Fate of superconductivity in disordered Dirac and semi-Dirac semimetals

The effect of weak non-magnetic disorder on $s$-wave superconductivity in ordinary metals can be formally described by the Abrikosov-Gorkov diagrammatic formalism. Within this formalism, the vertex correction is unimportant because the inequality $k_F l \gg 1$, where $k_F$ is the Fermi momentum and $l$ mean free path, is satisfied in ordinary metals with a large Fermi surface. In a Dirac semimetal that has discrete Fermi points, this inequality breaks down since $k_F \rightarrow 0$ and the vertex correction could be important. We incorporate the vertex correction into the Abrikosov-Gorkov formalism and then apply this approach to examine how superconductivity is affected by random chemical potential in two- and three-dimensional Dirac semimetals, as well as two-dimensional semi-Dirac semimetal. In the clean limit, superconductivity is formed only when the pairing interaction strength is greater than some critical value in these materials. Adding random chemical potential promotes superconductivity by generating a finite zero-energy density of states. In slightly disordered three-dimensional Dirac semimetal, the critical strength is reduced, but remains finite. In the rest cases, there is no quantum critical point and superconductivity is induced by arbitrarily weak attraction. Including the vertex correction does not change the above qualitative result. However, the vertex correction plays a vital role in the precise determination of the superconducting gap. Actually, the gap could be severely underestimated if the vertex correction is neglected. Bilayer graphene is quite special because its zero-energy density of states is nonzero despite the existence of Fermi points. Due to this peculiar property, superconductivity is always suppressed by random chemical potential.

