The phase diagram of a two-dimensional dirty tilted Dirac semimetal
TThe phase diagram of a two-dimensional dirty tilted Dirac semimetal
Yu-Li Lee ∗ Department of Physics, National Changhua University of Education, Changhua, Taiwan, R.O.C.
Yu-Wen Lee † Department of Applied Physics, Tunghai University, Taichung, Taiwan, R.O.C. (Dated: April 2, 2019)We investigate the effects of quenched disorder on a non-interacting tilted Dirac semimetal intwo dimensions. Depending on the magnitude of the tilting parameter, the system can have eitherFermi points (type-I) or Fermi lines (type-II). In general, there are three different types of disordersfor Dirac fermions in two dimensions, namely, the random scalar potential, the random vectorpotentials along and perpendicular to the tilting direction, and the random mass. We study theeffects of weak disorder in terms of the renormalization group, which is performed by integratingout the modes with large energies, instead of large momenta. Since the parametrization of thelow-energy degrees of freedom depends on the structure of the Fermi surface, the resulting one-looprenormalization-group equations depend on the type of tilted Dirac fermions. Whenever the disorderis a marginal perturbation, we examine its role on low-energy physics by a mean-field approximationof the replica field theory or the first-order Born approximation. Based on our analysis, we suggestthe phase diagrams of a two-dimensional tilted Dirac fermion in the presence of different types ofdisorder.
I. INTRODUCTION
The nodal semimetals, including the Dirac and theWeyl fermions, in solid-state materials have attractedintense theoretical and experimental interests in recentyears . On account of the linear quasiparticle dis-persion, which results in a vanishing density of states(DOS) at the Fermi level, and the non-trivial topologi-cal properties, these materials exhibit many interestingphenomena that are different from the ordinary metalsdescribed by the Fermi-liquid (FL) theory. Examplesof two-dimensional (2D) Dirac semimetals (DSMs) in-clude the graphene and the surface states of a three-dimensional (3D) topological insulator . Very recently,the Weyl semimetals (WSMs) have also been detectedexperimentally in the non-centrosymmetric but time-reversal preserving materials, such as TaAs, NbAs, TaP,and NbP .Due to the lack of a fundamental Lorentz symmetry,the spectra of DSMs (WSMs) realized in solid-state ma-terials do not have to be isotropic. In particular, they canbe tilted . Depending on the magnitude of the tiltingangle, there are two types of DSMs (WSMs). For type-IDSMs (WSMs), the Dirac (Weyl) cone is only moderatelytilted such that they still have a point-like Fermi surfaceat the Dirac (Weyl) node. When the tilting angle is largeenough, the electron and hole Fermi surfaces can coexistwith the band-touching Dirac/Weyl nodes. This leadsto a new kind of materials, which are commonly referredto as type-II DSMs (WSMs) . In three dimensions, thetilted Weyl cones were proposed to be realized in a ma-terial WTe , a spin-orbit coupled fermionic superfluidwith the Fulde-Ferrell ground state , or a cold-atom op-tical lattice . On the other hand, in two dimensions,the tilted Dirac cones were proposed to be realized in amechanically deformed graphene and the organic com- pound α -(BEDT-TTF) I . Recently, type-II Diracfermions are experimentally discovered in two materials:PdTe and PtTe .Since the disorder is ubiquitous in condensed-mattersystems, its role on the nodal semimetals is an inter-esting topic from both the theoretical and experimentalperspectives. For the non-tilting case, in a pioneeringwork , Fradkin showed that unlike the usual FL, a 3DDSM is stable against the presence of a weak randomscalar potential. When the disorder strength is beyondsome critical value, there is a quantum phase transition(QPT) which separates the DSM from a diffusive metal(DM) with a non-zero DOS at the Fermi level. In twodimensions, the system behaves more like an ordinarydisordered FL. That is, the ground state is always local-ized so that the system is an insulator.The effects of a random scalar potential on the type-IWSM have been studied in Refs. 25 and 26. The re-sults are similar to those of the untilted case. That is,the semimetallic phase remains stable for weak disorder.However, the presence of tilt decreases the region occu-pied by the semimetallic phase due to the reduction ofthe critical disorder strength for the QPT to the DM. Inthe mean time, the disorder increases effective tilt of thequasiparticle excitations in the semimetallic phase.In the present paper, we would like to study theground state of a non-interacting 2D tilted Dirac fermionsin the presence of quenched disorder. We adopt therenormalization-group (RG) method which is performedby integrating out disorder at each order in the perturba-tion theory. It is known from the study of the FL theorythat the RG transformation must scale toward the Fermisurface, instead of the origin in the momentum space .In the previous study of the Coulomb interaction effectson the tilted Dirac fermions , we have employed a reg-ularization scheme in which the modes with large en- a r X i v : . [ c ond - m a t . d i s - nn ] A p r ergies, not the large momenta, are integrated out. Fortype-I Dirac fermions, this method yields the same re-sults as those by integrating out the modes with largemomenta. This is because the Fermi surface is point-likeso that large momenta imply large energies. For type-IIDirac fermions, however, this scheme is necessary sincethe Fermi surface becomes extended.For 2D Dirac fermions, there are three types of dis-order: the random scalar potential (RSP), the randomvector potentials along and perpendicular to the tiltingdirection (referred to as the x -RVP and y -RVP, respec-tively), and the random mass (RM) . For type-I DSMs,the effects of all three types of disorder have been ex-amined in Ref. 30 by an RG analysis of a replica fieldtheory. Although we have performed the RG transfor-mations on different objects and adopted different regu-larization schemes (and thus the resulting RG equationsmay be distinct), the RG flows of various types of disor-der strengths in type-I DSMs are similar. However, theinterpertation of the resulting ground state is distinct insome situations (see below). On the other hand, the ef-fects of quenched disorder on type-II DSMs has not beenstudied before. Our main findings are summarized inFigs. 5, 8, 10, and Table I. We describe them briefly inthe following.(i) For the weak RSP or x -RVP, the fermion-disordercoupling flows to strong disorder strength at low ener-gies for both types of Dirac fermions. We assert thatthe corresponding ground states are insulating for bothcases. The phase diagram is shown in Fig. 5. For type-IDSMs, our result is in contrast with the previous work ,where it was claimed, based on the analysis of the fermionspectrum of the kinetic energy part of the renormalizedHamiltonian, that the ground state should be a DM witha bulk Fermi arc. In our opinion, this claim can be madeonly when the fermion-disorder strength is marginal orirrelevant.(ii) For the weak y -RVP, the fermion-disorder couplingin type-I Dirac fermions is marginal and the resultingground state is a semimetal (SM) with dynamical crit-ical exponent z >
1. These results are consistent withthose of Ref. 30. However, we further perform a replicamean-field analysis to study the effects of the marginalfermion-disorder coupling, which shows that there is acritical disorder strength, beyond which, we obtain a so-lution corresponding to the DM. It follows from the gen-eral consideration on the fluctuation effects around themean-field solution, which are described by a 2D gen-eralized nonlinear σ model , we assert that this DMis unstable toward an insulating state. Thus, there aretwo phases for the type-I DSM: the SM at weak disor-der and the insulating phase beyond the critical disorderstrength. Since the critical disorder strength is a decreas-ing function of the tilting angle, the SM is, in fact, fragileat moderate magnitude of the tilting angle.For type-II Dirac fermions, the fermion-disorder cou-pling flows to strong disorder strength at low energies.Hence, we expect that the resulting ground state is insu- lating. By combining these results, the phase diagram inthe presence of weak y -RVP is shown in Fig. 8.(iii) Finally, for the weak RM, the fermion-disordercoupling in the type-I DSM is marginally irrelevant.Moreover, the effective tilt is suppressed at low energiesso that the ground state is an untilted DSM. These areidentical to the results of Ref. 30.On the other hand, for type-II Dirac fermions, thefermion-disorder coupling is marginal and the dynami-cal critical exponent z >
