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Correlated interaction effects in three-dimensional semi-Dirac semimetal

Jing-Rong Wang,1 Wei Li,2, ∗ and Chang-Jin Zhang1, 3, †

1High Magnetic Field Laboratory of Anhui Province,

Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions,

Chinese Academy of Sciences, Hefei 230031, China2Key Laboratory of Materials Physics, Institute of Solid State Physics,

Chinese Academy of Sciences, Hefei 230031, China3Institute of Physical Science and Information Technology, Anhui University, Hefei 230601, China

Understanding the correlation effects in unconventional topological materials, in which the fermionexcitations 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-Diracsemimetal with an unusual fermion dispersion, that is linear along two directions but quadraticalong the third one. Based on renormalization group theory, we find all of 11 unstable fixed pointsincluding 5 quantum critical points, 5 bicritical points, and one tricritical point. The physicalessences of the quantum critical points are determined by analyzing the susceptibility exponents forall of the source terms in particle-hole and particle-particle channels. We also verify phase diagramof 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 helpfulfor us to understand the physical properties of candidate materials for three-dimensional semi-Diracsemimetal such as ZrTe5.

I. INTRODUCTION

The past 15 years have witnessed that study abouttopological materials becomes one of the most impor-tant fields in condensed matter physics [1–9]. Topo-logical materials have wide potential implications aselectronic devices due to their some fascinating physi-cal properties. In some topological materials, such asDirac semimetal (DSM) including Cd3As2 and Na3Bi,and Weyl semimetal (WSM) including TaAs, TaP, NbAs,and NbP, the low-energy fermion excitations are Diracfermions or Weyl fermions which resemble the elementaryparticles in high energy physics. Thus, these materialsprovide a platform to verify some important concepts inhigh energy physics.

Besides Dirac and Weyl fermions, there could be un-conventional fermions with unusual dispersion in topo-logical materials. In double- (triple-) WSM, the fermiondispersion is quadratic (cubic) along two directions butlinear along the third one [11, 12]. At the topologicalquantum critical point (QCP) between DSM or WSMand band insulator, the fermion dispersion is mixture oflinear and quadratic ones [13, 14]. Higher spin fermionswith multiband crossing have also attracted a lot of inter-est recently [15–17]. Spin-1 chiral fermions characterizedby combination of a Dirac-like band and a flat band withthree-bands crossing, and spin-3/2 chiral fermions dis-playing a birefringent spectrum with two distinct Fermivelocities have been observed recently [18–22].

The correlation effects in Dirac and Weyl fermion sys-tems are extensively studied, and are well understood

∗Corresponding author: wliustc@theory.issp.ac.cn†Corresponding author: zhangcj@hmfl.ac.cn

relatively [23–34]. The influence of many-body interac-tion on unconventional fermion systems also attractedmuch interest and is an important topic. There have beenstudies about influence of long-range Coulomb interac-tion [37–51], short-range four-fermion interaction [52–56], and quantum fluctuation of order parameter [57–60] on some unconventional fermion systems. Thesestudies reveled many novel behaviors, such as vari-ous quantum phase transitions, non-Fermi liquid behav-iors, anisotropic screening effect etc. These studies alsoshowed that the correlation effects in unconventionalfermion systems depend on the fermion dispersion subtly.However, there are still a lot of open questions about

the correlation effects in unconventional fermion sys-tems. In this article, we pay attention to the influ-ence of short-range four-fermion interactions on three-dimensional (3D) semi-DSM, in which the fermion dis-persion is linear along two directions but quadratic alongthe third one as shown in Fig. 1. 3D semi-DSM statecould be realized at the topological QCP between 3DDSM and band insulator [14].The rest of paper is structured as follows. The model

is defined in Sec. II. In Sec. III, we show the results basedon renormalization group (RG) theory. The results aresummarized and some discussion is given in Sec. IV. Thedetailed derivation of the RG equations and susceptibil-ity exponents for the source terms and careful numericalanalysis are presented in Appendices.

II. MODEL

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

S0 =

∫

dω

2π

d3k

(2π)3Ψ(ω,k) [iωγ0 +H(k)] Ψ(ω,k), (1)

2

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

where the Hamiltonian density H(k) is given by

H(k) = ivγ1k1 + ivk2γ2 + iAk23γ3, (2)

with v and A being model parameters. Ψ is four com-ponent spinor, and Ψ = Ψ†γ0. The gamma matrices aredefined as γ0 = τ2 ⊗ σ0, γ1 = τ3 ⊗ σ1, γ2 = τ3 ⊗ σ2,γ3 = τ3⊗σ3, and γ5 = τ1⊗σ0, where τ1,2,3 and σ1,2,3 arePauli matrices. The gamma matrices satisfy the anticom-mutation relation {γµ, γν} = 2δµν for µ, ν = 0, 1, 2, 3, 5.The energy dispersion of fermions takes the form E(k) =

±√

v2k2⊥ +A2k43 where k2⊥ = k21 + k22 . Density of states

(DOS) is given by ρ(ω) ∝ ω3/2/(v√A), which vanishes

at the Fermi level.Due to the vanishing DOS, the weak four-fermion inter-

action is irrelevant in 3D semi-DSM. However, if the four-fermion interaction is strong enough, the system couldbe 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 ofthem are linearly independent. Here, we consider theinteracting Lagrangian as following

Lint = g1(

Ψγ0Ψ)2

+ g2(

ΨΨ)2

+ g4(

Ψγ0γ5Ψ)2

+g5(

Ψiγ5Ψ)2

+ g3z(

Ψγ0γ3Ψ)2

. (3)

In this article, we study the influence of four-fermion in-teractions on 3D semi-DSM through the RG method [61].

III. RG RESULTS

As shown in the Appendices, we firstly calculate all ofthe corrections from the one-loop Feynman diagrams, byemploying a momentum shell bΛ <

√

v2k2⊥ +A2k43 < Λ,

where b = e−ℓ with ℓ being the RG running parame-ter. Then, we consider these corrections, and performRG transformations to restore the original form of theactions. Accordingly, we obtain the RG equations for ga,which can be formally written as

dgadℓ

= −3

2ga + Fa (g1, g2, g4, g5, g3z) , (4)

where a = 1, 2, 4, 5, 3z, The concrete expressions of Fa

can be found in the Appendices.

TABLE I: There are 5 QCPs among the 11 non-trivial un-stable fixed points. The corresponding order parameters forthe five QCPs are shown in second rows.

FP1 FP2 FP3 FP4 FP5Order parameter ∆2 ∆5 ∆2/∆5 ∆7z ∆8z

Solving the equations

dgadℓ

∣

∣

∣

∣

(g1,g2,g4,g5,g3z)=(g∗

1,g∗

2,g∗

4,g∗

5,g∗

3z)

= 0, (5)

we get 12 fixed points, including the trivial Gauss fixedpoint (g∗1 , g

∗2 , g

∗4 , g

∗5 , g

∗3z) = (0, 0, 0, 0, 0) and 11 non-trivial

fixed points

FPi : (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 ga in the vicinity of a

fixed point, we obtain

dδgadℓ

=∑

b

Mabδgb, (7)

where δga = ga − g∗a. M is five dimension square ma-trix, 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 correspondingto a stable (unstable) eigendirection [32, 34]. There is oneunstable direction for QCP, and there are two and threeunstable directions for bicritical point (BCP) and tricrit-ical point (TCP) respectively. Substituting the values ofg∗a at each fixed point into the expression of M , we cal-culate the corresponding eigenvalues of M . We find thatFP1, 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 ν is deter-

mined by the inverse of the corresponding positive eigen-value of M . For the five QCPs, ν always satisfies

ν−1 = 1.5. (8)

In order to determine the physical essences of theQCPs, we analyze the RG flows of all the fermion bi-linear source terms in particle-hole and particle-particlechannels. The source terms in particle-hole channel canbe generally written as

Ss = ∆X

∫

dω

2π

d3k

(2π)3Ψ(ω,k)ΓXΨ(ω,k) . (9)

There are 12 choices for the matrix ΓX , which corre-sponds to 12 different order parameters in particle-holechannel. The source terms in particle-particle channeltake the general form

Ss = ∆Y

∫

dω

2π

d3k

(2π)3Ψ†(ω,k)ΓY Ψ

∗ (ω,k) . (10)

3

FIG. 2: (a)-(e): Flows of g1, g2, g4, g5, and g3z with differentinitial conditions. g1,0 = 0, g4,0 = 0, g5,0 = 0, and g3z,0 = 0are taken.

