrections and perform RG transformations
to restore the original form of the source terms. Accordingly, through the RG
transformations, we obtain the equations

$\displaystyle\bar{\beta}_{X,Y}=H_{X,Y}\left(g_{1},g_{2},g_{4},g_{5},g_{3z}\right),$
(11)

where

$\displaystyle\bar{\beta}_{X,Y}=\frac{d\ln\left(\Delta_{X,Y}\right)}{d\ell}-1.$
(12)

$\bar{\beta}_{X,Y}$ is usually termed as susceptibility exponent or anomalous
dimension for the fermion-bilinear source term. $H_{X,Y}$ are functions of
$g_{a}$ with $a=1,2,4,5,3z$. The concrete expressions of $H_{X,Y}$ are shown
in Appendices. For a QCP, substituting the values of $g_{a}$ at the QCP into
Eq. (11), and finding the largest one among all of $\bar{\beta}_{X,Y}$, we can
determine the physical meaning of the QCP.

Figure 3: Flows of $\bar{\beta}_{X,Y}$which approach positive infinity and
ratios between $\bar{\beta}_{X,Y}$. (a) and (b): $g_{1,0}=0.15$,
$g_{2,0}=1.3$, $g_{4,0}=0.46$, $g_{5,0}=-0.56$, and $g_{3z,0}=0.055$ are
taken; (c) and (d): $g_{2,0}=0.14$, $g_{2,0}=-0.59$, $g_{4,0}=0.42$,
$g_{5,0}=1.36$, and $g_{3z,0}=0.07$ are taken; (e) and (f): $g_{1,0}=1.5$,
$g_{2,0}=0$, $g_{3,0}=0$, $g_{5,0}=0.2$, and $g_{3z,0}=0$ are taken.

For FP1, $\bar{\beta}_{2}$ takes the largest value. It represents that this
fixed point is corresponding to the QCP to a state in which the order
parameter $\Delta_{2}=\langle\bar{\Psi}\Psi\rangle$ acquires finite value. The
physical meaning of $\Delta_{2}$ is scalar mass.

For FP2, $\bar{\beta}_{5}$ is the largest one. It means that this fixed point
stands for the QCP to a state in which the order parameter
$\Delta_{5}=\langle\bar{\Psi}i\gamma_{5}\Psi\rangle$ becomes finite. The
physical meaning of $\Delta_{5}$ corresponds to pseudoscalar mass.

For FP3, $\bar{\beta}_{2}$ and $\bar{\beta}_{5}$ are largest simultaneously.
It indicates that the fixed point corresponds to the QCP to a phase in which
both of $\Delta_{2}$ and $\Delta_{5}$ become finite. The parameter of this
phase can be written as
$\langle\bar{\Psi}\left(\cos(\theta)+i\gamma_{5}\sin(\theta)\right)\Psi\rangle$.
This phase represents an axionic insulator Roy16 .

For FP4, $\bar{\beta}_{7z}$ takes the largest value. It signifies that this
fixed point is corresponding to the QCP to a state in which the order
parameter $\Delta_{7z}=\langle\bar{\Psi}i\gamma_{5}\gamma_{3}\Psi\rangle$
becomes finite. $\Delta_{7z}$ stands for axial magnetization along $z$ axis.

For FP5, $\bar{\beta}_{8z}$ is the largest one. It suggests that this fixed
point represents the QCP to a state with finite order parameter
$\Delta_{8z}=\langle\bar{\Psi}i\gamma_{3}\Psi\rangle$ . The physical meaning
of $\Delta_{8z}$ is current along $z$ axis.

For convenience, we summarize the corresponding order parameters for the 5
QCPs in Table 1.

For general given initial conditions that is decided by
$(g_{1,0},g_{2,0},g_{4,0},g_{5,0},g_{3z,0})$, we determine the corresponding
phase through the flows of four-fermion coupling parameters $g_{a}$ and flows
of susceptibility exponents $\bar{\beta}_{X,Y}$. We show the flows of $g_{1}$,
$g_{2}$, $g_{4}$, $g_{5}$, and $g_{3z}$ under several initial conditions in
Fig. 2. If $g_{a}$ with $a=1,2,4,5,3z$ approach to zero, it represents that
the system is still in SM phase. If $|g_{a}|$ with $a=1,2,4,5,3z$ flow to
infinity at a finite running parameter $\ell_{c}$, the system becomes unstable
to a new phase.

Figure 4: Phase diagrams on the planes of two initial values of four-fermion
coupling strength. (a) $g_{2,0}$ and $g_{5,0}$; (b) $g_{2,0}$ and $g_{1,0}$;
(c) $g_{2,0}$ and $g_{4,0}$; (d) $g_{2,0}$ and $g_{3z,0}$; (e) $g_{5,0}$ and
$g_{1,0}$; (f) $g_{5,0}$ and $g_{4,0}$; (g) $g_{5,0}$ and $g_{3z,0}$; (h)
$g_{1,0}$ and $g_{4,0}$; (i) $g_{1,0}$ and $g_{3z,0}$; (j) $g_{4,0}$ and
$g_{3z,0}$; (k) $g_{5,0}$ and $g_{3z,0}$ In (a)-(j), the initial values of
rest four-fermion coupling parameters are taken as zero. For example,
$g_{1,0}=0$, $g_{4,0}=0$ and $g_{3z,0}=0$ are taken in (a). In (k),
$g_{1,0}=-2.3$, $g_{2,0}=0$, and $g_{4,0}=-0.62$ are taken.

In order to determine the physical essence of the new phase, we calculate the
flows of the susceptibility exponents $\bar{\beta}_{X,Y}$ and compare them.
For three general initial conditions, the flows of $\bar{\beta}_{X,Y}$ and the
ratio between them are presented in Figs. 3(a)-3(f). Here we only show the
susceptibility exponents that approach to positive infinity in Figs. 3(a),
3(c), and 3(e) respectively. For the initial condition corresponding to Figs.
3(a) and 3(b), we can find that $\bar{\beta_{2}}$ approaches to positive
infinity most quickly. It means that scalar mass $\Delta_{2}$ is generated in
the new phase. For the initial condition corresponding to Figs. 3(c) and 3(d),
$\bar{\beta_{5}}$ flows to positive infinity with the largest speed. It
indicates that pseudoscalar mass $\Delta_{5}$ becomes finite in the new phase.
For the initial condition corresponding to Figs. 3(e) and 3(f), we can find
that $\bar{\beta}_{2}$ and $\bar{\beta}_{5}$ approach to positive infinity
most quickly and $\bar{\beta}_{5}/\bar{\beta}_{2}\rightarrow 1$. It represents
that the system becomes to a new phase that $\Delta_{2}$ and $\Delta_{5}$
acquire finite value simultaneously. Namely, the system becomes to axionic
insulating phase.

The phase diagrams on the planes composed by initial values of two four-
fermion coupling parameters are shown in Fig. 4. Different phases are marked
by different colors. In Figs. 4(a)-4(j), we show all the 10 phase diagrams on
the planes composed by initial values of two coupling parameters chosen from
the five linearly independent coupling parameters. The initial values of rest
of three coupling parameters are taken as zero. Taken Fig. 4(a) composed by
$g_{2,0}$ and $g_{5,0}$ as an example, we can notice that there are five
phases, SM, insulator with scalar mass $\Delta_{2}$, insulator characterized
by pseudoscalar mass $\Delta_{5}$, axionic insulating phase, and a phase with
current along $z$ axis $\Delta_{8z}$. In Fig. 4(k), we present the phase
diagram composed by composed by $g_{5,0}$ and $g_{3z,0}$. In this phase
diagram, $g_{1,0}$, $g_{2,0}$ and $g_{4,0}$ are taken as proper values so that
the phase with axial magnetization along $z$ axis $\Delta_{7z}$ appears in the
phase diagram.

Recently, Sur and Roy studied the quantum critical behaviors in the vicinity
of possible QCPs in various SMs with different dispersions generally Sur19 .
According to their studies, in 3D semi-DSM, the Yukawa coupling between
quantum fluctuation of order parameter and fermion excitations becomes
irrelevant in the low energy regime. Thus, the fermions should take Fermi
liquid behaviors in the vicinity of a QCP between SM phase and a symmetry
breaking phase in 3D semi-DSM. Concretely, the residue of fermions $Z_{f}$
approaches to a finite constant value in the lowest energy limit, and the
Landau damping rate of fermions $\Gamma(\omega)$ satisfies

$\displaystyle\lim_{\omega\rightarrow
0}\frac{\Gamma(\omega)}{\omega}\rightarrow 0.$ (13)

Additionally, under the influence of quantum fluctuation of order parameter,
the observable quantities DOS $\rho$, specific heat $C_{v}$, compressibility
$\kappa$, optical conductivities within the $xy$ and along $z$ axis
$\sigma_{\bot\bot}$ and $\sigma_{zz}$ should respectively still take the
behaviors

$\displaystyle\rho(\omega)\sim\omega^{3/2},\qquad C_{v}(T)\sim
T^{5/2},\qquad\kappa(T)\sim T^{3/2},$
$\displaystyle\sigma_{\bot\bot}(\omega)\sim\omega^{1/2},\qquad\sigma_{zz}(\omega)\sim\omega^{3/2},$
(14)

which are qualitatively same as the ones for free fermions.

##  IV Summary

In summary, we perform comprehensive studies about the influence of four-
fermion interactions on 3D semi-DSM through RG theory. We find 11 unstable
fixed points and show that five fixed points are QCPs, five fixed points are
BCPs, and the rest one is a TCP. The physical essence of the QCPs are
determined carefully by analyzing the scalings of fermion bilinear source
terms. The phase diagrams for general initial conditions are also presented
through detailed numerical calculations of flows of four-fermion couplings and
susceptibility exponents.

According to the theoretical study by Yang and Nagaosa Yang14 , 3D semi-DSM
state can be realized at the topological QCP between 3D DSM and band
insulator. Through magneto-optics and magneto-transport, Yuan _et al._
observed the evidence of 3D semi-DSM phase in ZrTe5 Yuan17 . The subsequent
study about magnetotransport properties of ZrTe5 under hydrostatic pressure
also supports the existence of 3D semi-DSM phase ZhangJingLei17 . Recent
measurements of optical spectroscopy in ZrTe5 are also consistent with 3D
semi-DSM phase Martino19 ; SantoCottin20 . 3D semi-DSM state was also realized
in pressed Cd3As2 ZhangCengXiuFaXian17 . We expect our theoretical results are
helpful for understanding the physical properties of these candidate materials
of 3D semi-DSM.

## ACKNOWLEDGEMENTS

J.R.W. is grateful to Prof. G.-Z. Liu for the valuable discussions. We
acknowledge the support from the National Key R&D Program of China under
Grants 2016YFA0300404, the National Natural Science Foundation of China under
Grants 11974356, and U1832209, and the Collaborative Innovation Program of
Hefei Science Center CAS under Grant 2019HSC-CIP002. A portion of this work
was supported by the High Magnetic Field Laboratory of Anhui Province under
Grant AHHM-FX-2020-01.

##  Appendix A Fierz Identity

###  A.1 Fierz identity for 3D DSM

For 3D DSM, the interacting Lagrangian density can be written as Roy16

$\displaystyle\mathcal{L}_{\mathrm{int}}$ $\displaystyle=$ $\displaystyle
g_{1}\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}+g_{2}\left(\bar{\Psi}\Psi\right)^{2}+g_{3}\sum_{j=1}^{3}\left(\bar{\Psi}\gamma_{0}\gamma_{j}\Psi\right)^{2}+g_{4}\left(\bar{\Psi}\gamma_{0}\gamma_{5}\right)^{2}+g_{5}\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}+g_{6}\sum_{\left<lk\right>}\left(\bar{\Psi}i\gamma_{l}\gamma_{k}\Psi\right)^{2}$
(15)
$\displaystyle+g_{7}\sum_{j=1}^{3}\left(\bar{\Psi}i\gamma_{5}\gamma_{j}\Psi\right)^{2}+g_{8}\sum_{j=1}^{3}\left(\bar{\Psi}i\gamma_{j}\Psi\right)^{2},$

where

$\displaystyle\sum_{\left<lk\right>}\left(\bar{\Psi}i\gamma_{l}\gamma_{k}\Psi\right)^{2}$
$\displaystyle=$
$\displaystyle\left[\left(\bar{\Psi}i\gamma_{2}\gamma_{3}\Psi\right)^{2}+\left(\bar{\Psi}i\gamma_{3}\gamma_{1}\Psi\right)^{2}+\left(\bar{\Psi}i\gamma_{1}\gamma_{2}\Psi\right)^{2}\right].$
(16)

There are eight kinds of four-fermion couplings. However, not all of them are
linearly independent, due to the constraint by Fierz identity Roy16 ; Herbut09
; Szabo21 .