1. By calculating the fermionself-energy within the first-order Born approximation, wefind that the quasiparticles acquire a non-zero mean-freetime. This suggests that this state is a DM. Based onthe conventional wisdom , this DM is unstable in thepresence of arbitrarily weak disorder and the ground stateis insulating.Since in the presence of weak RM, the ground state oftype-I Dirac fermions is an untilted DSM and the type-IIDirac fermion is in an insulating phase, we expect thatthere is a DSM-Insulator transition upon varying the tilt-ing angle for a fixed disorder strength. The schematicphase diagram at weak disorder is shown in Fig. 10.The organization of the rest of the paper is as follows.The model is defined and discussed in Sec. II. We presentthe one-loop RG equations and its implications in Secs.III, IV, and V for the RSP (and x -RVP), y -RVP, and RM,respectively. The last section is devoted to conclusivediscussions. The derivation of the one-loop RG equationsare put in the appendix. II. THE MODEL
We first introduce the minimal model of a disorderedtilted DSM whose Hamiltonian is given by H = H + H diswhere H = (cid:88) ξ,σ, p ˜ ψ † ξσ ( p )( ξwv p + ξv p σ + v p σ ) ˜ ψ ξσ ( p ) , (1)describes a non-interacting tilted DSM . Here ξ = ± σ = ± σ , , are the Pauli matrices describ-ing the conduction-valence band degrees of freedom. Thefermionic fields ˜ ψ ξσ ( p ) and ˜ ψ † ξσ ( p ) obey the canonical an-ticommutation relations. Without loss of generality, wetake the velocities v , v >
0. The dimensionless quan-tity w is called the tilting parameter. The Dirac cone istilted along the x -axis when w (cid:54) = 0. | w | < | w | > H is invariant against the particle-hole(PH) transformation˜ ψ ξσ ( p ) → σ ˜ ψ ∗ ξσ ( − p ) , (2)when the chemical potential µ = 0. This PH symmetryforbids terms like ˜ ψ † ξσ ˜ ψ ξσ or ˜ ψ † ξσ σ / ˜ ψ ξσ .The spectrum of H is (cid:15) ± ( p ) = ξwv p ± (cid:113) v p + v p , (3)for each valley. Here we have set the energy of the Diracpoint to be zero. When µ = 0, the Fermi surface fortype-I Dirac fermions consists of a single point for eachvalley, while it consists of two straight lines:˜ p = ± ˜ w ˜ p , (4)for type-II Dirac fermions, where ˜ p a = v a p a with a = 1 , w = √ w −
1. One may regard each line as a branchof the Fermi surface, and thus the + and − signs arethe labels of the branches. The Fermi-surface topologychanges from | w | < | w | > | w | = 1 corresponds tothe Lifshitz transition point at which the Fermi surfacereduces to a single line, given by p = 0 for the presentmodel.The Hamiltonian H dis, describing the coupling be-tween the Dirac fermions and a random field A ( r ), isof the form H dis = − (cid:88) ξ,σ (cid:90) d xψ † ξσ Γ ψ ξσ A ( r ) , (5)where ψ ξσ ( r ) is the inverse Fourier transform of ˜ ψ ξσ ( p ).The random field A ( r ) is nonuniform and random inspace, but constant in time. Thus, it mixes up the mo-menta but not the frequencies. We further assume thatit is a quenched, Gaussian white-noise field with the cor-relation functions: (cid:104) A ( r ) (cid:105) = 0 , (cid:104) A ( r ) A ( r ) (cid:105) = ∆ δ ( r − r ) . (6)and the variance ∆ is chosen to be dimensionless.In two dimensions, there are three types of disorder ,corresponding to Γ = u σ , Γ = u , σ , , and Γ = u σ ,provided that the random field does not mix the Diracfermions with different spins and valley indices, where σ is the 2 × u i with i = 0 , , , u i has the dimension of speed. Γ = u σ ,Γ = u σ , Γ = u σ , and Γ = u σ describe the RSP,the x -RVP, the y -RVP, and the RM, respectively. For the2D materials like graphene, the RSP can be produced byadsorbed atoms and vacancies, the RVP comes from thespatial distortion of the 2D sheet by ripples and theRM can be introdcued by the underlying substrate . Al-though the RSP and RVP break the PH symmetry for agiven impurity configuration, they preserve this symme-try on average.We will see later that within our model, the RSP andthe x -RVP will mix at the one-loop order as long as w (cid:54) = 0. (That is, the RSP and the RVP in the tiltingdirection will generate each other under the RG trans-formations.) Thus, the two types of disorder must beconsidered together. On the other hand, the y -RVP and the RM can exist on its own without generating othertypes of disorder. Therefore, we will study the effects ofeach of them separately.The other effect arising from a non-zero tilting parame-ter w is that the term ψ † ξσ σ ∂ τ ψ ξσ will be generated .Thus, the working action S in the imaginary-time formu-lation can be written as S = (cid:88) ξ,σ (cid:90) dτ d x ( L + L i ) , (7)where L = ψ † ξσ [(1+ λσ ) ∂ τ − iξv ( w + σ ) ∂ − iv σ ∂ ] ψ ξσ , (8)describes the non-interacting tilted Dirac fermions and L i is the coupling to the random field. For the RSP or x -RVP L i = − ψ † ξσ Γ ψ ξσ A ( r ) , (9)with Γ = u σ + u σ , and L i = − u j ψ † ξσ σ j ψ ξσ A ( r ) , (10)for the y -RVP ( j = 2) and the RM ( j = 3).In the following, we would like to study the effects of L i on the system with the help of the RG. Instead ofintegrating out the random field A to obtain a replicafield theory, we will integrate out the disorder at eachorder in the perturbation theory. This provides us sometechnical advantages.To properly perform the RG transformations such thatthey scale toward the Fermi surface, we parametrize thelow-energy degrees of freedom by their energies and anadditional dimensionless parameter. Given an energy E ,the equal-energy curve is (cid:15) ± ( p ) = E . For type-I Diracfermions ( | w | < p = − ξw − w E + | E | − w cos θ , ˜ p = | E |√ − w sin θ , (11)where 0 ≤ θ < π . On the other hand, for type-II Diracfermions ( | w | > p = ξww − E ± | E | w − θ , ˜ p = | E |√ w − θ , (12)where −∞ < θ < + ∞ . The + and − signs correspondto the right and the left branches of the hyperbola, re-spectively.To proceed, we separate the fermion fields ψ ξσ into theslow and fast modes. The slow modes ψ ξσ< and the fastmodes ψ ξσ> contain excitations in the energy range | E | < Λ /s and the energy shell Λ /s < | E | < Λ, respectively,where Λ is the UV cutoff in energies and s = e l > E → e − l E , θ → θ , τ → e zl τ ,ψ ξσ< → Z − / ψ ψ ξσ , A → e − l A , (13)to bring the term ψ † ξσ< ∂ τ ψ ξσ< in the action back to theoriginal form. In this way, we obtain a set of one-loopRG equations for the parameters in the action S . We willlist the one-loop RG equations for each type of disorderin the following sections, and leave the details of theirderivation to the appendix. III. THE RSP AND x -RVPA. Type-I DSMs We first consider the RSP and x -RVP. For type-I Diracfermions, the renormalized parameters are given by w (cid:48) = w ,v (cid:48) , v , = 1 + (cid:20) z − − ∆(1 − wλ )( u + u − wu u )2 πv v (1 − w ) / (cid:21) l + O ( l ) ,λ (cid:48) = λ − ∆[( w + λ )( u + u ) − wλ ) u u ]2 πv v (1 − w ) / × (1 − wλ ) l + O ( l ) ,u (cid:48) = u + (cid:20) z − − ∆(1 − wλ )( u + u − wu u )2 πv v (1 − w ) / (cid:21) u l + ∆[ u − wu − wu u + (1 + 2 w ) u u ]2 πv v (1 − w ) / l + O ( l ) ,u (cid:48) = u + (cid:20) z − − ∆(1 − wλ )( u + u − wu u )2 πv v (1 − w ) / (cid:21) u l − ∆[ wu − w u + 3 wu u − (2 + w ) u u ]2 πv v (1 − w ) / l + O ( l ) . If we choose v , to be RG invariants, then we have z = 1 + ∆(1 − wλ )( u + u − wu u )2 πv v (1 − w ) / , (14) FIG. 1: The RG flow of γ l and γ l for type-I DSMs with w = 0 .