These semimetals share a common feature: the Fermi surface is composed of a number of zero-dimensional points. This leads to the fact that the fermion density of states (DOS) vanishes at the Fermi level, which is apparently different from conventional metals whose DOS takes a finite value at the Fermi level. This difference is responsible for many intriguing properties of semimetal materials that cannot occur in ordinary metals.
Cooper pairing instability in 2D DSM has attracted considerable interest in recent years . It has been proposed that phonon or plasmon may mediate an effective attraction between Dirac fermions [4,69,72]. For an undoped 2D Dirac semimetal, superconductivity can be realized only when the net attraction is sufficiently strong . This is quite different from the conventional metal superconductors in which even an arbitrarily weak attraction suffices to trigger Cooper pairing [90][91][92]. The difference is owing to the fact that the fermion DOS vanishes at zero energy, namely ρ(0) = 0. Cooper pairing instability may be achieved in other semimetals, including 2D semi-DSM [41,42], 3D DSM [93], 3D WSM [94,95], and 3D QSM [67]. In these materials, there is a threshold value for the strength of attraction.
In terms of industrial and commercial applications, the existence of a threshold for net attraction appears to be a negative result because it makes it difficult to realize intrinsic superconductivity in undoped DSMs. However, from the perspective of theoretical study, it provides us with a unique opportunity to investigate various novel properties that do not exist in ordinary metal superconductors. The critical value of attraction defines a genuine quantum critical point (QCP) between the semimetallic and superconducting phases at zero temperature, and the system exhibits a wealth of attractive quantum critical behaviors around such QCP. For instance, an effective space-time supersymmetry was recently argued to emerge at this QCP at sufficiently low energies [96][97][98][99].
An interesting problem is how superconductivity formed in various DSMs is affected by weak disorder. Among the possible disorders, random chemical potential is most frequently encountered in realistic materials. In the case of conventional metal superconductors, Anderson theorem states that weak random chemical potential does not alter the superconducting gap ∆ and the critical temperature T c if the pairing is s-wave [100][101][102][103][104]. This result can be obtained by the diagrammatic approach developed by Abrikosov and Gorkov (AG) [104,105]. For such result to be valid, an important precondition is that the low-energy fermion DOS is not sensitive to weak disorder [103,104]. While this condition is satisfied in ordinary metals with a large Fermi surface, it breaks down in DSMs that have only isolated Fermi points. Renormalization group (RG) analysis showed that random chemical potential is a relevant perturbation to 2D DSM [84,89,[106][107][108][109], thus arbitrarily weak disorder eventually flows to the strong coupling regime and then drives the system to enter into a compressible diffusive metal (CDM) state, in which disorder generates a finite scattering rate Γ. In addition, the fermions acquire a finite zero-energy DOS ρ(0) that is a function of Γ and vanishes as Γ → 0 [84,89,[106][107][108][109][110][111][112]. It is obvious that the zero-energy DOS of the 2D DSM is very sensitive to weak disorder, in stark contrast to the case of ordinary metals. As a consequence, weak random chemical potential might have significant influence on ∆ and T c . The question is whether these two quantities are enhanced or suppressed.
Recently, Nandkishore et al. [84] studied the effect of random chemical potential on s-wave superconductivity by taking the surface electronic state of a 3D topological insulator as an example. After solving the mean-field gap equation in combination with a RG analysis [84], they found that an arbitrarily weak attraction leads to superconductivity, which implies that superconductivity is enhanced. Later, Potirniche et al. [85] investigated the interplay of superconductivity and random chemical potential by considering a Hubbard model defined on honeycomb lattices with an on-site attractive coupling U by means of self-consistent Bogoliubov-de Gennes (BdG) equation method [85]. In the clean limit, they argued that the system remains a semimetal if U is smaller than certain critical value U c , but becomes superconducting when U exceeds U c . Disorder was claimed to result in a complicated behavior of superconductivity. In the strong coupling regime U > U c , disorder can suppress superconductivity. In the weak coupling regime U < U c , weak disorder enhances superconductivity, but strong disorder eventually destroys superconductivity.
In a recent work [89], the diagrammatic AG method was applied to study the impact of random chemical potential on superconductivity in the context of a 2D DSM, yielding results that are qualitatively consistent with those of Nandkishore et al. [84] and Potirniche et al. [85]. However, we should emphasize that the original AG method is actually invalid in a 2D DSM [84,89], because it entirely neglects the vertex correction to the fermiondisorder coupling. Such correction would be small if the inequality k F l ≫ 1, where k F is the Fermi momentum and l the mean free path, is satisfied. Since the disorder scattering rate Γ is inversely proportional to l, the above inequality can be re-expressed as k F ≫ Γ. In ordinary metals with a finite Fermi surface, the Fermi momentum is usually large and the scattering rate caused by weak disorder is small, thus this inequality is certainly valid. Nevertheless, the 2D DSM contains only Fermi points, which means k F → 0. Therefore, there is no such inequality as k F ≫ Γ. To examine the fate of superconductivity against random chemical potential, one needs to go beyond the original AG approximation and incorporate the vertex correction carefully. This is the first motivation of the present work.
We notice an important fact that the impact of random chemical potential in various semimetals is closely related to the dimensionality and the concrete form of fermion dispersion. For example, weak random chemical potential is irrelevant in a 3D DSM/WSM, but becomes relevant if its strength is sufficiently large. Accordingly, for a 3D DSM, there is a QCP between the semimetallic and CDM phases with varying disorder strength [113][114][115][116][117]. This behavior is apparently different from the case of 2D DSM, where an arbitrarily weak random chemical potential leads to CDM state. For semi-DSM [43,44], in which the energy dispersion is quadratical with one momental component but linear with another component, and 3D double- [52] and triple-WSMs [60], in which the energy dispersion is quadratic (cubic) in two components of momenta but linear in the third, a CDM state is achieved by an arbitrarily weak disorder, analogous to 2D DSM. The second motivation of this work is to examine whether the conclusion obtained previously in the case of 2D DSM is applicable to other semimetals.
We will show that arbitrarily weak attraction suffices to induce an s-wave superconductivity in 2D DSM when random chemical potential is present. If the attraction is weak, the magnitude of superconducting gap increases with growing disorder strength, but begins to decrease when the disorder strength becomes large enough. For relatively strong attraction, the superconducting gap is always suppressed by the increasing disorder strength. In case of weak attraction and weak disorder, the magnitude of superconducting gap obtained by solving the gap equations is drastically amplified once the fermiondisorder vertex correction is considered, which in turns leads to a higher T c . However, the qualitative influence of disorder on superconductivity is not altered.
The generalized AG method can be easily applied to other semimetal systems in which the inequality k F ≫ Γ breaks down. To make our analysis more comprehensive, and also to examine how superconductivity relies on the dimensionality and the geometry of the Fermi surface, we then study the fate of superconductivity in disordered 3D DSM and 2D semi-DSM, which both have a pointtouching band structure.
In the case of slightly disordered 3D DSM, the critical pairing interaction strength g c is smaller than that obtained in the clean limit, but remains finite. Thus, there is still a superconducting QCP, at which interesting quantum critical phenomena might occur. When the 3D DSM contains strong random chemical potential, the critical value g c vanishes, and superconductivity is formed by arbitrarily weak pairing interaction. An apparent conclusion is that superconductivity is promoted by random chemical potential.
In a 2D semi-DSM, the disorder effect on superconductivity is very similar to 2D DSM: superconductivity occurs only when the pairing interaction strength exceeds a threshold g c in the clean limit, but is triggered by arbitrarily weak attraction when random chemical potential is introduced. If both pairing interaction and disorder are weak, the vertex correction needs to be carefully taken into account as one is attempting to precisely determine ∆ and T c . However, the vertex correction does not alter the qualitative results obtained without including it in the gap equation.
Comparing to the above three semimetals, the bilayer graphene is special in that its Fermi surface is composed of discrete points but the DOS is finite at the Fermi level. It is found that the s-wave superconductivity in bilayer graphene is always suppressed gradually and slowly by random chemical potential. This is qualitatively different from the other three semimetals considered in this work. Such difference is rooted in the fact that the zero-energy DOS is nonzero only in bilayer graphene but vanishes otherwise.
The rest of paper is organized as follows. In Sec. II, we present the Hamiltonian for the Cooper pairing on 2D DSM, concretely the surface state of a 3D topological insulator. Then, the influence of disorder on Cooper pairing instability on 2D DSM is analyzed within AG method. Subsequently, we study the question beyond AG method by including the vertex correction selfconsistently. The influence of disorder on superconductivity on 3D DSM and 2D semi-DSM is, studied in Sec. III and Sec. IV respectively. In Sec. V, we analyze the impact of disorder on bilayer graphene in which the Fermi surface is composed by discrete points but the DOS at Fermi level is finite. We summarize our main results and give some discussions in Sec. VI.