The Fierz identity indicates that Roy16 ; Herbut09 ; Szabo21

$\displaystyle\left[\bar{\Psi}(x)M\Psi(x)\right]\left[\bar{\Psi}(y)N\Psi(y)\right]=-\frac{1}{16}\mathrm{Tr}\left[M\Gamma_{a}N\Gamma_{b}\right]\left[\bar{\Psi}(x)\Gamma_{a}\Psi(y)\right]\left[\bar{\Psi}(y)\Gamma_{b}\Psi(x)\right].$
(17)

For local interaction, we have $x=y$. Thus,

$\displaystyle\left[\bar{\Psi}(x)M\Psi(x)\right]\left[\bar{\Psi}(x)N\Psi(x)\right]=-\frac{1}{16}\mathrm{Tr}\left[M\Gamma_{a}N\Gamma_{b}\right]\left[\bar{\Psi}(x)\Gamma_{a}\Psi(x)\right]\left[\bar{\Psi}(x)\Gamma_{b}\Psi(x)\right].$
(18)

Repeat of indexes $a$ and $b$ in Eqs. (17) and (18) represents summation.
Substituting each four-fermion coupling in Eq. (15) into Eq. (18), we could
get eight equations, which can be compactly expressed by

$\displaystyle FX=0,$ (19)

where

$\displaystyle
X=\left(\begin{array}[]{c}\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}\\\
\left(\bar{\Psi}\Psi\right)^{2}\\\
\sum\limits_{j=1}^{3}\left(\bar{\Psi}\gamma_{0}\gamma_{j}\Psi\right)^{2}\\\
\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}\\\
\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}\\\
\sum\limits_{<lk>}\left(\bar{\Psi}i\gamma_{l}\gamma_{k}\Psi\right)^{2}\\\
\sum\limits_{j=1}^{3}\left(\bar{\Psi}i\gamma_{5}\gamma_{j}\Psi\right)^{2}\\\
\sum\limits_{j=1}^{3}\left(\bar{\Psi}i\gamma_{j}\Psi\right)^{2}\end{array}\right),$
(28)

and

$\displaystyle F=\left(\begin{array}[]{rrrrrrrr}5&1&1&1&1&1&1&1\\\
1&5&-1&-1&-1&1&1&-1\\\ 3&-3&3&-3&3&1&-1&1\\\ 1&-1&-1&5&-1&-1&1&1\\\
1&-1&1&-1&5&-1&1&-1\\\ 3&3&1&-3&-3&3&-1&1\\\ 3&3&-1&3&3&-1&3&-1\\\
3&-3&1&3&-3&1&-1&3\end{array}\right).$ (37)

It is easy to verify that rank of $F$ is 4, namely

$\displaystyle\mathrm{Rank}(F)=4.$ (38)

Then, the number of linearly independent couplings is

$\displaystyle 8-\mathrm{Rank}(F)=4.$ (39)

For convenience, we take the four couplings

$\displaystyle\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2},\qquad\left(\bar{\Psi}\Psi\right)^{2},\qquad\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2},\qquad\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2},$
(40)

as linearly independent couplings. The other couplings

$\displaystyle\sum\limits_{j=1}^{3}\left(\bar{\Psi}\gamma_{0}\gamma_{j}\Psi\right)^{2},\qquad\sum\limits_{<lk>}\left(\bar{\Psi}i\gamma_{l}\gamma_{k}\Psi\right)^{2},\qquad\sum\limits_{j=1}^{3}\left(\bar{\Psi}i\gamma_{5}\gamma_{j}\Psi\right)^{2},\qquad\sum\limits_{j=1}^{3}\left(\bar{\Psi}i\gamma_{j}\Psi\right)^{2},$
(41)

can be expressed by the four independent couplings shown in Eq. (40). In order
to obtain the concrete expressions for other couplings, we define

$\displaystyle\tilde{X}=\left(\begin{array}[]{c}\sum\limits_{j=1}^{3}\left(\bar{\Psi}\gamma_{0}\gamma_{j}\Psi\right)^{2}\\\
\sum\limits_{<lk>}\left(\bar{\Psi}i\gamma_{l}\gamma_{k}\Psi\right)^{2}\\\
\sum\limits_{j=1}^{3}\left(\bar{\Psi}i\gamma_{5}\gamma_{j}\Psi\right)^{2}\\\
\sum\limits_{j=1}^{3}\left(\bar{\Psi}i\gamma_{j}\Psi\right)^{2}\\\
\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}\\\
\left(\bar{\Psi}\Psi\right)^{2}\\\
\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}\\\
\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}\end{array}\right).$ (50)

It is easy to find that

$\displaystyle\tilde{F}\tilde{X}=0,$ (51)

where

$\displaystyle\tilde{F}=\left(\begin{array}[]{rrrrrrrr}1&1&1&1&5&1&1&1\\\
-1&1&1&-1&1&5&-1&-1\\\ 3&1&-1&1&3&-3&-3&3\\\ -1&-1&1&1&1&-1&5&-1\\\
1&-1&1&-1&1&-1&-1&5\\\ 1&3&-1&1&3&3&-3&-3\\\ -1&-1&3&-1&3&3&3&3\\\
1&1&-1&3&3&-3&3&-3\end{array}\right).$ (60)

Performing a series of similarity transformations for $\tilde{F}$,

$\displaystyle\tilde{F}\rightarrow\tilde{F}^{\prime},$ (61)

we obtain

$\displaystyle\tilde{F}^{\prime}\tilde{X}=0,$ (62)

where

$\displaystyle\tilde{F}^{\prime}=\left(\begin{array}[]{rrrrrrrr}1&0&0&0&1&-1&-1&2\\\
0&1&0&0&1&2&-1&-1\\\ 0&0&1&0&2&1&1&1\\\ 0&0&0&1&1&-1&2&-1\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\end{array}\right).$ (71)

Eq. (62) can be equivalently expressed by

$\displaystyle\sum\limits_{j=1}^{3}\left(\bar{\Psi}\gamma_{0}\gamma_{j}\Psi\right)^{2}$
$\displaystyle=$
$\displaystyle-\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}+\left(\bar{\Psi}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}-2\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2},$
(72)
$\displaystyle\sum\limits_{<lk>}\left(\bar{\Psi}i\gamma_{l}\gamma_{k}\Psi\right)^{2}$
$\displaystyle=$
$\displaystyle-\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}-2\left(\bar{\Psi}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2},$
(73)
$\displaystyle\sum\limits_{j=1}^{3}\left(\bar{\Psi}i\gamma_{5}\gamma_{j}\Psi\right)^{2}$
$\displaystyle=$
$\displaystyle-2\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}-\left(\bar{\Psi}\Psi\right)^{2}-\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}-\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2},$
(74)
$\displaystyle\sum\limits_{j=1}^{3}\left(\bar{\Psi}i\gamma_{j}\Psi\right)^{2}$
$\displaystyle=$
$\displaystyle-\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}+\left(\bar{\Psi}\Psi\right)^{2}-2\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}.$
(75)

###  A.2 Fierz identity for 3D semi-DSM

For 3D semi-DSM, the interacting Lagrangian density is described by

$\displaystyle\mathcal{L}_{\mathrm{int}}$ $\displaystyle=$ $\displaystyle
g_{1}\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}+g_{2}\left(\bar{\Psi}\Psi\right)^{2}+g_{3\bot}\sum_{j=1}^{2}\left(\bar{\Psi}\gamma_{0}\gamma_{j}\Psi\right)^{2}+g_{3z}\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}+g_{4}\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}+g_{5}\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}$
(76)
$\displaystyle+g_{6\bot}\sum_{<<lk>>}\left(\bar{\Psi}i\gamma_{l}\gamma_{k}\Psi\right)^{2}+g_{6z}\left(\bar{\Psi}i\gamma_{1}\gamma_{2}\Psi\right)^{2}+g_{7\bot}\sum_{j=1}^{2}\left(\bar{\Psi}i\gamma_{5}\gamma_{j}\Psi\right)^{2}+g_{7z}\left(\bar{\Psi}i\gamma_{5}\gamma_{3}\Psi\right)^{2}$
$\displaystyle+g_{8\bot}\sum_{j=1}^{2}\left(\bar{\Psi}i\gamma_{j}\Psi\right)^{2}+g_{8z}\left(\bar{\Psi}i\gamma_{3}\Psi\right)^{2},$

where

$\displaystyle\sum_{<<lk>>}\left(\bar{\Psi}i\gamma_{l}\gamma_{k}\Psi\right)^{2}=\left[\left(\bar{\Psi}i\gamma_{2}\gamma_{3}\Psi\right)^{2}+\left(\bar{\Psi}i\gamma_{3}\gamma_{1}\Psi\right)^{2}\right].$
(77)

As shown in Eq. (15), there are 8 four-fermion couplings for 3D DSM. However,
we consider 12 kinds of four-fermion couplings as shown in Eq. (76) for 3D
semi-DSM, due to the anisotropy of the fermion dispersion.

Substituting each four-fermion coupling in Eq. (76) into Eq. (18), we could
get 12 equations, which can be compactly expressed by

$\displaystyle FX=0,$ (78)

where

$\displaystyle
X=\left(\begin{array}[]{c}\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}\\\
\left(\bar{\Psi}\Psi\right)^{2}\\\
\sum\limits_{j=1}^{2}\left(\bar{\Psi}\gamma_{0}\gamma_{j}\Psi\right)^{2}\\\
\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\\\
\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}\\\
\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}\\\
\sum\limits_{<<lk>>}\left(\bar{\Psi}i\gamma_{l}\gamma_{k}\Psi\right)^{2}\\\
\left(\bar{\Psi}i\gamma_{1}\gamma_{2}\Psi\right)^{2}\\\
\sum\limits_{j=1}^{2}\left(\bar{\Psi}i\gamma_{5}\gamma_{j}\Psi\right)^{2}\\\
\left(\bar{\Psi}i\gamma_{5}\gamma_{3}\Psi\right)^{2}\\\
\sum\limits_{j=1}^{2}\left(\bar{\Psi}i\gamma_{j}\Psi\right)^{2}\\\
\left(\bar{\Psi}i\gamma_{3}\Psi\right)^{2}\end{array}\right),$ (91)

and

$\displaystyle F=\left(\begin{array}[]{rrrrrrrrrrrr}5&1&1&1&1&1&1&1&1&1&1&1\\\
1&5&-1&-1&-1&-1&1&1&1&1&-1&-1\\\ 1&-1&2&-1&-1&1&0&1&0&-1&0&1\\\
1&-1&-1&5&-1&1&1&-1&-1&1&1&-1\\\ 1&-1&-1&-1&5&-1&-1&-1&1&1&1&1\\\
1&-1&1&1&-1&5&-1&-1&1&1&-1&-1\\\ 1&1&0&1&-1&-1&2&-1&0&-1&0&1\\\
1&1&1&-1&-1&-1&-1&5&-1&1&1&-1\\\ 1&1&0&-1&1&1&0&-1&2&-1&0&-1\\\
1&1&-1&1&1&1&-1&1&-1&5&-1&1\\\ 1&-1&0&1&1&-1&0&1&0&-1&2&-1\\\
1&-1&1&-1&1&-1&1&-1&-1&1&-1&5\end{array}\right).$ (104)

It is easy to find that

$\displaystyle\mathrm{Rank}(F)=7.$ (105)

Then, the number of linearly independent couplings is

$\displaystyle 12-\mathrm{Rank}(F)=5.$ (106)

For convenience, we take the five couplings

$\displaystyle\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2},\qquad\left(\bar{\Psi}\Psi\right)^{2},\qquad\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2},\qquad\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2},\qquad\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2},$
(107)

as linearly independent couplings. The other couplings

$\displaystyle\sum\limits_{j=1}^{2}\left(\bar{\Psi}\gamma_{0}\gamma_{j}\Psi\right)^{2},\qquad\sum\limits_{<<lk>>}\left(\bar{\Psi}i\gamma_{l}\gamma_{k}\Psi\right)^{2},\qquad\left(\bar{\Psi}i\gamma_{1}\gamma_{2}\Psi\right)^{2},\qquad\sum\limits_{j=1}^{2}\left(\bar{\Psi}i\gamma_{5}\gamma_{j}\Psi\right)^{2},\qquad\left(\bar{\Psi}i\gamma_{5}\gamma_{3}\Psi\right)^{2},$
$\displaystyle\sum\limits_{j=1}^{2}\left(\bar{\Psi}i\gamma_{j}\Psi\right)^{2},\qquad\left(\bar{\Psi}i\gamma_{3}\Psi\right)^{2},$
(108)

can be expressed by the five independent couplings shown in Eq. (107). In
order to obtain the concrete expressions for other couplings, we define