3. The fixed line γ = wγ is IR unstable. which leads to λ (cid:48) = λ − ∆[( w + λ )( u + u ) − wλ ) u u ]2 πv (1 − w ) / × (1 − wλ ) l + O ( l ) ,u (cid:48) = u + ∆[ u − wu − wu u + (1 + 2 w ) u u ]2 πv (1 − w ) / l + O ( l ) ,u (cid:48) = u − ∆[ wu − w u + 3 wu u − (2 + w ) u u ]2 πv (1 − w ) / l + O ( l ) . For simplicity, we have set v = v = v . Consequently,we get the one-loop RG equations for λ , u , and u dλ l dl = [2(1 + wλ l ) γ l γ l − ( w + λ l )( γ l + γ l )] × (1 − wλ l ) , (15)and dγ l dl = ( γ l − wγ l )( γ l − wγ l γ l + γ l ) , (16) dγ l dl = − ( γ l − wγ l )( wγ l − γ l γ l + wγ l ) , (17)where the quantities with subscript l refer to those at thescale l , the ones without the subscript refer to the barevalues ( l = 0), and γ l = (cid:115) ∆2 π (1 − w ) / u l v are the dimensionless fermion-disorder couplings.The typical RG flow of γ l and γ l is depicted in Fig.1. Equations (16) and (17) have a fixed line γ = wγ .The RSP and x -RVP correspond to the lines with γ = 0and γ = 0, respectively. The RG flow for the RSP and x -RVP are described in the following .We first consider the RSP, i.e., γ = 0. If we start with γ > γ = 0, then for w > γ l will increase and γ l will decrease with increasing l . Hence, at low energies,we get γ l → + ∞ and γ l → −∞ . On the other hand, if FIG. 2: The RG flow of λ l for a type-I DSM in the presenceof RSP ( γ = 0 = λ ) with w = 0 . γ l . The flow of λ l stops when one of γ l and γ l becomesdivergent. we start with γ < γ = 0, then for w > γ l willdecrease and γ l will increase with increasing l . Hence,at low energies, we get γ l → −∞ and γ l → + ∞ . Thismeans that the type-I DSM is unstable in the presence ofweak RSP. Since the disorder strength becomes strong atlow energies, we expect that the resulting ground stateis insulating.Next, we consider the x -RVP, i.e., γ = 0. If we startwith γ = 0 and γ >
0, then for w > γ l will de-crease and γ l will increase with increasing l . Hence, atlow energies, we get γ l → −∞ and γ l → + ∞ . On theother hand, if we start with γ = 0 and γ <
0, thenfor w > γ l will increase and γ l will decrease withincreasing l . Hence, at low energies, we get γ l → + ∞ and γ l → −∞ . This means that the type-I DSM is un-stable in the presence of weak x -RVP. Since the disorderstrength becomes strong at low energies, we expect thatthe resulting ground state is also insulating.The RG flow of λ l in the presence of the RSP, withvarious values of γ , is shown in Fig. 2. We see that λ l →− η w at some critical value l c where one of γ l and γ l becomes divergent. For given w , the value of l c decreaseswith the increasing value of γ . The situation is similarfor the x -RVP.Although our RG scheme is different from that adoptedin Ref. 30, the RG flows of the fermion-disorder cou-plings γ l , γ l and the parameter λ l are similar. How-ever, our interpretation of the resulting physics is dis-tinct from that in Ref. 30. There, the authors consideronly the kinetic energy part of the renormalized Hamil-tonian and claim that the resulting phase is a DM witha bulk Fermi arc. In our opinion, this is justified onlywhen the fermion-disorder couplings are marginal or ir-relevant. Then, they can be regarded as perturbationsand the kinetic energy part of the renormalized Hamil-tonian dominates the low-energy physics. In the presentcase, the fermion-disorder couplings are relevant opera-tors, exhibiting runaway RG flows, so that the low-energyphysics is dominated by these terms. When the disorderpotential becomes strong, we expect that the electronsare localized at the minia of the potential and the sys- FIG. 3: The RG flow of γ l and γ l for type-II DSMs with w = 1 .