II. SUPERCONDUCTIVITY IN 2D DIRAC SEMIMETAL
In this section, we consider the case of 2D DSM and study the impact of random chemical potential on superconductivity by means of the AG approach and its proper generalization. The framework employed in this section is quite general and will be utilized in the following several sections to study the fate of superconductivity other disordered semimetals, including 3D DSM, 2D semi-DSM, and bilayer graphene.

A. Model Hamiltonian
We consider a single species of massless Dirac fermions that emerge, for instance, on the surface of a 3D topological insulator. Following the notations adopted in Ref. [84], we write the Hamiltonian in the form where µ is the chemical potential and g is the coupling constant for the BCS-type attractive interaction. At a finite µ, the Dirac semimetal has a finite Fermi surface, and many of its low-energy properties are very similar to ordinary metals. In the following, we only consider the most interesting case of zero chemical potential, µ = 0, corresponding to undoped semimetal. The spinor field is defined as ψ k = ( c ↑k , c ↓k ) T , whose conjugate is Moreover, σ 1,2 are the standard Pauli matrices, and σ 0 is the identity matrix. The superconducting order parameter is defined as The system enters into a superconducting phase when ∆ acquires a nonzero value. Our analysis starts from the partition function where β = 1/T and the Lagrangian is related to the Hamiltonian H via the Legendre transformation To express the action in a more compact form, it is convenient to introduce a four-component Nambu spinor: After decoupling the quartic attractive interaction by means of Hubbad-Stratonovich transformation, we can re-write the partition function as where L takes the form Here, we take ∆ = 0, corresponding to nonsuperconducting phase.
In the above expression, we have defined a fermion propagator G ωn,k ≡ G(ω n , k) that is given by where k ± = k x ± ik y . Within the Matsubara formalism, the fermion frequency is ω n = (2n + 1)πT with n being integers. The above expression of partition function is consistent with Ref. [84].