There are 6 choices for the matrix ΓY , which is corre-sponding 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 bythe four-fermion interactions as shown in Eq. (3). Then,we include these corrections and perform RG transfor-mations to restore the original form of the source terms.Accordingly, through the RG transformations, we obtainthe equations

βX,Y = HX,Y (g1, g2, g4, g5, g3z) , (11)

where

βX,Y =d ln (∆X,Y )

dℓ− 1. (12)

βX,Y is usually termed as susceptibility exponent oranomalous dimension for the fermion-bilinear sourceterm. HX,Y are functions of ga with a = 1, 2, 4, 5, 3z.The concrete expressions of HX,Y are shown in Appen-dices. For a QCP, substituting the values of ga at theQCP into Eq. (11), and finding the largest one among allof βX,Y , we can determine the physical meaning of theQCP.For FP1, β2 takes the largest value. It represents that

this fixed point is corresponding to the QCP to a statein which the order parameter ∆2 = 〈ΨΨ〉 acquires finitevalue. The physical meaning of ∆2 is scalar mass.For FP2, β5 is the largest one. It means that this fixed

point stands for the QCP to a state in which the orderparameter ∆5 = 〈Ψiγ5Ψ〉 becomes finite. The physicalmeaning of ∆5 corresponds to pseudoscalar mass.For FP3, β2 and β5 are largest simultaneously. It

indicates that the fixed point corresponds to the QCPto a phase in which both of ∆2 and ∆5 become fi-nite. The parameter of this phase can be written as

FIG. 3: Flows of βX,Y which approach positive infinity andratios between βX,Y . (a) and (b): g1,0 = 0.15, g2,0 = 1.3,g4,0 = 0.46, g5,0 = −0.56, and g3z,0 = 0.055 are taken; (c)and (d): g2,0 = 0.14, g2,0 = −0.59, g4,0 = 0.42, g5,0 = 1.36,and g3z,0 = 0.07 are taken; (e) and (f): g1,0 = 1.5, g2,0 = 0,g3,0 = 0, g5,0 = 0.2, and g3z,0 = 0 are taken.

〈Ψ (cos(θ) + iγ5 sin(θ)) Ψ〉. This phase represents an ax-ionic insulator [33].

For FP4, β7z takes the largest value. It signifies thatthis fixed point is corresponding to the QCP to a statein which the order parameter ∆7z = 〈Ψiγ5γ3Ψ〉 becomesfinite. ∆7z stands for axial magnetization along z axis.

For FP5, β8z is the largest one. It suggests that thisfixed point represents the QCP to a state with finite orderparameter ∆8z = 〈Ψiγ3Ψ〉 . The physical meaning of ∆8z

is current along z axis.

For convenience, we summarize the corresponding or-der parameters for the 5 QCPs in Table I.

For general given initial conditions that is decided by(g1,0, g2,0, g4,0, g5,0, g3z,0), we determine the correspond-ing phase through the flows of four-fermion coupling pa-rameters ga and flows of susceptibility exponents βX,Y .We show the flows of g1, g2, g4, g5, and g3z under severalinitial conditions in Fig. 2. If ga with a = 1, 2, 4, 5, 3zapproach to zero, it represents that the system is stillin SM phase. If |ga| with a = 1, 2, 4, 5, 3z flow to infin-ity at a finite running parameter ℓc, the system becomesunstable to a new phase.

In order to determine the physical essence of the newphase, we calculate the flows of the susceptibility expo-nents βX,Y and compare them. For three general initialconditions, the flows of βX,Y and the ratio between themare presented in Figs. 3(a)-3(f). Here we only show thesusceptibility exponents that approach to positive infin-ity in Figs. 3(a), 3(c), and 3(e) respectively. For theinitial condition corresponding to Figs. 3(a) and 3(b),we can find that β2 approaches to positive infinity mostquickly. It means that scalar mass ∆2 is generated inthe new phase. For the initial condition corresponding

4

FIG. 4: Phase diagrams on the planes of two initial values of four-fermion coupling strength. (a) g2,0 and g5,0; (b) g2,0 andg1,0; (c) g2,0 and g4,0; (d) g2,0 and g3z,0; (e) g5,0 and g1,0; (f) g5,0 and g4,0; (g) g5,0 and g3z,0; (h) g1,0 and g4,0; (i) g1,0 and g3z,0;(j) g4,0 and g3z,0; (k) g5,0 and g3z,0 In (a)-(j), the initial values of rest four-fermion coupling parameters are taken as zero. Forexample, g1,0 = 0, g4,0 = 0 and g3z,0 = 0 are taken in (a). In (k), g1,0 = −2.3, g2,0 = 0, and g4,0 = −0.62 are taken.

to Figs. 3(c) and 3(d), β5 flows to positive infinity withthe largest speed. It indicates that pseudoscalar mass∆5 becomes finite in the new phase. For the initial con-dition corresponding to Figs. 3(e) and 3(f), we can findthat β2 and β5 approach to positive infinity most quicklyand β5/β2 → 1. It represents that the system becomesto a new phase that ∆2 and ∆5 acquire finite value si-multaneously. Namely, the system becomes to axionicinsulating phase.

The phase diagrams on the planes composed by ini-tial values of two four-fermion coupling parameters areshown in Fig. 4. Different phases are marked by differ-ent colors. In Figs. 4(a)-4(j), we show all the 10 phasediagrams on the planes composed by initial values of twocoupling parameters chosen from the five linearly inde-pendent coupling parameters. The initial values of restof three coupling parameters are taken as zero. TakenFig. 4(a) composed by g2,0 and g5,0 as an example, wecan notice that there are five phases, SM, insulator withscalar mass ∆2, insulator characterized by pseudoscalarmass ∆5, axionic insulating phase, and a phase with cur-rent along z axis ∆8z . In Fig. 4(k), we present the phasediagram composed by composed by g5,0 and g3z,0. In thisphase diagram, g1,0, g2,0 and g4,0 are taken as proper val-ues so that the phase with axial magnetization along zaxis ∆7z appears in the phase diagram.

Recently, Sur and Roy studied the quantum criticalbehaviors in the vicinity of possible QCPs in variousSMs with different dispersions generally [59]. Accord-

ing to their studies, in 3D semi-DSM, the Yukawa cou-pling between quantum fluctuation of order parameterand fermion excitations becomes irrelevant in the lowenergy regime. Thus, the fermions should take Fermiliquid behaviors in the vicinity of a QCP between SMphase and a symmetry breaking phase in 3D semi-DSM.Concretely, the residue of fermions Zf approaches to afinite constant value in the lowest energy limit, and theLandau damping rate of fermions Γ(ω) satisfies

limω→0

Γ(ω)

ω→ 0. (13)

Additionally, under the influence of quantum fluctuationof order parameter, the observable quantities DOS ρ,specific heat Cv, compressibility κ, optical conductivi-ties within the xy and along z axis σ⊥⊥ and σzz shouldrespectively still take the behaviors

ρ(ω) ∼ ω3/2, Cv(T ) ∼ T 5/2, κ(T ) ∼ T 3/2,

σ⊥⊥(ω) ∼ ω1/2, σzz(ω) ∼ ω3/2, (14)

which are qualitatively same as the ones for free fermions.

IV. SUMMARY

In summary, we perform comprehensive studies aboutthe influence of four-fermion interactions on 3D semi-DSM through RG theory. We find 11 unstable fixed

5

points and show that five fixed points are QCPs, five fixedpoints are BCPs, and the rest one is a TCP. The physicalessence of the QCPs are determined carefully by analyz-ing the scalings of fermion bilinear source terms. Thephase diagrams for general initial conditions are also pre-sented through detailed numerical calculations of flows offour-fermion couplings and susceptibility exponents.According to the theoretical study by Yang and Na-

gaosa [14], 3D semi-DSM state can be realized at thetopological QCP between 3D DSM and band insulator.Through magneto-optics and magneto-transport, Yuanet al. observed the evidence of 3D semi-DSM phase inZrTe5 [62]. The subsequent study about magnetotrans-port properties of ZrTe5 under hydrostatic pressure alsosupports the existence of 3D semi-DSM phase [63]. Re-cent measurements of optical spectroscopy in ZrTe5 arealso consistent with 3D semi-DSM phase [64, 65]. 3Dsemi-DSM state was also realized in pressed Cd3As2 [66].

We expect our theoretical results are helpful for under-standing the physical properties of these candidate ma-terials of 3D semi-DSM.

ACKNOWLEDGEMENTS

J.R.W. is grateful to Prof. G.-Z. Liu for the valu-able discussions. We acknowledge the support from theNational Key R&D Program of China under Grants2016YFA0300404, the National Natural Science Founda-tion of China under Grants 11974356, and U1832209, andthe Collaborative Innovation Program of Hefei ScienceCenter CAS under Grant 2019HSC-CIP002. A portionof this work was supported by the High Magnetic FieldLaboratory of Anhui Province under Grant AHHM-FX-2020-01.