$\displaystyle\tilde{X}=\left(\begin{array}[]{c}\sum\limits_{j=1}^{2}\left(\bar{\Psi}\gamma_{0}\gamma_{j}\Psi\right)^{2}\\\
\sum\limits_{<<lk>>}\left(\bar{\Psi}i\gamma_{l}\gamma_{k}\Psi\right)^{2}\\\
\left(\bar{\Psi}i\gamma_{1}\gamma_{2}\Psi\right)^{2}\\\
\sum\limits_{j=1}^{2}\left(\bar{\Psi}i\gamma_{5}\gamma_{j}\Psi\right)^{2}\\\
\left(\bar{\Psi}i\gamma_{5}\gamma_{3}\Psi\right)^{2}\\\
\sum\limits_{j=1}^{2}\left(\bar{\Psi}i\gamma_{j}\Psi\right)^{2}\\\
\left(\bar{\Psi}i\gamma_{3}\Psi\right)^{2}\\\
\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}\\\
\left(\bar{\Psi}\Psi\right)^{2}\\\
\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}\\\
\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}\\\
\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\end{array}\right).$ (121)

It is easy to find that

$\displaystyle\tilde{F}\tilde{X}=0,$ (122)

where

$\displaystyle\tilde{F}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{rrrrrrrrrrrr}1&1&1&1&1&1&1&5&1&1&1&1\\\
-1&1&1&1&1&-1&-1&1&5&-1&-1&-1\\\ 2&0&1&0&-1&0&1&1&-1&-1&1&-1\\\
-1&1&-1&-1&1&1&-1&1&-1&-1&1&5\\\ -1&-1&-1&1&1&1&1&1&-1&5&-1&-1\\\
1&-1&-1&1&1&-1&-1&1&-1&-1&5&1\\\ 0&2&-1&0&-1&0&1&1&1&-1&-1&1\\\
1&-1&5&-1&1&1&-1&1&1&-1&-1&-1\\\ 0&0&-1&2&-1&0&-1&1&1&1&1&-1\\\
-1&-1&1&-1&5&-1&1&1&1&1&1&1\\\ 0&0&1&0&-1&2&-1&1&-1&1&-1&1\\\
1&1&-1&-1&1&-1&5&1&-1&1&-1&-1\end{array}\right).$ (135)

Carrying out a series of similarity transformations for $\tilde{F}$,

$\displaystyle\tilde{F}\rightarrow\tilde{F}^{\prime},$ (136)

we arrive

$\displaystyle\tilde{F}^{\prime}\tilde{X}=0,$ (137)

where

$\displaystyle\tilde{F}^{\prime}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{rrrrrrrrrrrr}1&0&0&0&0&0&0&1&-1&-1&2&1\\\
0&1&0&0&0&0&0&1&1&-1&0&1\\\ 0&0&1&0&0&0&0&0&1&0&-1&-1\\\
0&0&0&1&0&0&0&1&1&1&0&-1\\\ 0&0&0&0&1&0&0&1&0&0&1&1\\\
0&0&0&0&0&1&0&1&-1&1&0&1\\\ 0&0&0&0&0&0&1&0&0&1&-1&-1\\\
0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0\end{array}\right).$ (150)

Eq. (137) can be also written as

$\displaystyle\sum_{j=1}^{2}\left(\bar{\Psi}\gamma_{0}\gamma_{j}\Psi\right)^{2}$
$\displaystyle=$
$\displaystyle-\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}+\left(\bar{\Psi}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}-2\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}-\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2},$
(151)
$\displaystyle\sum\limits_{<<lk>>}\left(\bar{\Psi}i\gamma_{l}\gamma_{k}\Psi\right)^{2}$
$\displaystyle=$
$\displaystyle-\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}-\left(\bar{\Psi}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}-\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2},$
(152) $\displaystyle\left(\bar{\Psi}i\gamma_{1}\gamma_{2}\Psi\right)^{2}$
$\displaystyle=$
$\displaystyle-\left(\bar{\Psi}\Psi\right)^{2}+\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2},$
(153)
$\displaystyle\sum\limits_{j=1}^{2}\left(\bar{\Psi}i\gamma_{5}\gamma_{j}\Psi\right)^{2}$
$\displaystyle=$
$\displaystyle-\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}-\left(\bar{\Psi}\Psi\right)^{2}-\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2},$
(154) $\displaystyle\left(\bar{\Psi}i\gamma_{5}\gamma_{3}\Psi\right)^{2}$
$\displaystyle=$
$\displaystyle-\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}-\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}-\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2},$
(155)
$\displaystyle\sum\limits_{j=1}^{2}\left(\bar{\Psi}i\gamma_{j}\Psi\right)^{2}$
$\displaystyle=$
$\displaystyle-\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}+\left(\bar{\Psi}\Psi\right)^{2}-\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}-\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2},$
(156) $\displaystyle\left(\bar{\Psi}i\gamma_{3}\Psi\right)^{2}$
$\displaystyle=$
$\displaystyle-\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}.$
(157)

##  Appendix B Derivation of the RG equations for the strength of four-fermion
couplings

###  B.1 Self-energy of the fermions

The fermion propagator reads as

$\displaystyle G_{0}(i\omega,\mathbf{k})$ $\displaystyle=$
$\displaystyle-\frac{i\omega\gamma_{0}+iv\left(k_{1}\gamma_{1}+k_{2}\gamma_{2}\right)+iAk_{3}^{2}\gamma_{3}}{\omega^{2}+E_{\mathbf{k}}^{2}},$
(158)

where $E_{\mathbf{k}}=\sqrt{v^{2}k_{\bot}^{2}+A^{2}k_{3}^{4}}$ with
$k_{\bot}^{2}=k_{1}^{2}+k_{2}^{2}$. The self-energy of fermions resulting from
Fig. 5(a) takes the form

$\displaystyle\Sigma_{1}=\sum_{a}g_{a}\int\frac{d\omega}{2\pi}\int^{\prime}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\Gamma_{a}G_{0}(\omega,\mathbf{k})\Gamma_{a},$
(159)

where

$\displaystyle\sum_{a}\equiv\sum_{a=1,2,4,5,3z}.$ (160)

$\int^{\prime}$ represents that a momentum shell will be properly taken.
Figure 5(b) induces the self-energy of fermions as following

$\displaystyle\Sigma_{2}=\sum_{a}g_{a}\int\frac{d\omega}{2\pi}\int^{\prime}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\mathrm{Tr}\left[G_{0}(\omega,\mathbf{k})\Gamma_{a}\right].$
(161)

Substituting Eq. (158) into Eqs. (159) and (161), we obtain

$\displaystyle\Sigma_{1}$ $\displaystyle=$ $\displaystyle 0,$ (162)
$\displaystyle\Sigma_{2}$ $\displaystyle=$ $\displaystyle 0.$ (163)

It should be notice that a generated constant term in $\Sigma_{1}$ has been
discarded. The generated constant term in self-energy is also discarded in the
study about long-range Coulomb interaction in 3D semi-DSM Abrikosov72 .
According to Eqs. (162) and (163), the fermion propagator is not renormalized
by the four-fermion interactions to one-loop order.

Figure 5: Feynman diagrams for the self-energies of fermions induced by four-
fermion interactions. Solid line represents the fermion propagator, and wavy
line stands for the four-fermion interaction.

###  B.2 One-loop corrections for the four-fermion couplings

Figure 6: One-loop Feynman diagrams for the corrections to the four-fermion
couplings.

Fig. 6(a) leads to the correction

$\displaystyle V_{a}^{(1)}$ $\displaystyle=$
$\displaystyle-2g_{a}^{2}\left(\bar{\Psi}\Gamma_{a}\Psi\right)^{2}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\int^{\prime}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\mathrm{Tr}\left[\Gamma_{a}G_{0}(i\omega,\mathbf{k})\Gamma_{a}G_{0}(i\omega,\mathbf{k})\right].$
(164)

Fig. 6(b) results in the correction

$\displaystyle V_{a}^{(2)}=\sum_{b}V_{ab}^{(2)},$ (165)

where

$\displaystyle V_{a}^{(2)}$ $\displaystyle=$ $\displaystyle
4g_{a}g_{b}\left(\bar{\Psi}\Gamma_{a}\bar{\Psi}\right)\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\int^{\prime}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\left(\bar{\Psi}\Gamma_{b}G_{0}(i\omega,\mathbf{k})\Gamma_{a}G_{0}(i\omega,\mathbf{k})\Gamma_{b}\Psi\right).$
(166)

The Figs. 6(c) and 6(d) induce the correction

$\displaystyle V^{(3)+(4)}$ $\displaystyle=$ $\displaystyle\sum_{a}\sum_{a\leq
b}V_{ab}^{(3)+(4)},$ (167)

where

$\displaystyle
V_{ab}^{(3)+(4)}=4g_{a}g_{b}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\int^{\prime}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\left(\bar{\Psi}\Gamma_{a}G_{0}(i\omega,\mathbf{k})\Gamma_{b}\Psi\right)\bar{\Psi}\left[\Gamma_{b}G_{0}(i\omega,\mathbf{k})\Gamma_{a}+\Gamma_{a}G_{0}(-i\omega,-\mathbf{k})\Gamma_{b}\right]\Psi.$
(168)

Substituting Eq. (158) into Eq. (164), we obtain

$\displaystyle V_{a}^{(1)}$ $\displaystyle=$ $\displaystyle\delta
g_{a}^{(1)}\left(\bar{\Psi}\Gamma_{a}\Psi\right)^{2},$ (169)

where

$\displaystyle\delta g_{1}^{(1)}$ $\displaystyle=$ $\displaystyle 0,$ (170)
$\displaystyle\delta g_{2}^{(1)}$ $\displaystyle=$ $\displaystyle
g_{2}^{2}\frac{2\Lambda^{\frac{3}{2}}}{\pi^{2}v^{2}\sqrt{A}}\ell,$ (171)
$\displaystyle\delta g_{4}^{(1)}$ $\displaystyle=$ $\displaystyle 0,$ (172)
$\displaystyle\delta g_{5}^{(1)}$ $\displaystyle=$ $\displaystyle
g_{5}^{2}\frac{2\Lambda^{\frac{3}{2}}}{\pi^{2}v^{2}\sqrt{A}}\ell,$ (173)
$\displaystyle\delta g_{3z}^{(1)}$ $\displaystyle=$ $\displaystyle
g_{3z}^{2}\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell.$ (174)

Substituting Eq. (158) into Eqs. (165) and (166), we find that the
contribution from Fig. 6(b) can be written as

$\displaystyle V_{a}^{(2)}$ $\displaystyle=$ $\displaystyle\delta
g_{a}^{(2)}\left(\bar{\Psi}\Gamma_{a}\Psi\right)^{2},$ (175)

where

$\displaystyle\delta g_{1}^{(2)}$ $\displaystyle=$ $\displaystyle 0,$ (176)
$\displaystyle\delta g_{2}^{(2)}$ $\displaystyle=$
$\displaystyle\left(-g_{2}g_{1}-g_{2}^{2}+g_{2}g_{4}+g_{2}g_{5}+g_{2}g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{\pi^{2}v^{2}\sqrt{A}}\ell,$
(177) $\displaystyle\delta g_{4}^{(2)}$ $\displaystyle=$ $\displaystyle 0,$
(178) $\displaystyle\delta g_{5}^{(2)}$ $\displaystyle=$
$\displaystyle\left(-g_{5}g_{1}+g_{5}g_{2}+g_{5}g_{4}-g_{5}^{2}-g_{5}g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{\pi^{2}v^{2}\sqrt{A}}\ell,$
(179) $\displaystyle\delta g_{3z}^{(2)}$ $\displaystyle=$
$\displaystyle\left(-g_{3z}g_{1}+g_{3z}g_{2}+g_{3z}g_{4}-g_{3z}g_{5}-g_{3z}^{2}\right)\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell.$
(180)

Substituting Eq. (158) into Eq. (168), the contribution from Figs. 6(c) and
6(d) can be written as