3. Notice the fixed line at wγ = γ . tem is an insulator. B. Type-II DSMs
Next, we consider the type-II DSMs. Similar to type-I DSMs, we find that w (cid:48) = w to the one-loop order.Moreover, we choose the value of z to be z = 1 + ∆( w − λ )[ w ( u + u ) − u u ]2 π v v | w | ( w − , (18)so that both v and v are RG invariants. Thus, we mayset v = v = v for simplicity. Accordingly, the one-loopRG equations for λ , u , and u are dλ l dl = ( λ l − w )[(1 + wλ l )( γ l + γ l ) − w + λ l ) γ l γ l ] , (19)and dγ l dl = ( wγ l − γ l )( wγ l − γ l γ l + wγ l ) , (20) dγ l dl = − ( wγ l − γ l )( γ l − wγ l γ l + γ l ) , (21)where γ l = (cid:115) ∆2 π | w | ( w − u l v . Equations (20) and (21) have a fixed line wγ = γ . Thetypical RG flow of γ l and γ l is depicted in Fig. 3. Wenotice that the qualitative behaviors of the RG flow for γ l and γ l are similar for both types of DSMs. As aresult, similar to type-I DSMs, the ground state is aninsulator for type-II DSMs in the presence of weak RSPor x -RVP.The RG flow of λ l in the presence of the RSP, withvarious values of γ , is shown in Fig. 4. The qualita-tive behavior is similar to type-I DSMs. λ l → − η w atsome critical value l c where one of γ l and γ l becomesdivergent. For given w , the value of l c decreases withthe increasing value of γ . The case with the x -RVP issimilar. The only difference between type-I and type-II FIG. 4: The RG flow of λ l for a type-II DSM in the presenceof RSP ( γ = 0 = λ ) with w = 1 . γ l . The flow of λ l stops when one of γ l and γ l becomesdivergent.FIG. 5: The phase diagram of a non-interacting tilted DSMin the presence of RSP or x -RVP. γ and w are the (dimen-sionless) disorder strength and the tilting parameter, respec-tively. I denotes the insulator. Point A located at | w | = 1 and γ = 0 is the Lifshitz transition point, separating the type-Iand type-II DSMs in the absence of disorder. DSMs is that l c is smaller for the latter with the samevalue of γ .From the above analysis, we expect that the behaviorsof the system at finite disorder strength in the regionswith | w | < | w | > x -RVP,there is no phase transition from | w | < | w | > x -RVP is shownin Fig. 5. IV. THE y -RVPA. Type-I DSMs Next, we consider the y -RVP. For type-I DSMs, we findthat w (cid:48) = w to the one-loop order. If we choose z = 1 + ∆ u πv v (1 − w ) / (1 − wλ ) , (22)then v , v , and u are all marginal at the one-loop order.On the other hand, the one-loop RG equation for λ is dλ l dl = ∆ u πv v (1 − w ) / (1 − wλ l )( w − λ l ) . (23)The solution of Eq. (23) with the initial value λ = 0is given by λ l − wλ l − /w = w exp (cid:20) − ∆ u l πv v √ − w (cid:21) . (24)At low energies, i.e., l → + ∞ , we get λ ∗ = λ + ∞ = w .Inserting this value of λ l into Eq. (22), we get a non-universal dynamical exponent z given by z = 1+ η , where η = ∆ u πv v √ − w . (25)A non-zero value of λ ∗ will affect the dispersion relationof quasiparticles, which is determined by the poles ofthe single-particle propagator on the complex frequencyplane with the replacement v → v (cid:20) pk (cid:21) η , where p = | p | , k ∼ /a , and a is the lattce spacing.As a result, the dispersion relation of the quasiparticlesis given by E ± ( p ) = ± (cid:18) pk (cid:19) η (cid:115) v p + v − w p . (26)To sum up, in the presence of weak y -RVP, the systemis a SM with z > . To answerthis question, we employ the replica trick to map therenormalized action into a replica field theory and thenperform a mean-field analysis.The disorder-averaged replicated partition function Z in the imaginary-time formulation is given by Z = (cid:90) D [ A ] P [ A ] (cid:90) D [ χ ] D [ ¯ χ ] e − S − S i , where S = M (cid:88) a =1 (cid:88) ξ,n (cid:90) d x ¯ χ ξna [ − iω n (1 + wσ ) + ˆ h ξ ] χ ξna = (cid:88) ξ,n (cid:90) d x ¯ ψ ξn [ − iω n (1 + wσ ) + ˆ h ξ ] ⊗ I M ψ ξn ,S i = − u M (cid:88) a =1 (cid:88) ξ,n (cid:90) d x ¯ χ ξna σ χ ξna A = − u (cid:88) ξ,n (cid:90) d x ¯ ψ ξn σ ⊗ I M ψ ξn A , and P [ A ] = exp (cid:18) − (cid:90) d xA (cid:19) . In the above, ω n = (2 n + 1) πT , a is the replica index, I M is the unit matrix of dimension M in the replica space, ψ ξn = [ χ ξn , · · · , χ ξnM ] t , ¯ ψ ξn = [ ¯ χ ξn , · · · , ¯ χ ξnM ], andˆ h ξ = v (cid:18) pk (cid:19) η [ ξ ( w + σ ) p + σ p ] , in the momentum space. For simplicity, we have set v = v = v and dropped out the spin index σ . By integratingout the random field A ( r ), Z can be written as Z = (cid:90) D [ χ ] D [ ¯ χ ] exp (cid:20) − S + g (cid:90) d x ( ¯ΨΓΨ) (cid:21) , where g = (cid:112) ∆ u , Ψ = [Ψ , Ψ − ] t , ¯Ψ = [ ¯Ψ , ¯Ψ − ],Γ abmn ; ξξ (cid:48) = δ mn δ ab δ ξξ (cid:48) σ , andΨ ξ = [ · · · , ψ ξ , ψ ξ , ψ ξ − , · · · ] t , ¯Ψ ξ = [ · · · , ¯ ψ ξ , ¯ ψ ξ , ¯ ψ ξ − , · · · ] . To proceed, we make a Hubbard-Stratonovich trans-formation on the four-fermion coupling arising from theintegration over the random field:exp (cid:20) g (cid:90) d x ( ¯ΨΓΨ) (cid:21) = (cid:90) D [ Q ] exp (cid:26) − (cid:90) d x (cid:20)
12 tr Q − ig ¯Ψ Q ΓΨ (cid:21)(cid:27) , where Q † = Q . If we put an UV cutoff on the frequen-cies, i.e., − R ≤ n < R or | ω n | ≤ (2 R − πT , then thesymmetry group in the absence of the frequency term isU(2 RM ). Under the U(2 RM ) transformation,Ψ ξ → U Ψ ξ , ¯Ψ ξ → ¯Ψ ξ U † , (27)the Q field transforms as Q ξξ (cid:48) → U Q ξξ (cid:48) U † . (28)By integrating out the fermion fields, Z can be writtenas Z = (cid:90) D [ Q ] e − I [ Q ] , (29)where I [ Q ] = − tr ln (cid:104) i ˆ ω (1 + wσ ) − ˆ h + ig Q Γ (cid:105) + 12 (cid:90) d x tr Q . (30)In Eq. (30), ˆ ω abmn ; ξξ (cid:48) = ω n δ mn δ ab δ ξξ (cid:48) and ˆ h abmn ; ξξ (cid:48) =ˆ h ξ δ mn δ ab δ ξξ (cid:48) .We assume that the path integral is dominated by con-figurations of the Q field close to the homogeneous solu-tion Q of the saddle-point equation δI [ Q ] /δQ = 0. It isgiven by Q = ig (cid:90) d p (2 π ) tr[Γ ˆ G ( iω n , p )] , (31)where the trace is taken over the spinor space which de-scribes the conduction-valence band degrees of freedomand ˆ G − ( iω n , p ) = i ˆ ω (1 + wσ ) − ˆ h + ig Q Γ . (32)To solve Eq. (31), we try the ansatz g Q ξξ (cid:48) = α ξ Λ δ ξξ (cid:48) ,where Λ abmn = s n δ mn δ ab , s n = sgn( ω n ), and α ξ is a realconstant which may depend on the valley index ξ . Then, α ξ satisfies the equation α ξ = (cid:90) | p i | 