Appendix A: Fierz Identity

1. Fierz identity for 3D DSM

For 3D DSM, the interacting Lagrangian density can be written as [33]

Lint = g1(

Ψγ0Ψ)2

+ g2(

ΨΨ)2

+ g3

3∑

j=1

(

Ψγ0γjΨ)2

+ g4(

Ψγ0γ5)2

+ g5(

Ψiγ5Ψ)2

+ g6∑

〈lk〉

(

ΨiγlγkΨ)2

+g7

3∑

j=1

(

Ψiγ5γjΨ)2

+ g8

3∑

j=1

(

ΨiγjΨ)2

, (A1)

where

∑

〈lk〉

(

ΨiγlγkΨ)2

=[

(

Ψiγ2γ3Ψ)2

+(

Ψiγ3γ1Ψ)2

+(

Ψiγ1γ2Ψ)2]

. (A2)

There are eight kinds of four-fermion couplings. However, not all of them are linearly independent, due to theconstraint by Fierz identity [31, 33, 34].

The Fierz identity indicates that [31, 33, 34]

[

Ψ(x)MΨ(x)] [

Ψ(y)NΨ(y)]

= − 1

16Tr [MΓaNΓb]

[

Ψ(x)ΓaΨ(y)] [

Ψ(y)ΓbΨ(x)]

. (A3)

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

[

Ψ(x)MΨ(x)] [

Ψ(x)NΨ(x)]

= − 1

16Tr [MΓaNΓb]

[

Ψ(x)ΓaΨ(x)] [

Ψ(x)ΓbΨ(x)]

. (A4)

Repeat of indexes a and b in Eqs. (A3) and (A4) represents summation. Substituting each four-fermion coupling inEq. (A1) into Eq. (A4), we could get eight equations, which can be compactly expressed by

FX = 0, (A5)

6

where

X =

(

Ψγ0Ψ)2

(

ΨΨ)2

3∑

j=1

(

Ψγ0γjΨ)2

(

Ψγ0γ5Ψ)2

(

Ψiγ5Ψ)2

∑

<lk>

(

ΨiγlγkΨ)2

3∑

j=1

(

Ψiγ5γjΨ)2

3∑

j=1

(

ΨiγjΨ)2

, (A6)

and

F =

5 1 1 1 1 1 1 11 5 −1 −1 −1 1 1 −13 −3 3 −3 3 1 −1 11 −1 −1 5 −1 −1 1 11 −1 1 −1 5 −1 1 −13 3 1 −3 −3 3 −1 13 3 −1 3 3 −1 3 −13 −3 1 3 −3 1 −1 3

. (A7)

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

Rank(F ) = 4. (A8)

Then, the number of linearly independent couplings is

8− Rank(F ) = 4. (A9)

For convenience, we take the four couplings

(

Ψγ0Ψ)2

,(

ΨΨ)2

,(

Ψγ0γ5Ψ)2

,(

Ψiγ5Ψ)2

, (A10)

as linearly independent couplings. The other couplings

3∑

j=1

(

Ψγ0γjΨ)2

,∑

<lk>

(

ΨiγlγkΨ)2

,

3∑

j=1

(

Ψiγ5γjΨ)2

,

3∑

j=1

(

ΨiγjΨ)2

, (A11)

can be expressed by the four independent couplings shown in Eq. (A10). In order to obtain the concrete expressionsfor other couplings, we define

X =

3∑

j=1

(

Ψγ0γjΨ)2

∑

<lk>

(

ΨiγlγkΨ)2

3∑

j=1

(

Ψiγ5γjΨ)2

3∑

j=1

(

ΨiγjΨ)2

(

Ψγ0Ψ)2

(

ΨΨ)2

(

Ψγ0γ5Ψ)2

(

Ψiγ5Ψ)2

. (A12)

7

It is easy to find that

F X = 0, (A13)

where

F =

1 1 1 1 5 1 1 1−1 1 1 −1 1 5 −1 −13 1 −1 1 3 −3 −3 3

−1 −1 1 1 1 −1 5 −11 −1 1 −1 1 −1 −1 51 3 −1 1 3 3 −3 −3

−1 −1 3 −1 3 3 3 31 1 −1 3 3 −3 3 −3

. (A14)

Performing a series of similarity transformations for F ,

F → F ′, (A15)

we obtain

F ′X = 0, (A16)

where

F ′ =

1 0 0 0 1 −1 −1 20 1 0 0 1 2 −1 −10 0 1 0 2 1 1 10 0 0 1 1 −1 2 −10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

. (A17)

Eq. (A16) can be equivalently expressed by

3∑

j=1

(

Ψγ0γjΨ)2

= −(

Ψγ0Ψ)2

+(

ΨΨ)2

+(

Ψγ0γ5Ψ)2 − 2

(

Ψiγ5Ψ)2

, (A18)

∑

<lk>

(

ΨiγlγkΨ)2

= −(

Ψγ0Ψ)2 − 2

(

ΨΨ)2

+(

Ψγ0γ5Ψ)2

+(

Ψiγ5Ψ)2

, (A19)

3∑

j=1

(

Ψiγ5γjΨ)2

= −2(

Ψγ0Ψ)2 −

(

ΨΨ)2 −

(

Ψγ0γ5Ψ)2 −

(

Ψiγ5Ψ)2

, (A20)

3∑

j=1

(

ΨiγjΨ)2

= −(

Ψγ0Ψ)2

+(

ΨΨ)2 − 2

(

Ψγ0γ5Ψ)2

+(

Ψiγ5Ψ)2

. (A21)

2. Fierz identity for 3D semi-DSM

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

Lint = g1(

Ψγ0Ψ)2

+ g2(

ΨΨ)2

+ g3⊥

2∑

j=1

(

Ψγ0γjΨ)2

+ g3z(

Ψγ0γ3Ψ)2

+ g4(

Ψγ0γ5Ψ)2

+ g5(

Ψiγ5Ψ)2

+g6⊥∑

<<lk>>

(

ΨiγlγkΨ)2

+ g6z(

Ψiγ1γ2Ψ)2

+ g7⊥

2∑

j=1

(

Ψiγ5γjΨ)2

+ g7z(

Ψiγ5γ3Ψ)2

+g8⊥

2∑

j=1

(

ΨiγjΨ)2

+ g8z(

Ψiγ3Ψ)2

, (A22)

8

where

∑

<<lk>>

(

ΨiγlγkΨ)2

=[

(

Ψiγ2γ3Ψ)2

+(

Ψiγ3γ1Ψ)2]

. (A23)

As shown in Eq. (A1), there are 8 four-fermion couplings for 3D DSM. However, we consider 12 kinds of four-fermioncouplings as shown in Eq. (A22) for 3D semi-DSM, due to the anisotropy of the fermion dispersion.Substituting each four-fermion coupling in Eq. (A22) into Eq. (A4), we could get 12 equations, which can be

compactly expressed by

FX = 0, (A24)

where

X =

(

Ψγ0Ψ)2

(

ΨΨ)2

2∑

j=1

(

Ψγ0γjΨ)2

(

Ψγ0γ3Ψ)2

(

Ψγ0γ5Ψ)2

(

Ψiγ5Ψ)2

∑

<<lk>>

(

ΨiγlγkΨ)2

(

Ψiγ1γ2Ψ)2

2∑

j=1

(

Ψiγ5γjΨ)2

(

Ψiγ5γ3Ψ)2

2∑

j=1

(

ΨiγjΨ)2

(

Ψiγ3Ψ)2

, (A25)

and

F =

5 1 1 1 1 1 1 1 1 1 1 11 5 −1 −1 −1 −1 1 1 1 1 −1 −11 −1 2 −1 −1 1 0 1 0 −1 0 11 −1 −1 5 −1 1 1 −1 −1 1 1 −11 −1 −1 −1 5 −1 −1 −1 1 1 1 11 −1 1 1 −1 5 −1 −1 1 1 −1 −11 1 0 1 −1 −1 2 −1 0 −1 0 11 1 1 −1 −1 −1 −1 5 −1 1 1 −11 1 0 −1 1 1 0 −1 2 −1 0 −11 1 −1 1 1 1 −1 1 −1 5 −1 11 −1 0 1 1 −1 0 1 0 −1 2 −11 −1 1 −1 1 −1 1 −1 −1 1 −1 5

. (A26)

It is easy to find that

Rank(F ) = 7. (A27)

Then, the number of linearly independent couplings is

12− Rank(F ) = 5. (A28)

For convenience, we take the five couplings

(

Ψγ0Ψ)2

,(

ΨΨ)2

,(

Ψγ0γ5Ψ)2

,(

Ψiγ5Ψ)2

,(

Ψγ0γ3Ψ)2

, (A29)

9

as linearly independent couplings. The other couplings

2∑

j=1

(

Ψγ0γjΨ)2

,∑

<<lk>>

(

ΨiγlγkΨ)2

,(

Ψiγ1γ2Ψ)2

,

2∑

j=1

(

Ψiγ5γjΨ)2

,(

Ψiγ5γ3Ψ)2

,

2∑

j=1

(

ΨiγjΨ)2

,(

Ψiγ3Ψ)2

, (A30)

can be expressed by the five independent couplings shown in Eq. (A29). In order to obtain the concrete expressionsfor other couplings, we define