$\displaystyle V_{1,1}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{1}^{2}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{3}\Psi\right)^{2},$
(181) $\displaystyle V_{2,2}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{2}^{2}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{3}\Psi\right)^{2},$
(182) $\displaystyle V_{4,4}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{4}^{2}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{3}\Psi\right)^{2},$
(183) $\displaystyle V_{5,5}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{5}^{2}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{3}\Psi\right)^{2},$
(184) $\displaystyle V_{3z,3z}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{3z}^{2}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{3}\Psi\right)^{2},$
(185) $\displaystyle V_{1,2}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{1}g_{2}\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\sum_{j=1}^{2}\left(\bar{\Psi}\gamma_{0}\gamma_{j}\Psi\right)^{2},$
(186) $\displaystyle V_{1,4}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{1}g_{4}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{5}\gamma_{3}\Psi\right)^{2},$
(187) $\displaystyle V_{1,5}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{1}g_{5}\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\sum_{<<lk>>}\left(\bar{\Psi}i\gamma_{l}\gamma_{k}\Psi\right)^{2},$
(188) $\displaystyle V_{1,3z}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle 0,$
(189) $\displaystyle V_{2,4}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle-
g_{2}g_{4}\frac{\Lambda^{\frac{3}{2}}}{\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}+g_{2}g_{4}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{1}\gamma_{2}\Psi\right)^{2},$
(190) $\displaystyle V_{2,5}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle-
g_{2}g_{5}\frac{\Lambda^{\frac{3}{2}}}{\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}+g_{2}g_{5}\sum_{j=1}^{2}\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{5}\gamma_{j}\Psi\right)^{2},$
(191) $\displaystyle V_{2,3z}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle-
g_{2}g_{3z}\frac{\Lambda^{\frac{3}{2}}}{\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{3}\Psi\right)^{2},$
(192) $\displaystyle V_{4,5}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle-
g_{4}g_{5}\frac{\Lambda^{\frac{3}{2}}}{\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}\Psi\right)^{2}+g_{4}g_{5}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2},$
(193) $\displaystyle V_{4,3z}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle-
g_{4}g_{3z}\frac{\Lambda^{\frac{3}{2}}}{\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{1}\gamma_{2}\Psi\right)^{2}+g_{4}g_{3z}\sum_{j=1}^{2}\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}\gamma_{0}\gamma_{j}\Psi\right)^{2}+g_{4}g_{3z}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2},$
(194) $\displaystyle V_{5,3z}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{5}g_{3z}\sum_{j=1}^{2}\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{j}\Psi\right)^{2}+g_{5}g_{3z}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}.$
(195)

Using the relations shown in Eqs. (151)-(157), we further get

$\displaystyle V_{1,1}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{0}^{2}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left[-\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\right],$
(196) $\displaystyle V_{2,2}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{2}^{2}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left[-\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\right],$
(197) $\displaystyle V_{4,4}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{4}^{2}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left[-\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\right],$
(198) $\displaystyle V_{5,5}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{5}^{2}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left[-\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\right],$
(199) $\displaystyle V_{3z,3z}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{3z}^{2}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left[-\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\right],$
(200) $\displaystyle V_{1,2}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{1}g_{2}\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left[-\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}+\left(\bar{\Psi}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}-2\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}-\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\right],$
(201) $\displaystyle V_{1,4}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{1}g_{4}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left[-\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}-\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}-\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\right],$
(202) $\displaystyle V_{1,5}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{1}g_{5}\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left[-\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}-\left(\bar{\Psi}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}-\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\right],$
(203) $\displaystyle V_{1,3z}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle 0,$
(204) $\displaystyle V_{2,4}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{2}g_{4}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left[-\left(\bar{\Psi}\Psi\right)^{2}-4\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\right],$
(205) $\displaystyle V_{2,5}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{2}g_{5}\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left[-\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}-\left(\bar{\Psi}\Psi\right)^{2}-\frac{7}{2}\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\right],$
(206) $\displaystyle V_{2,3z}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle-
g_{2}g_{3z}\frac{\Lambda^{\frac{3}{2}}}{\pi^{2}v^{2}\sqrt{A}}\ell\left[-\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\right],$
(207) $\displaystyle V_{4,5}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{4}g_{5}\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left[-5\left(\bar{\Psi}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\right]\,$
(208) $\displaystyle V_{4,3z}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{4}g_{3z}\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left[-\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}+\frac{7}{2}\left(\bar{\Psi}\Psi\right)^{2}+\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}-4\left(\bar{\Psi}i\gamma_{5}\Psi\right)^{2}-\frac{7}{2}\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\right],$
(209) $\displaystyle V_{5,3z}^{(3)+(4)}$ $\displaystyle=$ $\displaystyle
g_{5}g_{3z}\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left[-\left(\bar{\Psi}\gamma_{0}\Psi\right)^{2}+\left(\bar{\Psi}\Psi\right)^{2}-\frac{1}{2}\left(\bar{\Psi}\gamma_{0}\gamma_{5}\Psi\right)^{2}-\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right)^{2}\right].$
(210)

Thus, the contribution from Figs. 6(c) and 6(d) is given by

$\displaystyle V^{(3)+(4)}$ $\displaystyle=$
$\displaystyle\sum_{a=1,2,4,5,3z}\sum_{a\leq b}V_{ab}^{(3)+(4)}$ (211)
$\displaystyle=$ $\displaystyle\sum_{a=1,2,4,5,3z}\delta
g_{a}^{(3)+(4)}\left(\bar{\Psi}\Gamma_{a}\Psi\right)^{2},$

where

$\displaystyle\delta g_{1}^{(3)+(4)}$ $\displaystyle=$
$\displaystyle\left(-g_{1}g_{2}-\frac{1}{2}g_{1}g_{4}-g_{1}g_{5}-g_{2}g_{5}-g_{4}g_{3z}-g_{5}g_{3z}\right)\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell,$
(212) $\displaystyle\delta g_{2}^{(3)+(4)}$ $\displaystyle=$
$\displaystyle\left(g_{1}g_{2}-g_{1}g_{5}-\frac{1}{2}g_{2}g_{4}-g_{2}g_{5}-\frac{5}{2}g_{4}g_{5}+\frac{7}{2}g_{4}g_{3z}+g_{5}g_{3z}\right)\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell,$
(213) $\displaystyle\delta g_{4}^{(3)+(4)}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{2}g_{1}^{2}-\frac{1}{2}g_{2}^{2}-\frac{1}{2}g_{4}^{2}-\frac{1}{2}g_{5}^{2}-\frac{1}{2}g_{3z}^{2}+g_{1}g_{2}+g_{1}g_{5}-\frac{7}{2}g_{2}g_{5}+\frac{5}{2}g_{2}g_{3z}+g_{4}g_{3z}-\frac{1}{2}g_{5}g_{3z}\right)$
(214)
$\displaystyle\times\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell,$
$\displaystyle\delta g_{5}^{(3)+(4)}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}g_{1}^{2}+\frac{1}{2}g_{2}^{2}+\frac{1}{2}g_{4}^{2}+\frac{1}{2}g_{5}^{2}+\frac{1}{2}g_{3z}^{2}-2g_{1}g_{2}-\frac{1}{2}g_{1}g_{4}-2g_{2}g_{4}-\frac{5}{2}g_{2}g_{3z}-4g_{4}g_{3z}\right)\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell,$
(215) $\displaystyle\delta g_{3z}^{(3)+(4)}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}g_{1}^{2}+\frac{1}{2}g_{2}^{2}+\frac{1}{2}g_{4}^{2}+\frac{1}{2}g_{5}^{2}+\frac{1}{2}g_{3z}^{2}-g_{1}g_{2}-\frac{1}{2}g_{1}g_{4}-g_{1}g_{5}+\frac{1}{2}g_{2}g_{4}+g_{2}g_{5}-\frac{5}{2}g_{2}g_{3z}+\frac{1}{2}g_{4}g_{5}\right.$
(216)
$\displaystyle\left.-\frac{7}{2}g_{4}g_{3z}-g_{5}g_{3z}\right)\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell.$

From the above results, we obtain

$\displaystyle\delta g_{a}$ $\displaystyle=$ $\displaystyle\delta
g_{a}^{(1)}+\delta g_{a}^{(2)}+\delta g_{a}^{(3)+(4)}.$ (217)

Concretely,

$\displaystyle\delta g_{1}$ $\displaystyle=$
$\displaystyle\left(-g_{1}g_{2}-\frac{1}{2}g_{1}g_{4}-g_{1}g_{5}-g_{2}g_{5}-g_{4}g_{3z}-g_{5}g_{3z}\right)\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell,$
(218) $\displaystyle\delta g_{2}$ $\displaystyle=$
$\displaystyle\left(\frac{5}{2}g_{2}^{2}-\frac{3}{2}g_{1}g_{2}-g_{1}g_{5}+2g_{2}g_{4}+\frac{3}{2}g_{2}g_{5}+\frac{5}{2}g_{2}g_{3z}-\frac{5}{2}g_{4}g_{5}+\frac{7}{2}g_{4}g_{3z}+g_{5}g_{3z}\right)\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell,$
(219) $\displaystyle\delta g_{4}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{2}g_{1}^{2}-\frac{1}{2}g_{2}^{2}-\frac{1}{2}g_{4}^{2}-\frac{1}{2}g_{5}^{2}-\frac{1}{2}g_{3z}^{2}+g_{1}g_{2}+g_{1}g_{5}-\frac{7}{2}g_{2}g_{5}+\frac{5}{2}g_{2}g_{3z}+g_{4}g_{3z}-\frac{1}{2}g_{5}g_{3z}\right)\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell,$
(220) $\displaystyle\delta g_{5}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}g_{1}^{2}+\frac{1}{2}g_{2}^{2}+\frac{1}{2}g_{4}^{2}+3g_{5}^{2}+\frac{1}{2}g_{3z}^{2}-2g_{1}g_{2}-\frac{1}{2}g_{1}g_{4}-\frac{5}{2}g_{1}g_{5}-2g_{2}g_{4}+\frac{5}{2}g_{2}g_{5}-\frac{5}{2}g_{2}g_{3z}+\frac{5}{2}g_{4}g_{5}-4g_{4}g_{3z}\right.$
(221)
$\displaystyle\left.-\frac{5}{2}g_{5}g_{3z}\right)\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell,$
$\displaystyle\delta g_{3z}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}g_{1}^{2}+\frac{1}{2}g_{2}^{2}+\frac{1}{2}g_{4}^{2}+\frac{1}{2}g_{5}^{2}+g_{3z}^{2}-g_{1}g_{2}-\frac{1}{2}g_{1}g_{4}-g_{1}g_{5}-\frac{1}{2}g_{1}g_{3z}+\frac{1}{2}g_{2}g_{4}+g_{2}g_{5}-2g_{2}g_{3z}+\frac{1}{2}g_{4}g_{5}\right.$
(222)
$\displaystyle\left.-3g_{4}g_{3z}-\frac{3}{2}g_{5}g_{3z}\right)\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell.$

###  B.3 Scaling transformations

The free action of fermions is

$\displaystyle
S_{\Psi}=\int\frac{d\omega}{2\pi}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\bar{\Psi}(\omega,\mathbf{k})\left(i\omega\gamma_{0}+ivk_{1}\gamma_{1}+ivk_{2}\gamma_{2}+iAk_{3}^{2}\gamma_{3}\right)\Psi(\omega,\mathbf{k}).$
(223)

The fermion self-energy induced by four-fermion interactions to one-loop order
vanishes. Thus, the form of action $S_{\Psi}$ is not changed. Employing the
transformations

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\omega^{\prime}e^{-\ell},$ (224) $\displaystyle k_{1}$
$\displaystyle=$ $\displaystyle k_{1}^{\prime}e^{-\ell},$ (225) $\displaystyle
k_{2}$ $\displaystyle=$ $\displaystyle k_{2}^{\prime}e^{-\ell},$ (226)
$\displaystyle k_{3}$ $\displaystyle=$ $\displaystyle
k_{3}^{\prime}e^{-\frac{\ell}{2}},$ (227) $\displaystyle v$ $\displaystyle=$
$\displaystyle v^{\prime},$ (228) $\displaystyle A$ $\displaystyle=$
$\displaystyle A^{\prime},$ (229) $\displaystyle\Psi$ $\displaystyle=$
$\displaystyle\Psi^{\prime}e^{\frac{9}{4}\ell},$ (230)

the action becomes

$\displaystyle
S_{\Psi^{\prime}}=\int\frac{d\omega^{\prime}}{2\pi}\frac{d^{3}\mathbf{k}^{\prime}}{(2\pi)^{3}}\bar{\Psi}^{\prime}(\omega^{\prime},\mathbf{k}^{\prime})\left(i\omega^{\prime}\gamma_{0}+iv^{\prime}k_{1}^{\prime}\gamma_{1}+iv^{\prime}k_{2}^{\prime}\gamma_{2}+iA^{\prime}k_{3}^{\prime
2}\gamma_{3}\right)\Psi^{\prime}(\omega^{\prime},\mathbf{k}^{\prime}),$ (231)

which has the same form as the original action.