0. Moreover, we have changed the variable s n p → s n p . The momentumintegral is divergent, and an UV cutoff k in momenta is introduced. We notice that this equation has real solutions.Furthermore, α ξ is independent of ξ , and thus we will set α ξ → α . Defining the dimensionless quantity ˜ α = α/ ( vk ),the above equation becomes ˜ α = (cid:90) D d x (2 π ) ig /v ( i ˜ α − r η x )(1 − w ) r η x + ( r η x − i ˜ α ) , (33)where r = (cid:112) x + x and D = { ( x , x ) || x | , | x | ≤ } . Equation (33) has a trivial solution ˜ α = 0. We would liketo search for a non-trivial real solution if it exists. It is clear that if ˜ α is a solution of Eq. (33), then − ˜ α is also asolution. Without loss of generality, we take ˜ α ≥ α = (cid:90) D d x (2 π ) g /v ˜ α [ r η x + ˜ α − (1 − w ) r η x ][(1 − w ) r η x + r η x − ˜ α ] + 4 r η x ˜ α . Now the right hand side of this equation becomes real. Therefore, a non-trivial solution satisfies this equation1 = 4 tπ (cid:90) dx dx r η x + ˜ α − (1 − w ) r η x [(1 − w ) r η x + r η x − ˜ α ] + 4 r η x ˜ α , (34) FIG. 6: The behavior of I ( s ) in the range 0 ≤ s ≤ t with | w | = 0 . 3, where s = ˜ α . For reference, I ( s ) = 1 is indicated by the dotted line. where t = g / (2 πv ) measures the disorder strength and η = t/ √ − w . Since vk can be regarded as the largestenergy scale in this problem, we must have ˜ α < s = ˜ α : I ( s )= (cid:90) dx dx (4 t/π )[ s + r η x − (1 − w ) r η x ][(1 − w ) r η x + r η x − s ] + 4 sr η x . Figure 6 shows the function I ( s ) in the range 0 ≤ s ≤ t with | w | = 0 . 3. We see that forgiven | w | , there exists a critical value t c ( | w | ) such thatwe get a nontrivial solution of ˜ α when t > t c ( | w | ). Onthe other hand, there is only a trivial solution ˜ α = 0when t < t c ( | w | ). Moreover, for a fixed value of | w | , thenontrivial solution ˜ α , if it exists, is an increasing functionof t .The mean-field solution with ˜ α (cid:54) = 0 has a nonvanishingspectral density at the Fermi level, and thus correspondsto the DM phase. Since the U(2 RM ) symmetry is bro-ken down to U( RM ) × U( RM ) when ˜ α (cid:54) = 0, there will beGoldstone modes according to the Goldstone theorem.The DM phase is stable only when it survives the fluctu-ations of these Goldstone modes. The latter is describedby a certain type of generalized non-linear σ models. TheRG analysis of the generalized non-linear σ model indi-cates that the DM phase in d = 2 is unstable toward aninsulator . On the other hand, the mean-field solutionwith ˜ α = 0 corresponds to the SM phase with z > z > y -RVP when t < t c ( | w | ).For given | w | , the critical value t c ( | w | ) is determinedby setting ˜ α = 0 in Eq. (34), yielding1 = 4 tπ (cid:90) dx dx r η x − (1 − w ) r η x [(1 − w ) r η x + r η x ] . (35) FIG. 7: The critical (dimensionless) disorder strength t c as afunction of | w | in the range 0 ≤ | w | < Equation (35) can be solved numerically, and the resultis shown in Fig. 7. We see that t c is a monotonouslydecreasing function of | w | . Moreover, t c → | w | → B. Type-II DSMs Now we consider the type-II DSMs. To the one-looporder, we find that w (cid:48) = w . It we choose z to be z = 1 + ∆ u w ( w − λ )2 π v v | w | ( w − w ( w − λ ) , (36)then both v and v are RG invariants, and λ (cid:48) = λ + ∆ u (1 − wλ )( w − λ )2 π v v | w | ( w − l + O ( λ ) ,u (cid:48) = u + ∆ u π v v | w | l + O ( λ ) . As a result, w and v , are marginal to the one-loop order.On the other hand, the one-loop RG equations for λ and u are dλ l dl = γ l (1 − wλ )( w − λ )2( w − , (37) dγ l dl = γ l , (38)where γ l = ∆ | w | π ( u l /v ) and for simplicity, we have set v = v = v . From Eq. (38), we see that the u term is arelevant perturbation. That is, the pure type-II DSM isunstable in the presence of weak y -RVP, and the groundstate is supposed to be an insulator. This is in contrastwith type-I DSMs in the presence of weak y -RVP. Conse-quently, we expect the occurrence of a QPT upon varyingthe value of | w | for a given disorder strength. A schematicphase diagram in the presence of y -RVP is shown in Fig.8. The phase boundary between the SM and insulatoris obtained from Fig. 7. According to the mean-fieldtheory, the SM-insulator transition is continuous. FIG. 8: The schematic phase diagram of a non-interactingtilted DSM in the presence of y -RVP. γ and w are the (di-mensionless) disorder strength and the tilting parameter, re-spectively. Point A located at | w | = 1 and γ = 0 is the Lifshitztransition point, separating the type-I and type-II DSMs inthe absence of disorder. V. THE RMA. Type-I DSMs Finally, we consider the RM. To the one-loop order,we find that w (cid:48) = w . If we choose z to be z = 1 + ∆ u πv v (1 − w ) / (1 − wλ ) , (39)then both v and v are RG invariants, and λ (cid:48) = λ + ∆ u πv (1 − w ) / (1 − wλ )( w − λ ) l + O ( l ) ,u (cid:48) = u − ∆ u πv √ − w l + O ( l ) . In the last two equations, we have set v = v = v forsimplicity. Hence, the one-loop RG equations for λ and u are given by dλ l dl = ∆ u πv (1 − w ) / (1 − wλ l )( w − λ l ) , (40)and du l dl = − ∆ u l πv √ − w , (41)respectively. Equation (41) has only one fixed point u =0, with z = 1. Since the right hand side in Eq. (41) isnegative, this fixed point is IR stable. In other words,the RM term is marginally irrelevant at weak disorder.Consequently, the type-I DSM is stable against the weakRM disorder.0 FIG. 9: The dispersion relations E ± ( p ) of quasiparticlesaround a Dirac node with the valley index ξ = 1 in a type-IIDSM in the presence of RM, with w = 1 . z = 1 . E ± ( p )are measured in units of | w | vk and the momentum p is mea-sured in units of k . For simplicity, we have set v = v = v . To determine the fate of λ l , we have to solve Eqs.