X =

2∑

j=1

(

Ψγ0γjΨ)2

∑

<<lk>>

(

ΨiγlγkΨ)2

(

Ψiγ1γ2Ψ)2

2∑

j=1

(

Ψiγ5γjΨ)2

(

Ψiγ5γ3Ψ)2

2∑

j=1

(

ΨiγjΨ)2

(

Ψiγ3Ψ)2

(

Ψγ0Ψ)2

(

ΨΨ)2

(

Ψγ0γ5Ψ)2

(

Ψiγ5Ψ)2

(

Ψγ0γ3Ψ)2

. (A31)

It is easy to find that

F X = 0, (A32)

where

F =

1 1 1 1 1 1 1 5 1 1 1 1−1 1 1 1 1 −1 −1 1 5 −1 −1 −12 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 −11 −1 −1 1 1 −1 −1 1 −1 −1 5 10 2 −1 0 −1 0 1 1 1 −1 −1 11 −1 5 −1 1 1 −1 1 1 −1 −1 −10 0 −1 2 −1 0 −1 1 1 1 1 −1

−1 −1 1 −1 5 −1 1 1 1 1 1 10 0 1 0 −1 2 −1 1 −1 1 −1 11 1 −1 −1 1 −1 5 1 −1 1 −1 −1

. (A33)

Carrying out a series of similarity transformations for F ,

F → F ′, (A34)

we arrive

F ′X = 0, (A35)

10

where

F ′ =

1 0 0 0 0 0 0 1 −1 −1 2 10 1 0 0 0 0 0 1 1 −1 0 10 0 1 0 0 0 0 0 1 0 −1 −10 0 0 1 0 0 0 1 1 1 0 −10 0 0 0 1 0 0 1 0 0 1 10 0 0 0 0 1 0 1 −1 1 0 10 0 0 0 0 0 1 0 0 1 −1 −10 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0

. (A36)

Eq. (A35) can be also written as

2∑

j=1

(

Ψγ0γjΨ)2

= −(

Ψγ0Ψ)2

+(

ΨΨ)2

+(

Ψγ0γ5Ψ)2 − 2

(

Ψiγ5Ψ)2 −

(

Ψγ0γ3Ψ)2

, (A37)

∑

<<lk>>

(

ΨiγlγkΨ)2

= −(

Ψγ0Ψ)2 −

(

ΨΨ)2

+(

Ψγ0γ5Ψ)2 −

(

Ψγ0γ3Ψ)2

, (A38)

(

Ψiγ1γ2Ψ)2

= −(

ΨΨ)2

+(

Ψiγ5Ψ)2

+(

Ψγ0γ3Ψ)2

, (A39)

2∑

j=1

(

Ψiγ5γjΨ)2

= −(

Ψγ0Ψ)2 −

(

ΨΨ)2 −

(

Ψγ0γ5Ψ)2

+(

Ψγ0γ3Ψ)2

, (A40)

(

Ψiγ5γ3Ψ)2

= −(

Ψγ0Ψ)2 −

(

Ψiγ5Ψ)2 −

(

Ψγ0γ3Ψ)2

, (A41)

2∑

j=1

(

ΨiγjΨ)2

= −(

Ψγ0Ψ)2

+(

ΨΨ)2 −

(

Ψγ0γ5Ψ)2 −

(

Ψγ0γ3Ψ)2

, (A42)

(

Ψiγ3Ψ)2

= −(

Ψγ0γ5Ψ)2

+(

Ψiγ5Ψ)2

+(

Ψγ0γ3Ψ)2

. (A43)

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

1. Self-energy of the fermions

The fermion propagator reads as

G0(iω,k) = − iωγ0 + iv (k1γ1 + k2γ2) + iAk23γ3ω2 + E2

k

, (B1)

where Ek =√

v2k2⊥ +A2k43 with k2⊥ = k21 + k22 . The self-energy of fermions resulting from Fig. 5(a) takes the form

Σ1 =∑

a

ga

∫

dω

2π

∫ ′ d3k

(2π)3ΓaG0(ω,k)Γa, (B2)

where∑

a

≡∑

a=1,2,4,5,3z

. (B3)

∫ ′represents that a momentum shell will be properly taken. Figure 5(b) induces the self-energy of fermions as

following

Σ2 =∑

a

ga

∫

dω

2π

∫ ′ d3k

(2π)3Tr [G0(ω,k)Γa] . (B4)

11

Substituting Eq. (B1) into Eqs. (B2) and (B4), we obtain

Σ1 = 0, (B5)

Σ2 = 0. (B6)

It should be notice that a generated constant term in Σ1 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 [38]. According toEqs. (B5) and (B6), the fermion propagator is not renormalized by the four-fermion interactions to one-loop order.

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

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

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

Fig. 6(a) leads to the correction

V (1)a = −2g2a

(

ΨΓaΨ)2

∫ +∞

−∞

dω

2π

∫ ′ d3k

(2π)3Tr [ΓaG0(iω,k)ΓaG0(iω,k)] . (B7)

Fig. 6(b) results in the correction

V (2)a =

∑

b

V(2)ab , (B8)

where

V (2)a = 4gagb

(

ΨΓaΨ)

∫ +∞

−∞

dω

2π

∫ ′ d3k

(2π)3(

ΨΓbG0(iω,k)ΓaG0(iω,k)ΓbΨ)

. (B9)

12

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

V (3)+(4) =∑

a

∑

a≤b

V(3)+(4)ab , (B10)

where

V(3)+(4)ab = 4gagb

∫ +∞

−∞

dω

2π

∫ ′ d3k

(2π)3(

ΨΓaG0(iω,k)ΓbΨ)

Ψ [ΓbG0(iω,k)Γa + ΓaG0(−iω,−k)Γb] Ψ. (B11)

Substituting Eq. (B1) into Eq. (B7), we obtain

V (1)a = δg(1)a

(

ΨΓaΨ)2

, (B12)

where

δg(1)1 = 0, (B13)

δg(1)2 = g22

2Λ3

2

π2v2√Aℓ, (B14)

δg(1)4 = 0, (B15)

δg(1)5 = g25

2Λ3

2

π2v2√Aℓ, (B16)

δg(1)3z = g23z

2Λ3

2

5π2v2√Aℓ. (B17)

Substituting Eq. (B1) into Eqs. (B8) and (B9), we find that the contribution from Fig. 6(b) can be written as

V (2)a = δg(2)a

(

ΨΓaΨ)2

, (B18)

where

δg(2)1 = 0, (B19)

δg(2)2 =

(

−g2g1 − g22 + g2g4 + g2g5 + g2g3z) Λ

3

2

π2v2√Aℓ, (B20)

δg(2)4 = 0, (B21)

δg(2)5 =

(

−g5g1 + g5g2 + g5g4 − g25 − g5g3z) Λ

3

2

π2v2√Aℓ, (B22)

δg(2)3z =

(

−g3zg1 + g3zg2 + g3zg4 − g3zg5 − g23z) Λ

3

2

5π2v2√Aℓ. (B23)

Substituting Eq. (B1) into Eq. (B11), the contribution from Figs. 6(c) and 6(d) can be written as

V(3)+(4)1,1 = g21

Λ3

2

5π2v2√Aℓ(

Ψiγ3Ψ)2

, (B24)

V(3)+(4)2,2 = g22

Λ3

2

5π2v2√Aℓ(

Ψiγ3Ψ)2

, (B25)

V(3)+(4)4,4 = g24

Λ3

2

5π2v2√Aℓ(

Ψiγ3Ψ)2

, (B26)

V(3)+(4)5,5 = g25

Λ3

2

5π2v2√Aℓ(

Ψiγ3Ψ)2

, (B27)

V(3)+(4)3z,3z = g23z

Λ3

2

5π2v2√Aℓ(

Ψiγ3Ψ)2

, (B28)

13

V(3)+(4)1,2 = g1g2

2Λ3

2

5π2v2√Aℓ

2∑

j=1

(

Ψγ0γjΨ)2

, (B29)

V(3)+(4)1,4 = g1g4

Λ3

2

5π2v2√Aℓ(

Ψiγ5γ3Ψ)2

, (B30)

V(3)+(4)1,5 = g1g5

2Λ3

2

5π2v2√Aℓ

∑

<<lk>>

(

ΨiγlγkΨ)2

, (B31)

V(3)+(4)1,3z = 0, (B32)

V(3)+(4)2,4 = −g2g4

Λ3

2

π2v2√Aℓ(

Ψiγ5Ψ)2

+ g2g4Λ

3

2

5π2v2√Aℓ(

Ψiγ1γ2Ψ)2

, (B33)