The original action of four-fermion interactions takes the form

$\displaystyle S_{\Psi^{4}}$ $\displaystyle=$
$\displaystyle\sum_{a=1,2,4,5,3z}g_{a}\int\frac{d\omega_{1}}{2\pi}\frac{d^{3}\mathbf{k}_{1}}{(2\pi)^{3}}\frac{d\omega_{2}}{2\pi}\frac{d^{3}\mathbf{k}_{2}}{(2\pi)^{3}}\frac{d\omega_{3}}{2\pi}\frac{d^{3}\mathbf{k}_{3}}{(2\pi)^{3}}\bar{\Psi}(\omega_{1},\mathbf{k}_{1})\Gamma_{a}\Psi(\omega_{2},\mathbf{k}_{2})\bar{\Psi}(\omega_{3},k_{3})\Gamma_{a}$
(232)
$\displaystyle\times\Psi(\omega_{1}-\omega_{2}+\omega_{3},\mathbf{k}_{1}-\mathbf{k}_{2}+\mathbf{k}_{3}).$

Including the one-loop order correction, the action becomes

$\displaystyle S_{\Psi^{4}}$ $\displaystyle=$
$\displaystyle\sum_{a=1,2,4,5,3z}\left(g_{a}+\delta
g_{a}\right)\int\frac{d\omega_{1}}{2\pi}\frac{d^{3}\mathbf{k}_{1}}{(2\pi)^{3}}\frac{d\omega_{2}}{2\pi}\frac{d^{3}\mathbf{k}_{2}}{(2\pi)^{3}}\frac{d\omega_{3}}{2\pi}\frac{d^{3}\mathbf{k}_{3}}{(2\pi)^{3}}\bar{\Psi}(\omega_{1},\mathbf{k}_{1})\Gamma_{a}\Psi(\omega_{2},\mathbf{k}_{2})\bar{\Psi}(\omega_{3},k_{3})\Gamma_{a}$
(233)
$\displaystyle\times\Psi(\omega_{1}-\omega_{2}+\omega_{3},\mathbf{k}_{1}-\mathbf{k}_{2}+\mathbf{k}_{3}).$

Utilizing the transformations Eqs. (224)-(227) and (230), we get

$\displaystyle S_{\Psi^{\prime 4}}$ $\displaystyle=$
$\displaystyle\sum_{a=1,2,4,5,3z}\left(g_{a}+\delta
g_{a}\right)e^{-\frac{3}{2}\ell}\int\frac{d\omega_{1}^{\prime}}{2\pi}\frac{d^{3}\mathbf{k}_{1}^{\prime}}{(2\pi)^{3}}\frac{d\omega_{2}^{\prime}}{2\pi}\frac{d^{3}\mathbf{k}_{2}^{\prime}}{(2\pi)^{3}}\frac{d\omega_{3}^{\prime}}{2\pi}\frac{d^{3}\mathbf{k}_{3}^{\prime}}{(2\pi)^{3}}\bar{\Psi}^{\prime}(\omega_{1}^{\prime},\mathbf{k}_{1}^{\prime})\Gamma_{a}\Psi^{\prime}(\omega_{2}^{\prime},\mathbf{k}_{2}^{\prime})\bar{\Psi}^{\prime}(\omega_{3}^{\prime},k_{3}^{\prime})\Gamma_{a}$
(234)
$\displaystyle\times\Psi^{\prime}(\omega_{1}^{\prime}-\omega_{2}^{\prime}+\omega_{3}^{\prime},\mathbf{k}_{1}^{\prime}-\mathbf{k}_{2}^{\prime}+\mathbf{k}_{3}^{\prime}).$

Let

$\displaystyle g_{a}^{\prime}$ $\displaystyle=$
$\displaystyle\left(g_{a}+\delta g_{a}\right)e^{-\frac{3}{2}\ell}\approx
g_{a}-\frac{3}{2}g_{a}\ell+\delta g_{a},$ (235)

we obtain

$\displaystyle S_{\Psi^{\prime 4}}$ $\displaystyle=$
$\displaystyle\sum_{a=1,2,4,5,3z}g_{a}^{\prime}\int\frac{d\omega_{1}^{\prime}}{2\pi}\frac{d^{3}\mathbf{k}_{1}^{\prime}}{(2\pi)^{3}}\frac{d\omega_{2}^{\prime}}{2\pi}\frac{d^{3}\mathbf{k}_{2}^{\prime}}{(2\pi)^{3}}\frac{d\omega_{3}^{\prime}}{2\pi}\frac{d^{3}\mathbf{k}_{3}^{\prime}}{(2\pi)^{3}}\bar{\Psi}^{\prime}(\omega_{1}^{\prime},\mathbf{k}_{1}^{\prime})\Gamma_{a}\Psi^{\prime}(\omega_{2}^{\prime},\mathbf{k}_{2}^{\prime})\bar{\Psi}^{\prime}(\omega_{3}^{\prime},k_{3}^{\prime})\Gamma_{a}$
$\displaystyle\times\Psi^{\prime}(\omega_{1}^{\prime}-\omega_{2}^{\prime}+\omega_{3}^{\prime},\mathbf{k}_{1}^{\prime}-\mathbf{k}_{2}^{\prime}+\mathbf{k}_{3}^{\prime})$

which recovers the original form of the action.

From Eq. (235), we get the RG equation for $g_{a}$ as following

$\displaystyle\frac{dg_{a}}{d\ell}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}g_{a}+\frac{d\delta g_{a}}{d\ell}.$ (236)

Substituting Eqs. (218)-(222) into Eq. (236), we find

$\displaystyle\frac{dg_{1}}{d\ell}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}g_{1}-\frac{2}{5}g_{1}\left(g_{2}+\frac{1}{2}g_{4}+g_{5}\right)-\frac{2}{5}\left(g_{2}g_{5}+g_{4}g_{3z}+g_{5}g_{3z}\right),$
(237) $\displaystyle\frac{dg_{2}}{d\ell}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}g_{2}+g_{2}^{2}+g_{2}\left(-\frac{3}{5}g_{1}+\frac{4}{5}g_{4}+\frac{3}{5}g_{5}+g_{3z}\right)-\frac{2}{5}g_{1}g_{5}+g_{4}\left(-g_{5}+\frac{7}{5}g_{3z}\right)+\frac{2}{5}g_{5}g_{3z},$
(238) $\displaystyle\frac{dg_{4}}{d\ell}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}g_{4}-\frac{1}{5}g_{4}^{2}-\frac{1}{5}\left(g_{1}^{2}+g_{2}^{2}+g_{5}^{2}+g_{3z}^{2}\right)+\frac{2}{5}g_{4}g_{3z}+\frac{2}{5}g_{1}\left(g_{2}+g_{5}\right)+g_{2}\left(-\frac{7}{5}g_{5}+g_{3z}\right)-\frac{1}{5}g_{5}g_{3z},$
(239) $\displaystyle\frac{dg_{5}}{d\ell}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}g_{5}+\frac{6}{5}g_{5}^{2}+\frac{1}{5}\left(g_{1}^{2}+g_{2}^{2}+g_{4}^{2}+g_{3z}^{2}\right)+g_{5}\left(-g_{1}+g_{2}+g_{4}-g_{3z}\right)-\frac{2}{5}g_{1}\left(2g_{2}+\frac{1}{2}g_{4}\right)-g_{2}\left(\frac{4}{5}g_{4}+g_{3z}\right)$
(240) $\displaystyle-\frac{8}{5}g_{4}g_{3z},$
$\displaystyle\frac{dg_{3z}}{d\ell}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}g_{3z}+\frac{2}{5}g_{3z}^{2}+\frac{1}{5}\left(g_{1}^{2}+g_{2}^{2}+g_{4}^{2}+g_{5}^{2}\right)-\frac{2}{5}g_{3z}\left(\frac{1}{2}g_{1}+2g_{2}+3g_{4}+\frac{3}{2}g_{5}\right)-\frac{2}{5}g_{1}\left(g_{2}+\frac{1}{2}g_{4}+g_{5}\right)$
(241)
$\displaystyle+\frac{2}{5}g_{2}\left(\frac{1}{2}g_{4}+g_{5}\right)+\frac{1}{5}g_{4}g_{5}.$

The redefinition

$\displaystyle\frac{\Lambda^{\frac{3}{2}}g_{a}}{\pi^{2}v^{2}\sqrt{A}}\rightarrow
g_{a},$ (242)

has been employed.

##  Appendix C Susceptibility of source terms

We consider the Lagrangian for the source terms as following

$\displaystyle\mathcal{L}_{s}$ $\displaystyle=$
$\displaystyle\Delta_{1}\bar{\Psi}\gamma_{0}\Psi+\Delta_{2}\bar{\Psi}\Psi+\Delta_{3\bot}\sum_{j=1}^{2}\bar{\Psi}\gamma_{0}\gamma_{j}\Psi+\Delta_{3z}\bar{\Psi}\gamma_{0}\gamma_{3}\Psi+\Delta_{4}\bar{\Psi}\gamma_{0}\gamma_{5}\Psi+\Delta_{5}\bar{\Psi}i\gamma_{5}\Psi+\Delta_{6\bot}\sum_{<<lk>>}\left(\bar{\Psi}i\gamma_{l}\gamma_{k}\Psi\right)$
(243)
$\displaystyle+\Delta_{6z}\bar{\Psi}i\gamma_{1}\gamma_{2}\Psi+\Delta_{7\bot}\sum_{j=1}^{2}\bar{\Psi}i\gamma_{5}\gamma_{j}\Psi+\Delta_{7z}\bar{\Psi}i\gamma_{5}\gamma_{3}\Psi+\Delta_{8\bot}\sum_{j=1}^{2}\bar{\Psi}i\gamma_{j}\Psi+\Delta_{8z}\bar{\Psi}i\gamma_{3}\Psi+\Delta_{S}\Psi^{{\dagger}}i\gamma_{0}\gamma_{5}\gamma_{2}\Psi^{*}$
$\displaystyle+\Delta_{op}\Psi^{{\dagger}}i\gamma_{0}\gamma_{2}\Psi^{*}+\Delta_{V,1}\Psi^{{\dagger}}\gamma_{3}\Psi^{*}+\Delta_{V,2}\Psi^{{\dagger}}i\gamma_{0}\gamma_{5}\Psi^{*}+\Delta_{V,3}\Psi^{{\dagger}}\gamma_{1}\Psi^{*}+\Delta_{V,0}\Psi^{{\dagger}}i\gamma_{0}\gamma_{2}\gamma_{3}\Psi^{*}.$

Figure 7: (a) and (b): One-loop Feynman diagrams for the corrections to the
source terms in particle-hole channel. (c): One-loop Feynman diagram for the
correction to the source terms in particle-particle channel.

###  C.1 One-loop order corrections for source terms in particle-hole channel

There are two one-loop Feynman diagrams lead to the correction for source
terms in particle-hole channel. The one-loop correction for the source term
$\Delta_{X}$ from Fig. 7(a) is given by

$\displaystyle
W_{\Delta_{X}}^{(1)}=-2\Delta_{X}g_{X}\left(\bar{\Psi}\Gamma_{X}\Psi\right)\sum_{a=1,2,4,5,3z}g_{a}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\int^{\prime}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\mathrm{Tr}\left[\Gamma_{X}G_{0}(i\omega,\mathbf{k})\Gamma_{a}G_{0}(i\omega,\mathbf{k})\right].$
(244)

The one-loop correction for the source term $\Delta_{X}$ resulting from Fig.
7(b) can be written as

$\displaystyle
W_{\Delta_{X}}^{(2)}=2\Delta_{X}\sum_{a=1,2,4,5,3z}g_{a}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\int^{\prime}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\left(\bar{\Psi}\Gamma_{a}G_{0}(i\omega,\mathbf{k})\Gamma_{X}G_{0}(i\omega,\mathbf{k})\Gamma_{a}\Psi\right).$
(245)