(40) and (41). By introducing the dimensionless cou-pling γ l = ∆ π √ − w ( u l /v ) , the solution with the barevalue λ = 0 is λ l − wλ l − /w = w √ γ l . (42)From Eq. (42), we find that λ ∗ = λ + ∞ = w . Using thisvalue of λ l , the dispersion relation of quasiparticles nearthe Dirac point is given by E ± ( p ) = ± (cid:115) v p + v − w p . (43)We see that the quasiparticles can be described by theDirac fermions with an untilted and anisotropic Diraccone. This is consistent with Ref. 30.To sum up, the ground state at | w | < B. Type-II DSMs Now we consider type-II DSMs. To the one-loop order,we find that w (cid:48) = w . If we choose z to be z = 1 + ∆ u w ( w − λ )2 π v v | w | ( w − , (44) then we get v (cid:48) , = v , , u (cid:48) = u , and λ (cid:48) = λ + ∆ u (1 − wλ )( w − λ )2 π v v | w | ( w − l + O ( λ ) . As a result, w , v , , and u are all marginal to the one-loop order. On the other hand, the one-loop RG equationfor λ is dλ l dl = ∆ u (1 − wλ )( w − λ )2 π v v | w | ( w − . (45)The solution of Eq. (45) with the initial value λ = 0 is λ l − /wλ l − w = 1 w exp (cid:18) − ∆ u l π v v | w | (cid:19) . (46)Hence, we get λ ∗ = λ + ∞ = 1 /w . Inserting this valueof λ l into Eq. (44), we obtain the dynamical exponent z = 1 + η , where η = ∆ u π v v | w | . (47)For λ = λ ∗ , the dispersion relation of quasiparticles is E ± ( p ) = w (cid:18) pk (cid:19) η (cid:18) ξv p ± v √ w − p (cid:19) , (48)and k ∼ /a . A typical form of E ± ( p ) is plotted inFig. 9. We see that the quasiparticles at low energies arenot described by the Dirac fermions anymore. However,it is still a FL with an open Fermi surface given by p = ± ξ (cid:112) w − v /v ) p , (49)which consists of two straight lines for each valley.To determine the fate of this FL in the presence of aweak marginal fermion-disorder coupling, we determinethe physical properties in terms of the perturbation the-ory. This is valid when the disorder strength is weak sincethe fermion-disorder coupling is marginal. In particular,we calculate the one-loop fermion self-energy:Σ ξσ ( ip ) = − ∆ u (cid:90) d p (2 π ) ip − ξwv ( p/k ) η p [ p + iξwv ( p/k ) η p ] + [ − ip /w + ξv ( p/k ) η p ] + v ( p/k ) η p − ∆ u σ (cid:90) d p (2 π ) ip /w − ξv ( p/k ) η p [ p + iξwv ( p/k ) η p ] + [ − ip /w + ξv ( p/k ) η p ] + v ( p/k ) η p . ip → p + i + , the retarded slef-energy Σ rξσ ( p ) is given byΣ rξσ ( p ) = ∆ u k π v (1 − /w ) (cid:16) σ w (cid:17)(cid:90) d x p − ξwr η x − (cid:113) w w − r η x + i + + ∆ u k π v (1 − /w ) (cid:16) σ w (cid:17)(cid:90) d x p − ξwr η x + (cid:113) w w − r η x + i + , where x i = p i /k with i = 1 , r = (cid:112) x + x , and ˜ p = p / ( vk ). For simplicity, we have set v = v = v . As a result,its imaginary part is of the formImΣ rξσ ( p ) = − ∆ u k πv (1 − /w ) (cid:16) σ w (cid:17)(cid:90) d xδ (cid:32) ˜ p − ξwr η x − (cid:114) w w − r η x (cid:33) − ∆ u k πv (1 − /w ) (cid:16) σ w (cid:17)(cid:90) d xδ (cid:32) ˜ p − ξwr η x + (cid:114) w w − r η x (cid:33) . Setting ˜ p = 0, we find thatImΣ rξσ (0) = − ∆ u k | w | πv ( w − (cid:16) σ w (cid:17)(cid:90) d x r η δ (cid:18) x + η w ξ √ w − x (cid:19) − ∆ u k | w | πv ( w − (cid:16) σ w (cid:17)(cid:90) d x r η δ (cid:18) x − η w ξ √ w − x (cid:19) = − ∆ u k | w | − η πv ( w − − η/ (cid:16) σ w (cid:17)(cid:90) + ∞ dx x η . Since η < D/ ( vk )where D = O ( vk ) is the band width. Without loss ofgenerality, we choose D = vk . (A different choice of theratio D/ ( vk ) corresponds to the redefinition of the barevalue u .) Thus, we haveImΣ rξσ (0) = − ∆ u k | w | − η πv ( w − − η/ (cid:16) σ w (cid:17)(cid:90) dx x η = − ∆ u k | w | − η πv (1 − η )( w − − η/ (cid:16) σ w (cid:17) . This result implies that to the one-loop order, thesingle-particle Green function for quasiparticles at lowfrequencies and small momenta is of the form G − ξ ( P ) = (cid:16) σ w (cid:17)(cid:20) − ip + ξv (cid:18) pk (cid:19) η p − sgn( p )2 τ (cid:21) + v (cid:18) pk (cid:19) η p σ , (50)in the imaginary-time formulation, where the mean freetime τ is given by1 τ = ∆ u k | w | − η πv (1 − η )( w − − η/ . (51)Since the quasiparticles acquire a nonzero mean free time,the system is a DM at weak disorder strength. Accord-ing to the scaling theory of localization , a DM phase in d = 2 is unstable in the presence of weak disorderand turns into an insulator. Alternatively, we can in-vestigate the role of the marginal fermion-disorder cou-pling by a replica mean-field theory, similar to what wehave done for type-I DSMs in the presence of y -RVP. Theabove perturbative calculation suggests that the mean-field equation always has a non-zero solution such thatthe quasiparticles acquire a nonvanishing mean free time.The fluctuations around this broken-symmetry solutionare described by a generalized nonlinear σ model. In twodimensions, the nonlinear σ model has only one phase– the disordered phase, corresponding to the insulatorwithin the present context. In any case, we reach theconclusion that the ground state is insulating for | w | > | w | < | w | > 1, weconclude that there is a QPT from | w | < | w | > y -direction in the present setup)becomes singular at the phase boundary. On the otherhand, if we approach the phase boundary from the sideof the insulator, we find that 1 /τ → + ∞ at the phaseboundary. All these imply that the quantum fluctuationsare strong around the line | w | = 1 and the starting point2 FIG. 10: The schematic phase diagram of a non-interactingtilted DSM in the presence of weak RM. γ and w are the(dimensionless) disorder strength and the tilting parameter,respectively. Point A located at | w | = 1 and γ = 0 is theLifshitz transition point, separating the type-I and type-IIDSMs in the absence of disorder. The phase boundary be-tween the DSM and insulating phase is schematic. When thedisorder strength is further increased, the DSM will turn intoan insulator. we have adopted, i.e., starting from either | w | < | w | > | w | = 1 line. VI. CONCLUSIONS AND DISCUSSIONS We study the effects of various types of quenched dis-order on the non-interacting tilted Dirac fermions in twodimensions with the help of the perturbative RG. Sincethe RG transformations must scale to the Fermi surface,we parametrize the low-energy degrees of freedom ac-cording to their energies so that we can integrate out themodes with large energies properly. When the Fermi sur-face is point-like, the results are consistent with those byintegrating out the modes with large momenta. On theother hand, the answers may be different when the Fermisurface is extended. Although we focus on the 2D tiltedDSMs, it is straightforward ro extend our method to thetilted WSMs in three dimensions.The relevancy of various fermion-disorder couplingsunder the RG transformations in both types of DSMs tothe one-loop order is summarized in Table I. Wheneverthe fermion-disorder coupling is