V(3)+(4)2,5 = −g2g5

Λ3

2

π2v2√Aℓ(

Ψγ0γ5Ψ)2

+ g2g5

2∑

j=1

2Λ3

2

5π2v2√Aℓ(

Ψiγ5γjΨ)2

, (B34)

V(3)+(4)2,3z = −g2g3z

Λ3

2

π2v2√Aℓ(

Ψiγ3Ψ)2

, (B35)

V(3)+(4)4,5 = −g4g5

Λ3

2

π2v2√Aℓ(

ΨΨ)2

+ g4g5Λ

3

2

5π2v2√Aℓ(

Ψγ0γ3Ψ)2

, (B36)

V(3)+(4)4,3z = −g4g3z

Λ3

2

π2v2√Aℓ(

Ψiγ1γ2Ψ)2

+ g4g3z

2∑

j=1

2Λ3

2

5π2v2√Aℓ(

Ψγ0γjΨ)2

+ g4g3zΛ

3

2

5π2v2√Aℓ(

Ψiγ5Ψ)2

,(B37)

V(3)+(4)5,3z = g5g3z

2∑

j=1

2Λ3

2

5π2v2√Aℓ(

ΨiγjΨ)2

+ g5g3zΛ

3

2

5π2v2√Aℓ(

Ψγ0γ5Ψ)2

. (B38)

Using the relations shown in Eqs. (A37)-(A43), we further get

V(3)+(4)1,1 = g20

Λ3

2

5π2v2√Aℓ[

−(

Ψγ0γ5Ψ)2

+(

Ψiγ5Ψ)2

+(

Ψγ0γ3Ψ)2]

, (B39)

V(3)+(4)2,2 = g22

Λ3

2

5π2v2√Aℓ[

−(

Ψγ0γ5Ψ)2

+(

Ψiγ5Ψ)2

+(

Ψγ0γ3Ψ)2]

, (B40)

V(3)+(4)4,4 = g24

Λ3

2

5π2v2√Aℓ[

−(

Ψγ0γ5Ψ)2

+(

Ψiγ5Ψ)2

+(

Ψγ0γ3Ψ)2]

, (B41)

V(3)+(4)5,5 = g25

Λ3

2

5π2v2√Aℓ[

−(

Ψγ0γ5Ψ)2

+(

Ψiγ5Ψ)2

+(

Ψγ0γ3Ψ)2]

, (B42)

V(3)+(4)3z,3z = g23z

Λ3

2

5π2v2√Aℓ[

−(

Ψγ0γ5Ψ)2

+(

Ψiγ5Ψ)2

+(

Ψγ0γ3Ψ)2]

, (B43)

V(3)+(4)1,2 = g1g2

2Λ3

2

5π2v2√Aℓ[

−(

Ψγ0Ψ)2

+(

ΨΨ)2

+(

Ψγ0γ5Ψ)2 − 2

(

Ψiγ5Ψ)2 −

(

Ψγ0γ3Ψ)2]

, (B44)

V(3)+(4)1,4 = g1g4

Λ3

2

5π2v2√Aℓ[

−(

Ψγ0Ψ)2 −

(

Ψiγ5Ψ)2 −

(

Ψγ0γ3Ψ)2]

, (B45)

V(3)+(4)1,5 = g1g5

2Λ3

2

5π2v2√Aℓ[

−(

Ψγ0Ψ)2 −

(

ΨΨ)2

+(

Ψγ0γ5Ψ)2 −

(

Ψγ0γ3Ψ)2]

, (B46)

V(3)+(4)1,3z = 0, (B47)

V(3)+(4)2,4 = g2g4

Λ3

2

5π2v2√Aℓ[

−(

ΨΨ)2 − 4

(

Ψiγ5Ψ)2

+(

Ψγ0γ3Ψ)2]

, (B48)

V(3)+(4)2,5 = g2g5

2Λ3

2

5π2v2√Aℓ

[

−(

Ψγ0Ψ)2 −

(

ΨΨ)2 − 7

2

(

Ψγ0γ5Ψ)2

+(

Ψγ0γ3Ψ)2]

, (B49)

V(3)+(4)2,3z = −g2g3z

Λ3

2

π2v2√Aℓ[

−(

Ψγ0γ5Ψ)2

+(

Ψiγ5Ψ)2

+(

Ψγ0γ3Ψ)2]

, (B50)

14

V(3)+(4)4,5 = g4g5

Λ3

2

5π2v2√Aℓ[

−5(

ΨΨ)2

+(

Ψγ0γ3Ψ)2]

(B51)

V(3)+(4)4,3z = g4g3z

2Λ3

2

5π2v2√Aℓ

[

−(

Ψγ0Ψ)2

+7

2

(

ΨΨ)2

+(

Ψγ0γ5Ψ)2 − 4

(

Ψiγ5Ψ)2 − 7

2

(

Ψγ0γ3Ψ)2]

, (B52)

V(3)+(4)5,3z = g5g3z

2Λ3

2

5π2v2√Aℓ

[

−(

Ψγ0Ψ)2

+(

ΨΨ)2 − 1

2

(

Ψγ0γ5Ψ)2 −

(

Ψγ0γ3Ψ)2]

. (B53)

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

V (3)+(4) =∑

a=1,2,4,5,3z

∑

a≤b

V(3)+(4)ab

=∑

a=1,2,4,5,3z

δg(3)+(4)a

(

ΨΓaΨ)2

, (B54)

where

δg(3)+(4)1 =

(

−g1g2 −1

2g1g4 − g1g5 − g2g5 − g4g3z − g5g3z

)

2Λ3

2

5π2v2√Aℓ, (B55)

δg(3)+(4)2 =

(

g1g2 − g1g5 −1

2g2g4 − g2g5 −

5

2g4g5 +

7

2g4g3z + g5g3z

)

2Λ3

2

5π2v2√Aℓ, (B56)

δg(3)+(4)4 =

(

−1

2g21 −

1

2g22 −

1

2g24 −

1

2g25 −

1

2g23z + g1g2 + g1g5 −

7

2g2g5 +

5

2g2g3z + g4g3z −

1

2g5g3z

)

× 2Λ3

2

5π2v2√Aℓ, (B57)

δg(3)+(4)5 =

(

1

2g21 +

1

2g22 +

1

2g24 +

1

2g25 +

1

2g23z − 2g1g2 −

1

2g1g4 − 2g2g4 −

5

2g2g3z − 4g4g3z

)

2Λ3

2

5π2v2√Aℓ, (B58)

δg(3)+(4)3z =

(

1

2g21 +

1

2g22 +

1

2g24 +

1

2g25 +

1

2g23z − g1g2 −

1

2g1g4 − g1g5 +

1

2g2g4 + g2g5 −

5

2g2g3z +

1

2g4g5

−7

2g4g3z − g5g3z

)

2Λ3

2

5π2v2√Aℓ. (B59)

From the above results, we obtain

δga = δg(1)a + δg(2)a + δg(3)+(4)a . (B60)

Concretely,

δg1 =

(

−g1g2 −1

2g1g4 − g1g5 − g2g5 − g4g3z − g5g3z

)

2Λ3

2

5π2v2√Aℓ, (B61)

δg2 =

(

5

2g22 −

3

2g1g2 − g1g5 + 2g2g4 +

3

2g2g5 +

5

2g2g3z −

5

2g4g5 +

7

2g4g3z + g5g3z

)

2Λ3

2

5π2v2√Aℓ, (B62)

δg4 =

(

−1

2g21 −

1

2g22 −

1

2g24 −

1

2g25 −

1

2g23z + g1g2 + g1g5 −

7

2g2g5 +

5

2g2g3z + g4g3z −

1

2g5g3z

)

2Λ3

2

5π2v2√Aℓ, (B63)

δg5 =

(

1

2g21 +

1

2g22 +

1

2g24 + 3g25 +

1

2g23z − 2g1g2 −

1

2g1g4 −

5

2g1g5 − 2g2g4 +

5

2g2g5 −

5

2g2g3z +

5

2g4g5 − 4g4g3z

−5

2g5g3z

)

2Λ3

2

5π2v2√Aℓ, (B64)

δg3z =

(

1

2g21 +

1

2g22 +

1

2g24 +

1

2g25 + g23z − g1g2 −

1

2g1g4 − g1g5 −

1

2g1g3z +

1

2g2g4 + g2g5 − 2g2g3z +

1

2g4g5

−3g4g3z −3

2g5g3z

)

2Λ3

2

5π2v2√Aℓ. (B65)

15

3. Scaling transformations

The free action of fermions is

SΨ =

∫

dω

2π

d3k

(2π)3Ψ(ω,k)

(

iωγ0 + ivk1γ1 + ivk2γ2 + iAk23γ3)

Ψ(ω,k). (B66)