Substituting Eq. (158) into Eq. (244), we find

$\displaystyle W_{\Delta_{1}}^{(1)}$ $\displaystyle=$ $\displaystyle 0,$ (246)
$\displaystyle W_{\Delta_{2}}^{(1)}$ $\displaystyle=$
$\displaystyle\Delta_{2}g_{2}\frac{2\Lambda^{\frac{3}{2}}}{\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}\Psi\right),$
(247) $\displaystyle W_{\Delta_{3\bot}}^{(1)}$ $\displaystyle=$ $\displaystyle
0,$ (248) $\displaystyle W_{3z}^{(1)}$ $\displaystyle=$
$\displaystyle\Delta_{3z}g_{3z}\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right),$
(249) $\displaystyle W_{\Delta_{4}}^{(1)}$ $\displaystyle=$ $\displaystyle 0,$
(250) $\displaystyle W_{\Delta_{5}}^{(1)}$ $\displaystyle=$
$\displaystyle\Delta_{5}g_{5}\frac{2\Lambda^{\frac{3}{2}}}{\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{5}\Psi\right),$
(251) $\displaystyle W_{\Delta_{6\bot}}^{(1)}$ $\displaystyle=$ $\displaystyle
0,$ (252) $\displaystyle W_{\Delta_{6z}}^{(1)}$ $\displaystyle=$
$\displaystyle 0,$ (253) $\displaystyle W_{\Delta_{7\bot}}^{(1)}$
$\displaystyle=$ $\displaystyle 0,$ (254) $\displaystyle
W_{\Delta_{7z}}^{(1)}$ $\displaystyle=$ $\displaystyle 0,$ (255)
$\displaystyle W_{\Delta_{8\bot}}^{(1)}$ $\displaystyle=$ $\displaystyle 0,$
(256) $\displaystyle W_{\Delta_{8z}}^{(1)}$ $\displaystyle=$ $\displaystyle
0.$ (257)

Substituting Eq. (158) into Eq. (245), we obtain

$\displaystyle W_{\Delta_{1}}^{(2)}$ $\displaystyle=$ $\displaystyle 0,$ (258)
$\displaystyle W_{\Delta_{2}}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Delta_{2}\left(-g_{1}-g_{2}+g_{4}+g_{5}+g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}\Psi\right),$
(259) $\displaystyle W_{\Delta_{3\bot}}^{(2)}$ $\displaystyle=$
$\displaystyle\Delta_{3\bot}\left(-g_{1}+g_{2}+g_{4}-g_{5}+g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\sum_{j=1}^{2}\left(\bar{\Psi}\gamma_{0}\gamma_{j}\Psi\right),$
(260) $\displaystyle W_{\Delta_{3z}}^{(2)}$ $\displaystyle=$
$\displaystyle\Delta_{3z}\left(-g_{1}+g_{2}+g_{4}-g_{5}-g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{10\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}\gamma_{0}\gamma_{3}\Psi\right),$
(261) $\displaystyle W_{\Delta_{4}}^{(2)}$ $\displaystyle=$ $\displaystyle 0,$
(262) $\displaystyle W_{\Delta_{5}}^{(2)}$ $\displaystyle=$
$\displaystyle\Delta_{5}\left(-g_{1}+g_{2}+g_{4}-g_{5}-g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{2\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{5}\Psi\right),$
(263) $\displaystyle W_{\Delta_{6\bot}}^{(2)}$ $\displaystyle=$
$\displaystyle\Delta_{6\bot}\left(-g_{1}-g_{2}+g_{4}+g_{5}-g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\sum_{<<lk>>}\left(\bar{\Psi}i\gamma_{l}\gamma_{k}\Psi\right),$
(264) $\displaystyle W_{\Delta_{6z}}^{(2)}$ $\displaystyle=$
$\displaystyle\Delta_{6z}\left(-g_{1}-g_{2}+g_{4}+g_{5}+g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{10\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{1}\gamma_{2}\Psi\right),$
(265) $\displaystyle W_{\Delta_{7\bot}}^{(2)}$ $\displaystyle=$
$\displaystyle\Delta_{7\bot}\left(-g_{1}-g_{2}-g_{4}-g_{5}+g_{3z}\right)\frac{3\Lambda^{\frac{3}{2}}}{10\pi^{2}v^{2}\sqrt{A}}\ell\sum_{j=1}^{2}\left(\bar{\Psi}i\gamma_{5}\gamma_{j}\Psi\right),$
(266) $\displaystyle W_{\Delta_{7z}}^{(2)}$ $\displaystyle=$
$\displaystyle-\Delta_{7z}\left(g_{1}+g_{2}+g_{4}+g_{5}+g_{3z}\right)\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{5}\gamma_{3}\Psi\right),$
(267) $\displaystyle W_{\Delta_{8\bot}}^{(2)}$ $\displaystyle=$
$\displaystyle\Delta_{8\bot}\left(-g_{1}+g_{2}-g_{4}+g_{5}-g_{3z}\right)\frac{3\Lambda^{\frac{3}{2}}}{10\pi^{2}v^{2}\sqrt{A}}\ell\sum_{j=1}^{2}\left(\bar{\Psi}i\gamma_{j}\Psi\right),$
(268) $\displaystyle W_{\Delta_{8z}}^{(2)}$ $\displaystyle=$
$\displaystyle\Delta_{8z}\left(-g_{1}+g_{2}-g_{4}+g_{5}+g_{3z}\right)\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell\left(\bar{\Psi}i\gamma_{3}\Psi\right).$
(269)

From

$\displaystyle W_{\Delta_{X}}=W_{\Delta_{X}}^{(1)}+W_{\Delta_{X}}^{(2)},$
(270)

we arrive

$\displaystyle
W_{\Delta_{X}}=\delta\Delta_{X}\left(\bar{\Psi}\Gamma_{X}\Psi\right).$ (271)

The parameters $\delta\Delta_{X}$ are given by

$\displaystyle\delta\Delta_{1}$ $\displaystyle=$ $\displaystyle 0,$ (272)
$\displaystyle\delta\Delta_{2}$ $\displaystyle=$
$\displaystyle\Delta_{2}\left(-g_{1}+3g_{2}+g_{4}+g_{5}+g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{2\pi^{2}v^{2}\sqrt{A}}\ell,$
(273) $\displaystyle\delta\Delta_{3\bot}$ $\displaystyle=$
$\displaystyle\Delta_{3\bot}\left(-g_{1}+g_{2}+g_{4}-g_{5}+g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell,$
(274) $\displaystyle\delta\Delta_{3z}$ $\displaystyle=$
$\displaystyle\Delta_{3z}\left(-g_{1}+g_{2}+g_{4}-g_{5}+3g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{10\pi^{2}v^{2}\sqrt{A}}\ell,$
(275) $\displaystyle\delta\Delta_{4}$ $\displaystyle=$ $\displaystyle 0,$
(276) $\displaystyle\delta\Delta_{5}$ $\displaystyle=$
$\displaystyle\Delta_{5}\left(-g_{1}+g_{2}+g_{4}+3g_{5}-g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{2\pi^{2}v^{2}\sqrt{A}}\ell,$
(277) $\displaystyle\delta\Delta_{6\bot}$ $\displaystyle=$
$\displaystyle\Delta_{6\bot}\left(-g_{1}-g_{2}+g_{4}+g_{5}-g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell,$
(278) $\displaystyle\delta\Delta_{6z}$ $\displaystyle=$
$\displaystyle\Delta_{6z}\left(-g_{1}-g_{2}+g_{4}+g_{5}+g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{10\pi^{2}v^{2}\sqrt{A}}\ell,$
(279) $\displaystyle\delta\Delta_{7\bot}$ $\displaystyle=$
$\displaystyle\Delta_{7\bot}\left(-g_{1}-g_{2}-g_{4}-g_{5}+g_{3z}\right)\frac{3\Lambda^{\frac{3}{2}}}{10\pi^{2}v^{2}\sqrt{A}}\ell,$
(280) $\displaystyle\delta\Delta_{7z}$ $\displaystyle=$
$\displaystyle\Delta_{7z}\left(-g_{1}-g_{2}-g_{4}-g_{5}-g_{3z}\right)\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell,$
(281) $\displaystyle\delta\Delta_{8\bot}$ $\displaystyle=$
$\displaystyle\Delta_{8\bot}\left(-g_{1}+g_{2}-g_{4}+g_{5}-g_{3z}\right)\frac{3\Lambda^{\frac{3}{2}}}{10\pi^{2}v^{2}\sqrt{A}}\ell,$
(282) $\displaystyle\delta\Delta_{8z}$ $\displaystyle=$
$\displaystyle\Delta_{8z}\left(-g_{1}+g_{2}-g_{4}+g_{5}+g_{3z}\right)\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell.$
(283)

###  C.2 One-loop order correction for source terms in particle-particle
channel

In particle-particle channel, to one-loop order, there is one Feynman diagram
as shown in Fig. 7(c) resulting in the correction to source terms. The
correction can be expresses as

$\displaystyle
W_{\Delta_{Y}}=2\Delta_{Y}\sum_{a=1,2,4,5,3z}g_{a}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\int^{\prime}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\left(\Psi^{{\dagger}}\Gamma_{a}^{T}G_{0}^{T}(i\omega,\mathbf{k})\Gamma_{Y}G_{0}(-i\omega,-\mathbf{k})\Gamma_{a}\Psi^{*}\right),$
(284)

where $T$ represents transposition. Substituting Eq. (158) into Eq. (284), we
get

$\displaystyle
W_{\Delta_{Y}}=\delta\Delta_{Y}\left(\Psi^{{\dagger}}\Gamma_{Y}\Psi\right),$
(285)

where

$\displaystyle\delta\Delta_{S}$ $\displaystyle=$
$\displaystyle\Delta_{S}\left(g_{1}-g_{2}+g_{4}+g_{5}-g_{3z}\right)\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell,$
(286) $\displaystyle\delta\Delta_{op}$ $\displaystyle=$
$\displaystyle\Delta_{op}\left(g_{1}+g_{2}+g_{4}-g_{5}+g_{3z}\right)\frac{2\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell,$
(287) $\displaystyle\delta\Delta_{V,1}$ $\displaystyle=$
$\displaystyle\Delta_{V,1}\left(g_{1}+g_{2}-g_{4}+g_{5}+g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell,$
(288) $\displaystyle\delta\Delta_{V,2}$ $\displaystyle=$
$\displaystyle\Delta_{V,2}\left(g_{1}+g_{2}-g_{4}+g_{5}+g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{5\pi^{2}v^{2}\sqrt{A}}\ell,$
(289) $\displaystyle\delta\Delta_{V,3}$ $\displaystyle=$
$\displaystyle\Delta_{V,3}\left(g_{1}+g_{2}-g_{4}+g_{5}-g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{2\pi^{2}v^{2}\sqrt{A}}\ell,$
(290) $\displaystyle\delta\Delta_{V,0}$ $\displaystyle=$
$\displaystyle\Delta_{V,0}\left(g_{1}-g_{2}-g_{4}-g_{5}+g_{3z}\right)\frac{\Lambda^{\frac{3}{2}}}{20\pi^{2}v^{2}\sqrt{A}}\ell.$
(291)

###  C.3 Derivation of the RG equations for source terms

In particle-hole channel, the bare action for the source terms is

$\displaystyle
S_{s}=\Delta_{X}\int\frac{d\omega}{2\pi}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\bar{\Psi}(\omega,\mathbf{k})\Gamma_{X}\Psi(\omega,\mathbf{k}).$
(292)

Considering the one-loop order corrections, we obtain

$\displaystyle S_{s}$ $\displaystyle=$
$\displaystyle\left(\Delta_{X}+\delta\Delta_{X}\right)\int\frac{d\omega}{2\pi}\frac{d^{3}\mathbf{k}}{(2\pi)^{3}}\bar{\Psi}(\omega,\mathbf{k})\Gamma_{X}\Psi(\omega,\mathbf{k}).$
(293)

Using the transformations Eqs. (224)-227) and (230), we can get

$\displaystyle S_{s}$ $\displaystyle=$
$\displaystyle\left(\Delta_{X}+\delta\Delta_{X}\right)e^{\ell}\int\frac{d\omega^{\prime}}{2\pi}\frac{d^{3}\mathbf{k}^{\prime}}{(2\pi)^{3}}\bar{\Psi}^{\prime}(\omega^{\prime},\mathbf{k}^{\prime})\Gamma_{X}\Psi^{\prime}(\omega^{\prime},\mathbf{k}^{\prime})$
(294) $\displaystyle\approx$
$\displaystyle\left(\Delta_{X}+\Delta_{X}\ell+\delta\Delta_{X}\right)\int\frac{d\omega^{\prime}}{2\pi}\frac{d^{3}\mathbf{k}^{\prime}}{(2\pi)^{3}}\bar{\Psi}^{\prime}(\omega^{\prime},\mathbf{k}^{\prime})\Gamma_{X}\Psi^{\prime}(\omega^{\prime},\mathbf{k}^{\prime}).$

Let

$\displaystyle\Delta_{X}^{\prime}=\Delta_{X}+\Delta_{X}\ell+\delta\Delta_{X},$
(295)

the action can be further written as

$\displaystyle S_{s}$ $\displaystyle=$
$\displaystyle\Delta_{X}^{\prime}\int\frac{d\omega^{\prime}}{2\pi}\frac{d^{3}\mathbf{k}^{\prime}}{(2\pi)^{3}}\bar{\Psi}^{\prime}(\omega^{\prime},\mathbf{k}^{\prime})\Gamma_{X}\Psi^{\prime}(\omega^{\prime},\mathbf{k}^{\prime}),$
(296)

which recovers the form of the original action. We can easily find that the RG
equation for $\Delta_{X}$ is