relevant, we extrapolateour one-loop RG equations to the strong disorder regimeand claim that the resulting phase is an insulator. Whenthe fermion-disorder coupling is marginal, we examineits role by using either the mean-field approximation ofa replica field theory or the first-order Born approxima-tion. If the fermion-disorder coupling is irrelevant, then disorder type-I type-IIRSP relevant relevant x -RVP relevant relevant y -RVP marginal relevantRM irrelevant marginalTABLE I: The relevancy of various fermion-disorder couplingsunder the RG transformations in both types of DSMs to theone-loop order. this phase is stable against the presence of weak disorder.When w (cid:54) = 0, the RSP and the x -RVP will generateeach other under the RG transformations even if one ofthe bare value is zero. Hence, we must consider themtogether when calculating the RG equations. In thepresence of the RSP or the x -RVP, we find that bothtypes of DSMs become insulators even at weak disor-der strength because the corresponding fermion-disordercoupling flows to strong disorder regime.For the y -RVP, we find that the system at | w | < z > 1. This SM is fragile since it becomes aninsulator at a moderate strength of disorder. Especially,the critical disorder strength vanishes as | w | → 1. On theother hand, the system is insulating at | w | > 1. Thus,we expect that there is a SM-insulator transition uponvarying | w | for a given disorder strength, which is con-tinuous according to our replica mean-field theory. Thecalculations of the critical exponents associated with thistransition are beyond the scope of the present work.For the weak RM, the system at | w | < | w | > 1. Thus, we expect thatthere is a DSM-insulator transition upon varying | w | for agiven disorder strength. The determination of the natureof this transition is beyond the scope of the present work.Moreover, on account of the strong fluctuations close tothe | w | = 1 line, our approach which starts from eitherside may fail, and there can exist other phases near the | w | = 1 line.For type-I DSMs, the effects of the quenched disorderhave been studied with a different type of RG scheme .For the y -RVP and RM, the phases we find at the weakdisorder are identical to the ones in Ref. 30. For theformer, we indicate that the SM may be unstable uponincreasing the disorder strength, which has not been ex-amined in Ref. 30. We further determine the criticaldisorder strength beyond which the SM becomes unsta-ble toward an insulator. The main difference between ourwork and Ref. 30 lies on the nature of the ground state oftype-I DSMs in the presence of RSP or x -RVP. Accordingto Ref. 30, the ground state is a DM with a bulk Fermiarc. This DM cannot be stable since the fermion-disordercoupling flows to the strong disorder regime at low ener-gies. One possibility in the strong disorder regime is aninsulating phase due to the random potential scattering.3 FIG. 11: The one-loop correction to the self-energy of Diracfermions (a) and the fermion-disorder coupling (b). The solidand the dashed lines correspond to the fermion propagatorand the disorder potential, respectively. Further studies, maybe numerics, are warranted to jus-tify the phase diagrams we have obtained in this work.In particular, the nature of the DSM-insulator transitionand the phases close to the transition line in the presenceof a weak RM are open questions. Since electrons carryelectric charges, the long-range Coulomb interaction isalways present. It is interesting to investigate how theelectron-electron interactions affect the phase diagrams.For type-I DSMs, this question has been studied in Ref.30. For type-II DSMs, however, this question remainsunanswered. Acknowledgments The works of Y.-W. Lee is supported by the Ministry ofScience and Technology, Taiwan, under the grant numberMOST 107-2112-M-029-002. Appendix A: Derivation of the one-loop RGequations Here we present the details of the derivation of theone-loop RG equations. To properly integrate out the modes with large energies, we have parametrized the low-energy degrees of freedom according to their energies, asshown in Eqs (11) and (12) for type-I and type-II DSMs,respectively. In terms of them, we write the involvedmomentum integrals as (cid:90) d ˜ p = 12 (cid:90) Λ0 EdE (1 − w ) / (cid:90) π dθ (1 − ξw cos θ )+ 12 (cid:90) − Λ | E | dE (1 − w ) / (cid:90) π dθ (1 + ξw cos θ ) , (A1)for type-I Dirac fermions, and (cid:90) d ˜ p = | w | (cid:90) Λ0 EdE ( w − / (cid:20)(cid:90) + ∞−∞ dθ (cosh θ + ξ/w )+ (cid:90) + ∞−∞ dθ (cosh θ − ξ/w ) (cid:21) + | w | (cid:90) − Λ | E | dE ( w − / (cid:20)(cid:90) + ∞−∞ dθ (cosh θ − ξ/w )+ (cid:90) + ∞−∞ dθ (cosh θ + ξ/w ) (cid:21) , (A2)for type-II Dirac fermions, where ˜ p i = v i p i with i = 1 , θ integrals for given E correspond to theintegrations over the right and the left branches of thehyperbola, respectively. In fact, it suffices to considerthe integrals over E > E < H at µ = 0. Hence, we define the momentumintegrals by Eqs. (A1) or (A2). This accounts for theprefactor 1 / ξσ ( K ) = − v v ∆ · · (cid:90) D d ˜ p (2 π ) Γ G (0) ξ ( ik , p )Γ= − ∆ v v (cid:90) D d ˜ p (2 π ) ( ik − ξw ˜ p )Γ ( k + iξw ˜ p ) + ( − iλk + ξ ˜ p ) + ˜ p − ∆ v v (cid:90) D d ˜ p (2 π ) Γ[( − iλk + ξ ˜ p ) σ + ˜ p σ ]Γ( k + iξw ˜ p ) + ( − iλk + ξ ˜ p ) + ˜ p = − ∆ v v (cid:90) D d ˜ p (2 π ) ( ik − ξw ˜ p )Γ ( k + iξw ˜ p ) + ( − iλk + ξ ˜ p ) + ˜ p − ∆ v v (cid:90) D d ˜ p (2 π ) ( − iλk + ξ ˜ p )Γ σ Γ( k + iξw ˜ p ) + ( − iλk + ξ ˜ p ) + ˜ p . where k and k are, respectively, the external frequency and the external momentum, D denotes the energy shell inthe range Λ /s < | E | < Λ, and G (0) ξ ( ip , p ) = 1 − ip + ξw ˜ p + ( − iλp + ξ ˜ p ) σ + ˜ p σ . l → D is symmetric underthe reflection ˜ p → − ˜ p . We see that Σ ξσ ( K ) depends only on k to the one-loop order, and we will denote it byΣ ξσ ( ik ). By performing the derivative expansion, we have for the RSP or x -RVPΣ ξσ ( ik ) = − ∆ v v ik I ( u + u − λu u − σ [ λ ( u + u ) − u u ])+ ∆ v v ξI [ w ( u + u ) − u u − σ ( u + u − wu u )] − ∆ v v w − λ ) ik I [ w ( u + u ) − u u − σ ( u + u − wu u )] + O ( k ) , and for the y -RVP or RM,Σ ξσ ( ik ) = − ∆ u / v v [ ik (1 + λσ ) I − ξ ( w + σ ) I + 2( w − λ ) ik ( w + σ ) I ] + O ( k ) , where I = (cid:90) D d ˜ p (2 π ) − w )˜ p + ˜ p ,I = (cid:90) D d ˜ p (2 π ) ˜ p (1 − w )˜ p + ˜ p ,I = (cid:90) D d ˜ p (2 π ) ˜ p [(1 − w )˜ p + ˜ p ] . The one-loop correction δS Γ to the fermion-disordercoupling is given by δS Γ = − ∆ v v (cid:90) D d ˜ p (2 π ) Γ G (0) ξ ( ik , p + k )Γ G (0) ξ ( ik , p )Γ= − ∆ v v (cid:90) D d ˜ p (2 π ) Γ G (0) ξ (0 , p )Γ G (0) ξ (0 , p )Γ + · · · , where · · · denotes the higher-order terms in powers of k and k , which will be ignored hereafter. For the RSP or x -RVP, we have δS Γ = − ∆ u v v ( I + 2 w I − wI σ )+ ∆ u v v [2 wI + ( I − I ) σ ]+ ∆ u u v v [2 wI + ( I − I ) σ ] − u u v v [ − wI + ( I + 2 w I ) σ ] − ∆ u u v v ( I + 2 w I − wI σ )+ 2∆ u u v v ( I − I + 2 wI σ ) . For the y -RVP and the RM, we find that δS Γ = ∆ u v v σ ( I − I ) , and δS Γ = ∆ u v v σ I , respectively, where I = (cid:90) D d ˜ p (2 π ) ˜ p [(1 − w )˜ p + ˜ p ] . The rest of the task is to calculate the four integrals I , · · · , I . The answers depend on the type of Diracfermions. We will calculate them separately in the fol-lowing. 1. Type-I DSMs We first consider type-I DSMs. In this case, we have I = l π √ − w [ F ( ξ, w ) + F ( − ξ, w )] ,I = l Λ8 π (1 − w ) / [ F ( ξ, w ) + F ( − ξ, w )] + O ( l ) ,I = l π (1 − w ) / [ F ( ξ, w ) + F ( − ξ, w )] ,I = l π √ − w [ F ( ξ, w ) + F ( − ξ, w )] , F ( ξ, w ) = (cid:90) π dθ − ξw cos θw − ξw cos θ + 1= 2 π ,F ( ξ, w ) = (cid:90) π dθ ( − ξw + cos θ )(1 − ξw cos θ ) w − ξw cos θ + 1= − πξw ,F ( ξ, w ) = (cid:90) π dθ ( − ξw + cos θ ) (1 − ξw cos θ )( w − ξw cos θ + 1) = π ,F ( ξ, w ) = (cid:90) π dθ sin θ (1 − ξw cos θ )( w − ξw cos θ + 1) = π . Consequently, we get I = l π √ − w ,I = 0 ,I = l π (1 − w ) / ,I = l π √ − w . For the RSP or x -RVP, we find that Σ ξσ ( iω ) =( − iω )(Σ σ + Σ σ ), whereΣ = ∆(1 − wλ )2 πv v (1 − w ) / ( u + u − wu u ) l , Σ = − ∆(1 − wλ )2 πv v (1 − w ) / [ w ( u + u ) − u u ] l , and δS Γ = V σ + V σ , where V = − ∆[ u − wu − wu u + (1 + 2 w ) u u ]2 πv v (1 − w ) / l ,V = ∆[ wu − w u + 3 wu u − (2 + w ) u u ]2 πv v (1 − w ) / l . Consequently, the Lagrangian density for the slow modesto the one-loop order is of the form L = (cid:88) ξ,σ ψ † ξσ< [(1 + Σ ) + ( λ + Σ ) σ ] ∂ τ ψ ξσ< − (cid:88) ξ,σ ψ † ξσ< [ iξv ( w + σ ) ∂ + iv σ ∂ ] ψ ξσ< − (cid:88) j =0 , ( u j − V j ) (cid:88) ξ,σ ψ † ξσ< σ j ψ ξσ< A ( r ) . We rescale the variables and fields according to Eq. 13to bring the term ψ † ξσ< ∂ τ ψ ξσ< back to the original form.Then, we have Z ψ = e l (1 + Σ ) , (A3) and the Lagrangian density becomes L = (cid:88) ξ,σ ψ † ξσ [1 + ( λ + Σ )(1 + Σ ) − σ ] ∂ τ ψ ξσ − Z − ψ e ( z +1) l (cid:88) ξ,σ ψ † ξσ [ iξv ( w + σ ) ∂ + iv σ ∂ ] ψ ξσ − Z − ψ e ( z +1) l (cid:88) j =0 , ( u j − V j ) (cid:88) ξ,σ ψ † ξσ σ j ψ ξσ A ( r ) . Therefore, the renormalized parameters are given by( wv ) (cid:48) = Z − ψ e ( z +1) l wv ,v (cid:48) , = Z − ψ e ( z +1) l v , ,λ (cid:48) = ( λ + Σ )(1 + Σ ) − ,u (cid:48) , = Z − ψ e ( z +1) l ( u , − V , ) , which give the equations in the main text.For the y -RVP,Σ = ∆ u πv v (1 − w ) / (1 − wλ ) l , Σ = ∆ u πv v (1 − w ) / w (1 − wλ ) l , and δS Γ = 0. On the other hand, for the RM,Σ = ∆ u πv v (1 − w ) / (1 − wλ ) l , Σ = ∆ u πv v (1 − w ) / w (1 − wλ ) l , and δS Γ = V σ , where V = ∆ u πv v √ − w l . By rescaling the variables and fields according to Eq. 13,we obtained the one-loop RG equations in the main text.6 2. Type-II DSMs Next, we consider type-II DSMs. In this case, we have I = − π (cid:90) ΛΛ /s dE/E √ w − K ( ξ, w ) + K ( − ξ, w )] − π (cid:90) − Λ /s − Λ dE/ | E |√ w − K ( ξ, w ) + K ( − ξ, w )] ,I = − π (cid:90) ΛΛ /s dE ( w − / [ K ( ξ, w ) − K ( − ξ, w )] − π (cid:90) − Λ /s − Λ dE ( w − / [ K ( − ξ, w ) − K ( ξ, w )]= 0 ,I = 18 π (cid:90) ΛΛ /s dE/E ( w − / [ K ( ξ, w ) + K ( − ξ, w )]+ 18 π (cid:90) − Λ /s − Λ dE/ | E | ( w − / [ K ( ξ, w ) + K ( − ξ, w )] ,I = 18 π (cid:90) ΛΛ /s dE/E √ w − K ( ξ, w ) + K ( − ξ, w )]+ 18 π (cid:90) − Λ /s − Λ dE/ | E |√ w − K ( ξ, w ) + K ( − ξ, w )] , where K ( ξ, w ) = (cid:90) + ∞−∞ dθ | w | (cosh θ + ξ/w ) w + 2 ξw cosh θ + 1 ,K ( ξ, w ) = (cid:90) + ∞−∞ dθ | w | ( ξw + cosh θ )(cosh θ + ξ/w ) w + 2 ξw cosh θ + 1 ,K ( ξ, w ) = (cid:90) + ∞−∞ dθ | w | ( ξw + cosh θ ) (cosh θ + ξ/w )( w + 2 ξw cosh θ + 1) ,K ( ξ, w ) = (cid:90) + ∞−∞ dθ | w | sinh θ (cosh θ + ξ/w )( w + 2 ξw cos θ + 1) . We notice that the integrals in K , K , and K are UVdivergent. By introducing the UV cutoff in θ , denotedby θ Λ , we obtain K ( ξ, w ) = η w ξ ( θ Λ − ln | w | ) ,K ( ξ, w ) ≈ e θ Λ | w | = K ( ξ, w ) . Since K is an odd function of ξ , we get I = 0.This UV arises from the linear approximation we havemade in the Hamiltonian. In real crystal, the size of theopen Fermi surface in type-II DSMs is restricted by thatof the first BZ. Hence, the value of θ Λ is determined bythe size of the first BZ. Suppose that the maximum valueof | p | is π/a where a is the lattice spacing at the scale l . Using the parametrization for ˜ p , we find that for given E | E |√ w − e θ Λ ≈ v πa ≡ D , leading to e θ Λ ≈ √ w − D | E | . Consequently, we get I = Dl π Λ | w | ( w − ,I = Dl π Λ | w | . In general, Λ = O ( D ) at the energy scale Λ. Withoutloss of generality, we set Λ = D and we get I = l π | w | ( w − , I = l π | w | . The choice of the ratio D/ Λ is arbitrary. But it willnot affect the low-energy physics. Different choices cor-respond to different bare values of fermion-disorder cou-plings.For the RSP or x -RVP, we find that Σ ξσ ( iω ) =( − iω )(Σ σ + Σ σ ), whereΣ = ∆( w − λ )[ w ( u + u ) − u u ]2 π v v | w | ( w − l , Σ = − ∆( w − λ )( u + u − wu u )2 π v v | w | ( w − l , and δS Γ = V σ + V σ , where V = − ∆[ w u − wu − wu u + ( w + 2) u u ]2 π v v | w | ( w − lV = ∆[ wu − u + 3 wu u − (1 + 2 w ) u u ]2 π v v | w | ( w − l . For the y -RVP, we haveΣ = ∆ u π v v | w | ( w − w ( w − λ ) l , Σ = ∆ u π v v | w | ( w − 1) ( w − λ ) l , and δS Γ = V σ , where V = − ∆ u π v v | w | l . Finally, for the RM,Σ = ∆ u π v v | w | ( w − w ( w − λ ) l , Σ = ∆ u π v v | w | ( w − 1) ( w − λ ) l , and δS Γ = 0. With the similar procedure, we obtain theone-loop RG equations in the main text.7 ∗ Electronic address: [email protected] † Electronic address: [email protected] O. Vafek and A. Vishwanath, Annu. Rev. Condens. MatterPhys. , 83 (2014). T.O. 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