The fermion self-energy induced by four-fermion interactions to one-loop order vanishes. Thus, the form of action SΨ

is not changed. Employing the transformations

ω = ω′e−ℓ, (B67)

k1 = k′1e−ℓ, (B68)

k2 = k′2e−ℓ, (B69)

k3 = k′3e− ℓ

2 , (B70)

v = v′, (B71)

A = A′, (B72)

Ψ = Ψ′e9

4ℓ, (B73)

the action becomes

SΨ′ =

∫

dω′

2π

d3k′

(2π)3Ψ′(ω′,k′)

(

iω′γ0 + iv′k′1γ1 + iv′k′2γ2 + iA′k′23 γ3)

Ψ′(ω′,k′), (B74)

which has the same form as the original action.The original action of four-fermion interactions takes the form

SΨ4 =∑

a=1,2,4,5,3z

ga

∫

dω1

2π

d3k1

(2π)3dω2

2π

d3k2

(2π)3dω3

2π

d3k3

(2π)3Ψ(ω1,k1)ΓaΨ(ω2,k2)Ψ(ω3, k3)Γa

×Ψ(ω1 − ω2 + ω3,k1 − k2 + k3). (B75)

Including the one-loop order correction, the action becomes

SΨ4 =∑

a=1,2,4,5,3z

(ga + δga)

∫

dω1

2π

d3k1

(2π)3dω2

2π

d3k2

(2π)3dω3

2π

d3k3

(2π)3Ψ(ω1,k1)ΓaΨ(ω2,k2)Ψ(ω3, k3)Γa

×Ψ(ω1 − ω2 + ω3,k1 − k2 + k3). (B76)

Utilizing the transformations Eqs. (B67)-(B70) and (B73), we get

SΨ′4 =∑

a=1,2,4,5,3z

(ga + δga) e− 3

2ℓ

∫

dω′1

2π

d3k′1

(2π)3dω′

2

2π

d3k′2

(2π)3dω′

3

2π

d3k′3

(2π)3Ψ′(ω′

1,k′1)ΓaΨ

′(ω′2,k

′2)Ψ

′(ω′3, k

′3)Γa

×Ψ′(ω′1 − ω′

2 + ω′3,k

′1 − k

′2 + k

′3). (B77)

Let

g′a = (ga + δga) e− 3

2ℓ ≈ ga −

3

2gaℓ+ δga, (B78)

we obtain

SΨ′4 =∑

a=1,2,4,5,3z

g′a

∫

dω′1

2π

d3k′1

(2π)3dω′

2

2π

d3k′2

(2π)3dω′

3

2π

d3k′3

(2π)3Ψ′(ω′

1,k′1)ΓaΨ

′(ω′2,k

′2)Ψ

′(ω′3, k

′3)Γa

×Ψ′(ω′1 − ω′

2 + ω′3,k

′1 − k

′2 + k

′3)

which recovers the original form of the action.From Eq. (B78), we get the RG equation for ga as following

dgadℓ

= −3

2ga +

dδgadℓ

. (B79)

16

Substituting Eqs. (B61)-(B65) into Eq. (B79), we find

dg1dℓ

= −3

2g1 −

2

5g1

(

g2 +1

2g4 + g5

)

− 2

5(g2g5 + g4g3z + g5g3z) , (B80)

dg2dℓ

= −3

2g2 + g22 + g2

(

−3

5g1 +

4

5g4 +

3

5g5 + g3z

)

− 2

5g1g5 + g4

(

−g5 +7

5g3z

)

+2

5g5g3z, (B81)

dg4dℓ

= −3

2g4 −

1

5g24 −

1

5

(

g21 + g22 + g25 + g23z)

+2

5g4g3z +

2

5g1 (g2 + g5) + g2

(

−7

5g5 + g3z

)

− 1

5g5g3z, (B82)

dg5dℓ

= −3

2g5 +

6

5g25 +

1

5

(

g21 + g22 + g24 + g23z)

+ g5 (−g1 + g2 + g4 − g3z)−2

5g1

(

2g2 +1

2g4

)

− g2

(

4

5g4 + g3z

)

−8

5g4g3z, (B83)

dg3zdℓ

= −3

2g3z +

2

5g23z +

1

5

(

g21 + g22 + g24 + g25)

− 2

5g3z

(

1

2g1 + 2g2 + 3g4 +

3

2g5

)

− 2

5g1

(

g2 +1

2g4 + g5

)

+2

5g2

(

1

2g4 + g5

)

+1

5g4g5. (B84)

The redefinition

Λ3

2 ga

π2v2√A

→ ga, (B85)

has been employed.

Appendix C: Susceptibility of source terms

We consider the Lagrangian for the source terms as following

Ls = ∆1Ψγ0Ψ+∆2ΨΨ +∆3⊥

2∑

j=1

Ψγ0γjΨ+∆3zΨγ0γ3Ψ+∆4Ψγ0γ5Ψ+∆5Ψiγ5Ψ+∆6⊥

∑

<<lk>>

(

ΨiγlγkΨ)

+∆6zΨiγ1γ2Ψ+∆7⊥

2∑

j=1

Ψiγ5γjΨ+∆7zΨiγ5γ3Ψ+∆8⊥

2∑

j=1

ΨiγjΨ+∆8zΨiγ3Ψ+∆SΨ†iγ0γ5γ2Ψ

∗

+∆opΨ†iγ0γ2Ψ

∗ +∆V,1Ψ†γ3Ψ

∗ +∆V,2Ψ†iγ0γ5Ψ

∗ +∆V,3Ψ†γ1Ψ

∗ +∆V,0Ψ†iγ0γ2γ3Ψ

∗. (C1)

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

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. Theone-loop correction for the source term ∆X from Fig. 7(a) is given by

W(1)∆X

= −2∆XgX(

ΨΓXΨ)

∑

a=1,2,4,5,3z

ga

∫ +∞

−∞

dω

2π

∫ ′ d3k

(2π)3Tr [ΓXG0(iω,k)ΓaG0(iω,k)] . (C2)

17

The one-loop correction for the source term ∆X resulting from Fig. 7(b) can be written as

W(2)∆X

= 2∆X

∑

a=1,2,4,5,3z

ga

∫ +∞

−∞

dω

2π

∫ ′ d3k

(2π)3(

ΨΓaG0(iω,k)ΓXG0(iω,k)ΓaΨ)

. (C3)

Substituting Eq. (B1) into Eq. (C2), we find

W(1)∆1

= 0, (C4)

W(1)∆2

= ∆2g22Λ

3

2

π2v2√Aℓ(

ΨΨ)

, (C5)

W(1)∆3⊥

= 0, (C6)

W(1)3z = ∆3zg3z

2Λ3

2

5π2v2√Aℓ(

Ψγ0γ3Ψ)

, (C7)

W(1)∆4

= 0, (C8)

W(1)∆5

= ∆5g52Λ

3

2

π2v2√Aℓ(

Ψiγ5Ψ)

, (C9)

W(1)∆6⊥

= 0, (C10)

W(1)∆6z

= 0, (C11)

W(1)∆7⊥

= 0, (C12)

W(1)∆7z

= 0, (C13)

W(1)∆8⊥

= 0, (C14)

W(1)∆8z

= 0. (C15)

Substituting Eq. (B1) into Eq. (C3), we obtain

W(2)∆1

= 0, (C16)

W(2)∆2

=1

2∆2 (−g1 − g2 + g4 + g5 + g3z)

Λ3

2

π2v2√Aℓ(

ΨΨ)

, (C17)

W(2)∆3⊥

= ∆3⊥ (−g1 + g2 + g4 − g5 + g3z)Λ

3

2

5π2v2√Aℓ

2∑

j=1

(

Ψγ0γjΨ)

, (C18)

W(2)∆3z

= ∆3z (−g1 + g2 + g4 − g5 − g3z)Λ

3

2

10π2v2√Aℓ(

Ψγ0γ3Ψ)

, (C19)

W(2)∆4

= 0, (C20)

W(2)∆5

= ∆5 (−g1 + g2 + g4 − g5 − g3z)Λ

3

2

2π2v2√Aℓ(

Ψiγ5Ψ)

, (C21)

W(2)∆6⊥

= ∆6⊥ (−g1 − g2 + g4 + g5 − g3z)Λ

3

2

5π2v2√Aℓ

∑

<<lk>>

(

ΨiγlγkΨ)

, (C22)

W(2)∆6z

= ∆6z (−g1 − g2 + g4 + g5 + g3z)Λ

3

2

10π2v2√Aℓ(

Ψiγ1γ2Ψ)

, (C23)

W(2)∆7⊥

= ∆7⊥ (−g1 − g2 − g4 − g5 + g3z)3Λ

3

2

10π2v2√Aℓ

2∑

j=1

(

Ψiγ5γjΨ)

, (C24)