$\displaystyle\frac{d\Delta_{X}}{d\ell}$ $\displaystyle=$
$\displaystyle\Delta_{X}+\frac{d\delta\Delta_{X}}{d\ell}.$ (297)

Performing similar rescaling transformations, we can get the RG equation for
source terms in particle-particle channel

$\displaystyle\frac{d\Delta_{Y}}{d\ell}$ $\displaystyle=$
$\displaystyle\Delta_{Y}+\frac{d\delta\Delta_{Y}}{d\ell}.$ (298)

Substituting Eqs. (272)-(283) into Eq. (297), and substituting Eqs.
(286)-(291) into Eq. (298), we get the RG equations

$\displaystyle\bar{\beta}_{1}$ $\displaystyle=$ $\displaystyle 0,$ (299)
$\displaystyle\bar{\beta}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-g_{1}+3g_{2}+g_{4}+g_{5}+g_{3z}\right),$ (300)
$\displaystyle\bar{\beta}_{3\bot}$ $\displaystyle=$
$\displaystyle\frac{1}{5}\left(-g_{1}+g_{2}+g_{4}-g_{5}+g_{3z}\right),$ (301)
$\displaystyle\bar{\beta}_{3z}$ $\displaystyle=$
$\displaystyle\frac{1}{10}\left(-g_{1}+g_{2}+g_{4}-g_{5}\right),$ (302)
$\displaystyle\bar{\beta}_{4}$ $\displaystyle=$ $\displaystyle 0,$ (303)
$\displaystyle\bar{\beta}_{5}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-g_{1}+g_{2}+g_{4}+3g_{5}-g_{3z}\right),$ (304)
$\displaystyle\bar{\beta}_{6\bot}$ $\displaystyle=$
$\displaystyle\frac{1}{5}\left(-g_{1}-g_{2}+g_{4}+g_{5}-g_{3z}\right),$ (305)
$\displaystyle\bar{\beta}_{6z}$ $\displaystyle=$
$\displaystyle\frac{1}{10}\left(-g_{1}-g_{2}+g_{4}+g_{5}+g_{3z}\right),$ (306)
$\displaystyle\bar{\beta}_{7\bot}$ $\displaystyle=$
$\displaystyle\frac{3}{10}\left(-g_{1}-g_{2}-g_{4}-g_{5}+g_{3z}\right),$ (307)
$\displaystyle\bar{\beta}_{7z}$ $\displaystyle=$
$\displaystyle\frac{2}{5}\left(-g_{1}-g_{2}-g_{4}-g_{5}-g_{3z}\right),$ (308)
$\displaystyle\bar{\beta}_{8\bot}$ $\displaystyle=$
$\displaystyle\frac{3}{10}\left(-g_{1}+g_{2}-g_{4}+g_{5}-g_{3z}\right),$ (309)
$\displaystyle\bar{\beta}_{8z}$ $\displaystyle=$
$\displaystyle\frac{2}{5}\left(-g_{1}+g_{2}-g_{4}+g_{5}+g_{3z}\right),$ (310)
$\displaystyle\bar{\beta}_{S}$ $\displaystyle=$
$\displaystyle\frac{2}{5}\left(g_{1}-g_{2}+g_{4}+g_{5}-g_{3z}\right),$ (311)
$\displaystyle\bar{\beta}_{op}$ $\displaystyle=$
$\displaystyle\frac{2}{5}\left(g_{1}+g_{2}+g_{4}-g_{5}+g_{3z}\right),$ (312)
$\displaystyle\bar{\beta}_{V,1}$ $\displaystyle=$
$\displaystyle\frac{1}{5}\left(g_{1}+g_{2}-g_{4}+g_{5}+g_{3z}\right),$ (313)
$\displaystyle\bar{\beta}_{V,2}$ $\displaystyle=$
$\displaystyle\frac{1}{5}\left(g_{1}+g_{2}-g_{4}+g_{5}+g_{3z}\right),$ (314)
$\displaystyle\bar{\beta}_{V,3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(g_{1}+g_{2}-g_{4}+g_{5}-g_{3z}\right),$ (315)
$\displaystyle\bar{\beta}_{V,0}$ $\displaystyle=$
$\displaystyle\frac{1}{20}\left(g_{1}-g_{2}-g_{4}-g_{5}+g_{3z}\right),$ (316)

where

$\displaystyle\bar{\beta}_{X,Y}=\frac{d\ln(\Delta_{X,Y})}{d\ell}-1.$ (317)

For convenience, we show the physical meaning of different order parameters
and corresponding fermion bilinear in Table 2

Table 2:  Physical meaning of different order parameters and the corresponding fermion bilinears.  Order Parameter | Fermion Bilinear | Physical meaning
---|---|---
$\Delta_{1}$ | $\bar{\Psi}\gamma_{0}\Psi$ | chemical potential
$\Delta_{2}$ | $\bar{\Psi}\Psi$ | scalar mass
$\Delta_{3\bot}$ | $\sum\limits_{j=1,2}\bar{\Psi}\gamma_{0}\gamma_{j}\Psi$ | spin-orbit coupling within $xy$ plane
$\Delta_{3z}$ | $\bar{\Psi}\gamma_{0}\gamma_{3}\Psi$ | spin-orbit coupling along $z$ axis
$\Delta_{4}$ | $\bar{\Psi}\gamma_{0}\gamma_{5}\Psi$ | axial chemical potential
$\Delta_{5}$ | $\bar{\Psi}i\gamma_{5}\Psi$ | pseudoscalar mass
$\Delta_{6\bot}$ | $\bar{\Psi}\left(i\gamma_{2}\gamma_{3}+\gamma_{3}\gamma_{1}\right)\Psi$ | magnetization within $xy$ plane
$\Delta_{6z}$ | $\bar{\Psi}i\gamma_{1}\gamma_{2}\Psi$ | magnetization along $z$ axis
$\Delta_{7\bot}$ | $\sum\limits_{j=1,2}\bar{\Psi}i\gamma_{5}\gamma_{j}\Psi$ | axial magnetization within $xy$ plane
$\Delta_{7z}$ | $\bar{\Psi}i\gamma_{5}\gamma_{3}\Psi$ | axial magnetization along $z$ axis
$\Delta_{8\bot}$ | $\sum\limits_{j=1,2}\bar{\Psi}i\gamma_{j}\Psi$ | current within $xy$ plane
$\Delta_{8z}$ | $\bar{\Psi}i\gamma_{3}\Psi$ | current along $z$ axis
$\Delta_{S}$ | $\Psi^{{\dagger}}i\gamma_{0}\gamma_{5}\gamma_{2}\Psi^{*}$ |  $s$-wave paring
$\Delta_{op}$ | $\Psi^{{\dagger}}i\gamma_{0}\gamma_{2}\Psi^{*}$ | odd-parity pairing
$\Delta_{V,1}$ | $\Psi^{{\dagger}}\gamma_{3}\Psi^{*}$ | vector pairing along $x$ axis
$\Delta_{V,2}$ | $\Psi^{{\dagger}}i\gamma_{0}\gamma_{5}\Psi^{*}$ | vector pairing along $y$ axis
$\Delta_{V,3}$ | $\Psi^{{\dagger}}\gamma_{1}\Psi^{*}$ | vector pairing along $z$ axis
$\Delta_{V,0}$ | $\Psi^{{\dagger}}i\gamma_{0}\gamma_{1}\gamma_{3}\Psi^{*}$ | temporal vector paring

##  Appendix D Numerical Results

###  D.1 Fixed points and their properties

Solving the RG equations for $g_{a}$ as shown in Eqs. (237)-(241), we obtained
the real roots as following

$\displaystyle\mathrm{FP0:}\qquad\left(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*}\right)$
$\displaystyle=$ $\displaystyle\left(0,0,0,0,0\right)$ (318)
$\displaystyle\mathrm{FP1:}\qquad\left(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*}\right)$
$\displaystyle=$
$\displaystyle\left(0.152019,1.25444,0.459247,-0.561711,0.0551435\right),$
(319)
$\displaystyle\mathrm{FP2:}\qquad\left(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*}\right)$
$\displaystyle=$
$\displaystyle\left(0.140905,-0.585585,0.418385,1.34668,0.06996\right),$ (320)
$\displaystyle\mathrm{FP3:}\qquad\left(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*}\right)$
$\displaystyle=$
$\displaystyle\left(-0.100015,0.575751,-0.61003,0.775675,0.199924\right),$
(321)
$\displaystyle\mathrm{FP4:}\qquad\left(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*}\right)$
$\displaystyle=$
$\displaystyle\left(-2.33263,0.,-0.610178,-1.72246,-1.72246\right),$ (322)
$\displaystyle\mathrm{FP5:}\qquad\left(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*}\right)$
$\displaystyle=$
$\displaystyle\left(0.126936,-0.463077,-0.854005,0.769245,1.23232\right),$
(323)
$\displaystyle\mathrm{FP6:}\qquad\left(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*}\right)$
$\displaystyle=$
$\displaystyle\left(0.0860014,1.37623,0.236822,-0.304132,0.0941005\right),$
(324)
$\displaystyle\mathrm{FP7:}\qquad\left(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*}\right)$
$\displaystyle=$
$\displaystyle\left(0.103947,-0.465334,0.29293,1.43995,0.0944817\right),$
(325)
$\displaystyle\mathrm{FP8:}\qquad\left(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*}\right)$
$\displaystyle=$
$\displaystyle\left(-3.33745,-1.22097,-1.28072,-1.32395,-0.102973\right),$
(326)
$\displaystyle\mathrm{FP9:}\qquad\left(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*}\right)$
$\displaystyle=$
$\displaystyle\left(-2.68181,1.06255,-1.97657,-1.35604,-2.41859\right),$ (327)
$\displaystyle\mathrm{FP10:}\qquad\left(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*}\right)$
$\displaystyle=$ $\displaystyle\left(0,0,-1.25,1.25,1.25\right),$ (328)
$\displaystyle\mathrm{FP11:}\qquad\left(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*}\right)$
$\displaystyle=$
$\displaystyle\left(-5.16737,0,-4.38982,-0.777544,-0.777544\right),$ (329)

$\mathrm{FP0}$ is the trivial Gauss fixed point.
$\mathrm{FP1}$-$\mathrm{FP11}$ are non-trivial fixed points.

Expanding the RG equations (237)-(241) in the vicinity of a fixed point
$\left(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*}\right)$, we find
that

$\displaystyle\frac{dG}{d\ell}$ $\displaystyle=$ $\displaystyle MG,$ (330)

where

$\displaystyle G=\left(\begin{array}[]{c}\delta g_{1}\\\ \delta g_{2}\\\
\delta g_{4}\\\ \delta g_{5}\\\ \delta g_{3z}\end{array}\right),$ (336)

with $\delta g_{a}=g_{a}-g_{a}^{*}$. The matrix $M$ is given by

$\displaystyle
M=\left(\begin{array}[]{ccccc}M_{11}&M_{12}&M_{13}&M_{14}&M_{15}\\\
M_{21}&M_{22}&M_{23}&M_{24}&M_{25}\\\ M_{31}&M_{32}&M_{33}&M_{34}&M_{35}\\\
M_{41}&M_{42}&M_{43}&M_{44}&M_{45}\\\
M_{51}&M_{52}&M_{53}&M_{54}&M_{55}\end{array}\right),$ (342)

where

$\displaystyle M_{11}$ $\displaystyle=$
$\displaystyle-\left(\frac{3}{2}+\frac{2}{5}g_{2}^{*}+\frac{1}{5}g_{4}^{*}+\frac{2}{5}g_{5}^{*}\right),$
(343) $\displaystyle M_{12}$ $\displaystyle=$
$\displaystyle-\frac{2}{5}\left(g_{1}^{*}+g_{5}^{*}\right),$ (344)
$\displaystyle M_{13}$ $\displaystyle=$
$\displaystyle-\left(\frac{1}{5}g_{1}^{*}+\frac{2}{5}g_{3z}^{*}\right),$ (345)
$\displaystyle M_{14}$ $\displaystyle=$
$\displaystyle-\frac{2}{5}\left(g_{1}^{*}+g_{2}^{*}+g_{3z}^{*}\right),$ (346)
$\displaystyle M_{15}$ $\displaystyle=$
$\displaystyle-\frac{2}{5}\left(g_{4}^{*}+g_{5}^{*}\right),$ (347)
$\displaystyle M_{21}$ $\displaystyle=$
$\displaystyle-\frac{3}{5}g_{2}^{*}-\frac{2}{5}g_{5}^{*},$ (348)
$\displaystyle M_{22}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}+2g_{2}^{*}-\frac{3}{5}g_{1}^{*}+\frac{4}{5}g_{4}^{*}+\frac{3}{5}g_{5}^{*}+g_{3z}^{*},$
(349) $\displaystyle M_{23}$ $\displaystyle=$
$\displaystyle\frac{4}{5}g_{2}^{*}-g_{5}^{*}+\frac{7}{5}g_{3z}^{*},$ (350)
$\displaystyle M_{24}$ $\displaystyle=$
$\displaystyle\frac{3}{5}g_{2}^{*}-\frac{2}{5}g_{1}^{*}-g_{4}^{*}+\frac{2}{5}g_{3z}^{*},$
(351) $\displaystyle M_{25}$ $\displaystyle=$ $\displaystyle
g_{2}^{*}+\frac{7}{5}g_{4}^{*}+\frac{2}{5}g_{5}^{*},$ (352) $\displaystyle
M_{31}$ $\displaystyle=$
$\displaystyle-\frac{2}{5}g_{1}^{*}+\frac{2}{5}g_{2}^{*}+\frac{2}{5}g_{5}^{*},$
(353) $\displaystyle M_{32}$ $\displaystyle=$
$\displaystyle-\frac{2}{5}g_{2}^{*}+\frac{2}{5}g_{1}^{*}-\frac{7}{5}g_{5}^{*}+g_{3z}^{*},$
(354) $\displaystyle M_{33}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}-\frac{2}{5}g_{4}^{*}+\frac{2}{5}g_{3z}^{*},$ (355)
$\displaystyle M_{34}$ $\displaystyle=$
$\displaystyle-\frac{2}{5}g_{5}^{*}+\frac{2}{5}g_{1}^{*}-\frac{7}{5}g_{2}^{*}-\frac{1}{5}g_{3z}^{*},$
(356) $\displaystyle M_{35}$ $\displaystyle=$
$\displaystyle-\frac{2}{5}g_{3z}^{*}+\frac{2}{5}g_{4}^{*}+g_{2}^{*}-\frac{1}{5}g_{5}^{*},$
(357) $\displaystyle M_{41}$ $\displaystyle=$
$\displaystyle\frac{2}{5}g_{1}^{*}-g_{5}^{*}-\frac{4}{5}g_{2}^{*}-\frac{1}{5}g_{4}^{*},$
(358) $\displaystyle M_{42}$ $\displaystyle=$
$\displaystyle\frac{2}{5}g_{2}^{*}+g_{5}^{*}-\frac{4}{5}g_{1}^{*}-\frac{4}{5}g_{4}^{*}-g_{3z}^{*},$
(359) $\displaystyle M_{43}$ $\displaystyle=$
$\displaystyle\frac{2}{5}g_{4}^{*}+g_{5}^{*}-\frac{1}{5}g_{1}^{*}-\frac{4}{5}g_{2}^{*}-\frac{8}{5}g_{3z}^{*},$
(360) $\displaystyle M_{44}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}+\frac{12}{5}g_{5}^{*}-g_{1}^{*}+g_{2}^{*}+g_{4}^{*}-g_{3z}^{*},$
(361) $\displaystyle M_{45}$ $\displaystyle=$
$\displaystyle\frac{2}{5}g_{3z}^{*}-g_{5}^{*}-g_{2}^{*}-\frac{8}{5}g_{4}^{*},$
(362) $\displaystyle M_{51}$ $\displaystyle=$
$\displaystyle\frac{2}{5}g_{1}^{*}-\frac{1}{5}g_{3z}^{*}-\frac{2}{5}g_{2}^{*}-\frac{1}{5}g_{4}^{*}-\frac{2}{5}g_{5}^{*},$
(363) $\displaystyle M_{52}$ $\displaystyle=$
$\displaystyle\frac{2}{5}g_{2}^{*}-\frac{4}{5}g_{3z}^{*}-\frac{2}{5}g_{1}^{*}+\frac{1}{5}g_{4}^{*}+\frac{2}{5}g_{5}^{*},$
(364) $\displaystyle M_{53}$ $\displaystyle=$
$\displaystyle\frac{2}{5}g_{4}^{*}-\frac{6}{5}g_{3z}^{*}-\frac{1}{5}g_{1}^{*}+\frac{1}{5}g_{2}^{*}+\frac{1}{5}g_{5}^{*},$
(365) $\displaystyle M_{54}$ $\displaystyle=$
$\displaystyle\frac{2}{5}g_{5}^{*}-\frac{3}{5}g_{3z}^{*}-\frac{2}{5}g_{1}^{*}+\frac{2}{5}g_{2}^{*}+\frac{1}{5}g_{4}^{*},$
(366) $\displaystyle M_{55}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}+\frac{4}{5}g_{3z}^{*}-\frac{1}{5}g_{1}^{*}-\frac{4}{5}g_{2}^{*}-\frac{6}{5}g_{4}^{*}-\frac{3}{5}g_{5}^{*}.$
(367)

From eigenvalues of $M$ at a fixed point
$(g_{1}^{*},g_{2}^{*},g_{4}^{*},g_{5}^{*},g_{3z}^{*})$, we can get the
properties of the fixed point. A negative (positive) eigenvalue is
corresponding to a stable (unstable) eigendirection Maciejko14 ; Szabo21 . For
quantum critical point (QCP), bicritical point (BCP), and tricritical point
(TCP), there is/are one, two, and three unstable direction(s) respectively.
For a QCP, the correlation length exponent is determined by the inverse of the
corresponding positive eigenvalue.

Substituting the values of $g_{a}^{*}$ at each fixed point into the expression
$M$, we calculate the corresponding eigenvalues of $M$. The eigenvalues for
the fixed points are shown in Table 3. For FP0, the eigenvalues of $M$ are
always negative, thus FP0 is a stable fixed point. We can find that there is
one positive eigenvalue for FP1, FP2 FP3, FP4, and FP5, and there are two
positive eigenvalues for FP6, FP7, FP8, FP9, and FP10, and three positive
eigenvalues for FP11. Thus, FP1, FP2, FP3, FP4, and FP5 are QCPs, FP6, FP7,
FP8, FP9, and FP10 are BCPs, and FP11 is a TCP.

Table 3: Eigenvalues of matrix $M$ at different fixed points  FP0 | FP1 | FP2 | FP3 | FP4 | FP5 | FP6 | FP7 | FP8 | FP9 | FP10 | FP11
---|---|---|---|---|---|---|---|---|---|---|---
-1.5 | -3.12277 | -3.05426 | -2.11269 | -4.19874 | -2.50996 | -2.98441 | -2.99224 | -5.70333 | -5.29866 | -2.25 | -9.30126
-1.5 | -2.7634 | -2.48233 | -1.42791 | -2.79748 | -2.05542 | -2.77106 | -2.49091 | -3.26902 | -3.36509 | -2.25 | -1.69424
-1.5 | -1.77432 | -1.76197 | -1.26511 | -2.46863 | -1.41052 | -1.80787 | -1.77301 | -1.46547 | -1.38636 | -1.47474 | 1.5
-1.5 | -0.412398 | -0.231803 | -1.18664 | -0.848268 | -1.06884 | 0.390411 | 0.224733 | 1.5 | 1.5 | 0.974745 | 3.33999
-1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 2.47601 | 2.17557 | 1.5 | 5.46863

It is easy to find that the correlation length exponent at the QCPs FP1, FP2,
FP3, FP4 and FP5 all satisfy

$\displaystyle\nu^{-1}$ $\displaystyle=$ $\displaystyle 1.5$ (368)

Substituting the values of $g_{a}^{*}$ with $i=1,2,4,5,3z$ into Eqs.
(299)-(316), we can get values of $\bar{\beta}_{X,Y}$ for different
$\Delta_{X,Y}$, which are shown in Tabel 4. For a QCP, the largest value of
$\beta_{X,Y}$ is marked by the bold style. It represents that the fixed point
is a QCP to the new state in which $\Delta_{X,Y}$ acquires finite value. FP1,
FP2, FP4, and FP5 are corresponding to QCPs to a state in which $\Delta_{2}$,
$\Delta_{5}$, $\Delta_{7z}$ and $\Delta_{8z}$ acquire finite value
respectively. For FP3, it stands for a QCP to a state in which $\Delta_{2}$
and $\Delta_{5}$ become finite generally. This state represents an axionic
insulator whose order parameter can be written as
$\langle\bar{\Psi}\left(\cos(\theta)+i\gamma_{5}\sin(\theta)\right)\Psi\rangle$
Roy16 .

Table 4: $\beta_{X,Y}$ at different fixed points. The largest value at a QCP is marked by the bold style. Notice that FP1, FP2, FP3, FP4, FP5 are QCPs.  | FP1 | FP2 | FP3 | FP4 | FP5 | FP6 | FP7 | FP8 | FP9 | FP10 | FP11
---|---|---|---|---|---|---|---|---|---|---|---
$\bar{\beta}_{1}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\bar{\beta}_{2}$ | 1.78199 | -0.0313166 | 1.09642 | -0.861228 | -0.184302 | 2.03474 | 0.163708 | -1.51656 | 0.0591369 | 0.625 | -0.388772
$\bar{\beta}_{3\bot}$ | 0.435705 | -0.316965 | -0.102003 | 0.344491 | -0.196188 | 0.385057 | -0.324365 | 0.411346 | 0.141049 | -0.25 | 0.155509
$\bar{\beta}_{3z}$ | 0.212338 | -0.165478 | -0.070994 | 0.344491 | -0.221326 | 0.183119 | -0.171631 | 0.21597 | 0.312383 | -0.25 | 0.155509
$\bar{\beta}_{4}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\bar{\beta}_{5}$ | -0.0893033 | 1.83099 | 1.09642 | -0.861228 | -0.184302 | 0.260278 | 1.97451 | -1.51656 | 0.0591369 | 0.625 | -0.388772
$\bar{\beta}_{6\bot}$ | -0.312814 | 0.427957 | -0.102003 | 0.344491 | -0.196188 | -0.324729 | 0.399958 | 0.411346 | 0.141049 | -0.25 | 0.155509
$\bar{\beta}_{6z}$ | -0.145378 | 0.227971 | -0.0110167 | -0.172246 | 0.14837 | -0.143544 | 0.218875 | 0.185078 | -0.413193 | 0.125 | -0.0777544
$\bar{\beta}_{7\bot}$ | -0.374656 | -0.375128 | -0.132437 | 0.882843 | 0.495967 | -0.390247 | -0.383104 | 2.11804 | 0.759982 | 0.375 | 2.86716
$\bar{\beta}_{7z}$ | -0.543656 | -0.556138 | -0.336522 | 2.55509 | -0.324568 | -0.59561 | -0.586391 | 2.90643 | 2.94818 | -0.5 | 4.44491
$\bar{\beta}_{8\bot}$ | 0.00789679 | 0.0395537 | 0.558464 | 0.882843 | -0.0597252 | 0.196553 | 0.144978 | 0.652867 | 2.03504 | 0.375 | 2.86716
$\bar{\beta}_{8z}$ | 0.0546439 | 0.108706 | 0.904558 | -0.20084 | 0.906224 | 0.337352 | 0.26889 | 0.788112 | 0.778521 | 1.5 | 3.20084
$\bar{\beta}_{S}$ | -0.504013 | 0.968638 | -0.284018 | -1.17712 | -0.290828 | -0.580657 | 0.883073 | -1.84727 | -1.86335 | -0.5 | -3.82288
$\bar{\beta}_{op}$ | 0.993025 | -0.521206 | -0.284018 | -1.17712 | -0.290828 | 0.838916 | -0.565572 | -1.84727 | -1.86335 | -0.5 | -3.82288
$\bar{\beta}_{V,1}$ | 0.0881294 | 0.110715 | 0.412273 | -1.03347 | 0.503886 | 0.203076 | 0.176024 | -0.940925 | -0.683464 | 0.75 | -0.466527
$\bar{\beta}_{V,2}$ | 0.0881294 | 0.110715 | 0.412273 | -1.03347 | 0.503886 | 0.203076 | 0.176024 | -0.940925 | -0.683464 | 0.75 | -0.466527
$\bar{\beta}_{V,3}$ | 0.16518 | 0.206828 | 0.830758 | -0.861228 | 0.027394 | 0.41359 | 0.345578 | -2.24934 | 0.709928 | 0.625 | -0.388772
$\bar{\beta}_{V,0}$ | -0.0472408 | -0.0484308 | -0.0320743 | -0.0861228 | 0.0953547 | -0.0564411 | -0.053456 | 0.019261 | -0.141517 | 0.0625 | -0.0388772

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