W(2)∆7z

= −∆7z (g1 + g2 + g4 + g5 + g3z)2Λ

3

2

5π2v2√Aℓ(

Ψiγ5γ3Ψ)

, (C25)

W(2)∆8⊥

= ∆8⊥ (−g1 + g2 − g4 + g5 − g3z)3Λ

3

2

10π2v2√Aℓ

2∑

j=1

(

ΨiγjΨ)

, (C26)

18

W(2)∆8z

= ∆8z (−g1 + g2 − g4 + g5 + g3z)2Λ

3

2

5π2v2√Aℓ(

Ψiγ3Ψ)

. (C27)

From

W∆X= W

(1)∆X

+W(2)∆X

, (C28)

we arrive

W∆X= δ∆X

(

ΨΓXΨ)

. (C29)

The parameters δ∆X are given by

δ∆1 = 0, (C30)

δ∆2 = ∆2 (−g1 + 3g2 + g4 + g5 + g3z)Λ

3

2

2π2v2√Aℓ, (C31)

δ∆3⊥ = ∆3⊥ (−g1 + g2 + g4 − g5 + g3z)Λ

3

2

5π2v2√Aℓ, (C32)

δ∆3z = ∆3z (−g1 + g2 + g4 − g5 + 3g3z)Λ

3

2

10π2v2√Aℓ, (C33)

δ∆4 = 0, (C34)

δ∆5 = ∆5 (−g1 + g2 + g4 + 3g5 − g3z)Λ

3

2

2π2v2√Aℓ, (C35)

δ∆6⊥ = ∆6⊥ (−g1 − g2 + g4 + g5 − g3z)Λ

3

2

5π2v2√Aℓ, (C36)

δ∆6z = ∆6z (−g1 − g2 + g4 + g5 + g3z)Λ

3

2

10π2v2√Aℓ, (C37)

δ∆7⊥ = ∆7⊥ (−g1 − g2 − g4 − g5 + g3z)3Λ

3

2

10π2v2√Aℓ, (C38)

δ∆7z = ∆7z (−g1 − g2 − g4 − g5 − g3z)2Λ

3

2

5π2v2√Aℓ, (C39)

δ∆8⊥ = ∆8⊥ (−g1 + g2 − g4 + g5 − g3z)3Λ

3

2

10π2v2√Aℓ, (C40)

δ∆8z = ∆8z (−g1 + g2 − g4 + g5 + g3z)2Λ

3

2

5π2v2√Aℓ. (C41)

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 thecorrection to source terms. The correction can be expresses as

W∆Y= 2∆Y

∑

a=1,2,4,5,3z

ga

∫ +∞

−∞

dω

2π

∫ ′ d3k

(2π)3(

Ψ†ΓTaG

T0 (iω,k)ΓY G0(−iω,−k)ΓaΨ

∗)

, (C42)

where T represents transposition. Substituting Eq. (B1) into Eq. (C42), we get

W∆Y= δ∆Y

(

Ψ†ΓY Ψ)

, (C43)

19

where

δ∆S = ∆S (g1 − g2 + g4 + g5 − g3z)2Λ

3

2

5π2v2√Aℓ, (C44)

δ∆op = ∆op (g1 + g2 + g4 − g5 + g3z)2Λ

3

2

5π2v2√Aℓ, (C45)

δ∆V,1 = ∆V,1 (g1 + g2 − g4 + g5 + g3z)Λ

3

2

5π2v2√Aℓ, (C46)

δ∆V,2 = ∆V,2 (g1 + g2 − g4 + g5 + g3z)Λ

3

2

5π2v2√Aℓ, (C47)

δ∆V,3 = ∆V,3 (g1 + g2 − g4 + g5 − g3z)Λ

3

2

2π2v2√Aℓ, (C48)

δ∆V,0 = ∆V,0 (g1 − g2 − g4 − g5 + g3z)Λ

3

2

20π2v2√Aℓ. (C49)

3. Derivation of the RG equations for source terms

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

Ss = ∆X

∫

dω

2π

d3k

(2π)3Ψ(ω,k)ΓXΨ(ω,k). (C50)

Considering the one-loop order corrections, we obtain

Ss = (∆X + δ∆X)

∫

dω

2π

d3k

(2π)3Ψ(ω,k)ΓXΨ(ω,k). (C51)

Using the transformations Eqs. (B67)-B70) and (B73), we can get

Ss = (∆X + δ∆X) eℓ∫

dω′

2π

d3k′

(2π)3Ψ′(ω′,k′)ΓXΨ′(ω′,k′)

≈ (∆X +∆Xℓ+ δ∆X)

∫

dω′

2π

d3k′

(2π)3Ψ′(ω′,k′)ΓXΨ′(ω′,k′). (C52)

Let

∆′X = ∆X +∆Xℓ+ δ∆X , (C53)

the action can be further written as

Ss = ∆′X

∫

dω′

2π

d3k′

(2π)3Ψ′(ω′,k′)ΓXΨ′(ω′,k′), (C54)

which recovers the form of the original action. We can easily find that the RG equation for ∆X is

d∆X

dℓ= ∆X +

dδ∆X

dℓ. (C55)

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

d∆Y

dℓ= ∆Y +

dδ∆Y

dℓ. (C56)

20

Substituting Eqs. (C30)-(C41) into Eq. (C55), and substituting Eqs. (C44)-(C49) into Eq. (C56), we get the RGequations

β1 = 0, (C57)

β2 =1

2(−g1 + 3g2 + g4 + g5 + g3z) , (C58)

β3⊥ =1

5(−g1 + g2 + g4 − g5 + g3z) , (C59)

β3z =1

10(−g1 + g2 + g4 − g5) , (C60)

β4 = 0, (C61)

β5 =1

2(−g1 + g2 + g4 + 3g5 − g3z) , (C62)

β6⊥ =1

5(−g1 − g2 + g4 + g5 − g3z) , (C63)

β6z =1

10(−g1 − g2 + g4 + g5 + g3z) , (C64)

β7⊥ =3

10(−g1 − g2 − g4 − g5 + g3z) , (C65)

β7z =2

5(−g1 − g2 − g4 − g5 − g3z) , (C66)

β8⊥ =3

10(−g1 + g2 − g4 + g5 − g3z) , (C67)

β8z =2

5(−g1 + g2 − g4 + g5 + g3z) , (C68)

βS =2

5(g1 − g2 + g4 + g5 − g3z) , (C69)

βop =2

5(g1 + g2 + g4 − g5 + g3z) , (C70)

βV,1 =1

5(g1 + g2 − g4 + g5 + g3z) , (C71)

βV,2 =1

5(g1 + g2 − g4 + g5 + g3z) , (C72)

βV,3 =1

2(g1 + g2 − g4 + g5 − g3z) , (C73)

βV,0 =1

20(g1 − g2 − g4 − g5 + g3z) , (C74)

where

βX,Y =d ln(∆X,Y )

dℓ− 1. (C75)

For convenience, we show the physical meaning of different order parameters and corresponding fermion bilinear inTable II

Appendix D: Numerical Results

1. Fixed points and their properties

Solving the RG equations for ga as shown in Eqs. (B80)-(B84), we obtained the real roots as following

FP0 : (g∗1 , g∗2 , g

∗4 , g

∗5 , g

∗3z) = (0, 0, 0, 0, 0) (D1)

FP1 : (g∗1 , g∗2 , g

∗4 , g

∗5 , g

∗3z) = (0.152019, 1.25444, 0.459247,−0.561711, 0.0551435) , (D2)

FP2 : (g∗1 , g∗2 , g

∗4 , g

∗5 , g

∗3z) = (0.140905,−0.585585, 0.418385, 1.34668, 0.06996) , (D3)

FP3 : (g∗1 , g∗2 , g

∗4 , g

∗5 , g

∗3z) = (−0.100015, 0.575751,−0.61003, 0.775675, 0.199924) , (D4)

21

TABLE II: Physical meaning of different order parameters and the corresponding fermion bilinears.

Order Parameter Fermion Bilinear Physical meaning∆1 Ψγ0Ψ chemical potential∆2 ΨΨ scalar mass∆3⊥

∑

j=1,2

Ψγ0γjΨ spin-orbit coupling within xy plane

∆3z Ψγ0γ3Ψ spin-orbit coupling along z axis∆4 Ψγ0γ5Ψ axial chemical potential∆5 Ψiγ5Ψ pseudoscalar mass∆6⊥ Ψ (iγ2γ3 + γ3γ1)Ψ magnetization within xy plane∆6z Ψiγ1γ2Ψ magnetization along z axis∆7⊥

∑

j=1,2

Ψiγ5γjΨ axial magnetization within xy plane

∆7z Ψiγ5γ3Ψ axial magnetization along z axis∆8⊥

∑

j=1,2

ΨiγjΨ current within xy plane

∆8z Ψiγ3Ψ current along z axis∆S Ψ†iγ0γ5γ2Ψ

∗ s-wave paring∆op Ψ†iγ0γ2Ψ

∗ odd-parity pairing∆V,1 Ψ†γ3Ψ

∗ vector pairing along x axis∆V,2 Ψ†iγ0γ5Ψ

∗ vector pairing along y axis∆V,3 Ψ†γ1Ψ

∗ vector pairing along z axis∆V,0 Ψ†iγ0γ1γ3Ψ

∗ temporal vector paring

FP4 : (g∗1 , g∗2 , g

∗4 , g

∗5 , g

∗3z) = (−2.33263, 0.,−0.610178,−1.72246,−1.72246) , (D5)

FP5 : (g∗1 , g∗2 , g

∗4 , g

∗5 , g

∗3z) = (0.126936,−0.463077,−0.854005, 0.769245, 1.23232) , (D6)

FP6 : (g∗1 , g∗2 , g

∗4 , g

∗5 , g

∗3z) = (0.0860014, 1.37623, 0.236822,−0.304132, 0.0941005) , (D7)

FP7 : (g∗1 , g∗2 , g

∗4 , g

∗5 , g

∗3z) = (0.103947,−0.465334, 0.29293, 1.43995, 0.0944817) , (D8)

FP8 : (g∗1 , g∗2 , g

∗4 , g

∗5 , g

∗3z) = (−3.33745,−1.22097,−1.28072,−1.32395,−0.102973) , (D9)

FP9 : (g∗1 , g∗2 , g

∗4 , g

∗5 , g

∗3z) = (−2.68181, 1.06255,−1.97657,−1.35604,−2.41859) , (D10)

FP10 : (g∗1 , g∗2 , g

∗4 , g

∗5 , g

∗3z) = (0, 0,−1.25, 1.25, 1.25) , (D11)

FP11 : (g∗1 , g∗2 , g

∗4 , g

∗5 , g

∗3z) = (−5.16737, 0,−4.38982,−0.777544,−0.777544) , (D12)

FP0 is the trivial Gauss fixed point. FP1-FP11 are non-trivial fixed points.

Expanding the RG equations (B80)-(B84) in the vicinity of a fixed point (g∗1 , g∗2 , g

∗4 , g

∗5 , g

∗3z), we find that

dG

dℓ= MG, (D13)

where

G =

δg1δg2δg4δg5δg3z

, (D14)

with δga = ga − g∗a. The matrix M is given by

M =

M11 M12 M13 M14 M15

M21 M22 M23 M24 M25

M31 M32 M33 M34 M35

M41 M42 M43 M44 M45

M51 M52 M53 M54 M55

, (D15)

22

where

M11 = −(

3

2+

2

5g∗2 +

1

5g∗4 +

2

5g∗5

)

, (D16)

M12 = −2

5(g∗1 + g∗5) , (D17)

M13 = −(

1

5g∗1 +

2

5g∗3z

)

, (D18)

M14 = −2

5(g∗1 + g∗2 + g∗3z) , (D19)

M15 = −2

5(g∗4 + g∗5) , (D20)

M21 = −3

5g∗2 −

2

5g∗5 , (D21)

M22 = −3

2+ 2g∗2 −

3

5g∗1 +

4

5g∗4 +

3

5g∗5 + g∗3z, (D22)

M23 =4

5g∗2 − g∗5 +

7

5g∗3z, (D23)

M24 =3

5g∗2 −

2

5g∗1 − g∗4 +

2

5g∗3z, (D24)

M25 = g∗2 +7

5g∗4 +

2

5g∗5 , (D25)

M31 = −2

5g∗1 +

2

5g∗2 +

2

5g∗5 , (D26)

M32 = −2

5g∗2 +

2

5g∗1 − 7

5g∗5 + g∗3z, (D27)

M33 = −3

2− 2

5g∗4 +

2

5g∗3z, (D28)

M34 = −2

5g∗5 +

2

5g∗1 − 7

5g∗2 −

1

5g∗3z, (D29)

M35 = −2

5g∗3z +

2

5g∗4 + g∗2 −

1

5g∗5 , (D30)

M41 =2

5g∗1 − g∗5 − 4

5g∗2 −

1

5g∗4 , (D31)

M42 =2

5g∗2 + g∗5 − 4

5g∗1 −

4

5g∗4 − g∗3z, (D32)

M43 =2

5g∗4 + g∗5 − 1

5g∗1 −

4

5g∗2 −

8

5g∗3z, (D33)

M44 = −3

2+

12

5g∗5 − g∗1 + g∗2 + g∗4 − g∗3z, (D34)

M45 =2

5g∗3z − g∗5 − g∗2 −

8

5g∗4 , (D35)

M51 =2

5g∗1 −

1

5g∗3z −

2

5g∗2 −

1

5g∗4 − 2

5g∗5 , (D36)

M52 =2

5g∗2 −

4

5g∗3z −

2

5g∗1 +

1

5g∗4 +

2

5g∗5 , (D37)

M53 =2

5g∗4 −

6

5g∗3z −

1

5g∗1 +

1

5g∗2 +

1

5g∗5 , (D38)

M54 =2

5g∗5 −

3

5g∗3z −

2

5g∗1 +

2

5g∗2 +

1

5g∗4 , (D39)

M55 = −3

2+

4

5g∗3z −

1

5g∗1 −

4

5g∗2 −

6

5g∗4 −

3

5g∗5 . (D40)

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 [32, 34]. 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.

23

Substituting the values of g∗a at each fixed point into the expression M , we calculate the corresponding eigenvaluesof M . The eigenvalues for the fixed points are shown in Table III. 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 III: 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

ν−1 = 1.5 (D41)

Substituting the values of g∗a with i = 1, 2, 4, 5, 3z into Eqs. (C57)-(C74), we can get values of βX,Y for different∆X,Y , which are shown in Tabel IV. For a QCP, the largest value of βX,Y is marked by the bold style. It representsthat the fixed point is a QCP to the new state in which ∆X,Y acquires finite value. FP1, FP2, FP4, and FP5 arecorresponding to QCPs to a state in which ∆2, ∆5, ∆7z and ∆8z acquire finite value respectively. For FP3, it standsfor a QCP to a state in which ∆2 and ∆5 become finite generally. This state represents an axionic insulator whoseorder parameter can be written as 〈Ψ (cos(θ) + iγ5 sin(θ)) Ψ〉 [33].

TABLE IV: βX,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 FP11β1 0 0 0 0 0 0 0 0 0 0 0β2 1.78199 -0.0313166 1.09642 -0.861228 -0.184302 2.03474 0.163708 -1.51656 0.0591369 0.625 -0.388772β3⊥ 0.435705 -0.316965 -0.102003 0.344491 -0.196188 0.385057 -0.324365 0.411346 0.141049 -0.25 0.155509β3z 0.212338 -0.165478 -0.070994 0.344491 -0.221326 0.183119 -0.171631 0.21597 0.312383 -0.25 0.155509β4 0 0 0 0 0 0 0 0 0 0 0β5 -0.0893033 1.83099 1.09642 -0.861228 -0.184302 0.260278 1.97451 -1.51656 0.0591369 0.625 -0.388772β6⊥ -0.312814 0.427957 -0.102003 0.344491 -0.196188 -0.324729 0.399958 0.411346 0.141049 -0.25 0.155509β6z -0.145378 0.227971 -0.0110167 -0.172246 0.14837 -0.143544 0.218875 0.185078 -0.413193 0.125 -0.0777544β7⊥ -0.374656 -0.375128 -0.132437 0.882843 0.495967 -0.390247 -0.383104 2.11804 0.759982 0.375 2.86716β7z -0.543656 -0.556138 -0.336522 2.55509 -0.324568 -0.59561 -0.586391 2.90643 2.94818 -0.5 4.44491β8⊥ 0.00789679 0.0395537 0.558464 0.882843 -0.0597252 0.196553 0.144978 0.652867 2.03504 0.375 2.86716β8z 0.0546439 0.108706 0.904558 -0.20084 0.906224 0.337352 0.26889 0.788112 0.778521 1.5 3.20084βS -0.504013 0.968638 -0.284018 -1.17712 -0.290828 -0.580657 0.883073 -1.84727 -1.86335 -0.5 -3.82288βop 0.993025 -0.521206 -0.284018 -1.17712 -0.290828 0.838916 -0.565572 -1.84727 -1.86335 -0.5 -3.82288βV,1 0.0881294 0.110715 0.412273 -1.03347 0.503886 0.203076 0.176024 -0.940925 -0.683464 0.75 -0.466527βV,2 0.0881294 0.110715 0.412273 -1.03347 0.503886 0.203076 0.176024 -0.940925 -0.683464 0.75 -0.466527βV,3 0.16518 0.206828 0.830758 -0.861228 0.027394 0.41359 0.345578 -2.24934 0.709928 0.625 -0.388772βV,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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