B. Clean case
The superconducting gap equation can be derived by integrating over fermion fields Ψ and Ψ † . For this purpose, the partition functions is further written as Performing the functional integration over Ψ and Ψ † yields It is then easy to get Through direct calculation, we obtain For a sample of volume V , the free energy density is Making variation with respective to infinitesimal change of ∆, δf δ∆ = 0, we finally obtain the gap equation: We have already fixed the phase factor of the gap function ∆, and in the following will take ∆ as a real variable. At zero temperature T = 0, the gap equation becomes dω 2π Performing the integration of ω and k, we obtain where Λ is the cutoff of the momentum. Setting ∆ = 0 yields the critical coupling g c0 = 2πv/Λ. The gap ∆ acquires a nonzero value only when g > g c0 , and there is a semimetal-superconducting QCP at g = g c0 [69,71,74,84,89].

C. Analysis without vertex correction
In the presence of random chemical potential, the dynamics of Dirac fermions is modified due to the disorder scattering. Consequently, the fermion propagator can be expressed by the matrix where A i ≡ A i (ω n ) with i = 1, 2, 3 are the renormalization functions induced by the disorder scattering. The propagators G F ωn,k and G ωn,k are connected with each other through the Dyson equation where where n imp is the impurity concentration and u is the strength of one single impurity. It is easy to verify that A 2 = 1, and that The gap equation is given by (a) g/g c0 =0.6, γ=0.2  Carrying out the integration over momenta, we get two coupled equations We now take the zero temperature limit, namely T = 0, and then make the re-scaling transformations ω vΛ → ω and ∆ vΛ → ∆, which leads us to Upon approaching the QCP, g → g c , so the above equations are simplified to Numerical results of Eq. (25) are shown in Fig. 1(a) and (b). In the zero-energy limit, ωA approaches to a constant Γ, which is the disorder scattering rate. If ω decreases from the energy scale of Γ, ωA approaches to a constant. Above the energy scale Γ, A → 1. The asymptotic behavior of A can be approximately described by The integral appearing in Eq. (26) is divergent, which implies that g c → 0. Therefore, even an arbitrarily weak attraction suffices to trigger Cooper pairing instability. In this regard, random chemical potential tends to promote the formation of Cooper pairing, in agreement with the result of Nandkishore et al. [84]. The dependence of the gap ∆ on g at different values of γ is displayed in Figs. 2(a) and (b). We present the relation between ∆ and γ with different values of g in Figs. 3(a) and (b). The figures exhibit that ∆ is enhanced by weak disorder and then suppressed gradually by strong disorder if the attraction is weak. However, for sufficiently large attraction, with g > g c , the gap ∆ is suppressed by dis-order monotonously. These results are qualitatively the same as that obtained by Potirniche et al. by using selfconsistent BdG equations [85].
The ω-dependence of A(ω), obtained from the solutions of Eq. (22) and Eq. (23), are shown in Fig. 4(a). The function A(ω) always approaches to a finite value in the low energy region, which is expected since the gap ∆ provides a cutoff. According to Fig. 4(a), if the pairing interaction is relatively weak, A(ω) goes to a larger constant value at low energies for smaller γ. If the pairing interaction is relatively strong, however, A(ω) takes a larger constant value at low energies for larger γ.

D. Beyond AG approximation
We emphasize that the gap equation analysis of the disorder effects based on the original AG formalism is actually problematic. The validity of AG method relies on a crucial assumption that k F l ≫ 1, which is equivalent to k F ≫ Γ since Γ ∼ 1/l. This inequality is satisfied in an ordinary metal in which the Fermi momentum is large and the scattering rate Γ can be made sufficiently small if random chemical potential is supposed to be weak enough. For a 2D DSM, however, we know that k F = 0. Additionally, even an arbitrarily weak random chemical potential is able to drive 2D DSM to become a CDM [106][107][108][109][110][111][112], which in turn generates a finite Γ. Thus, the above inequality certainly breaks down, and the vertex correction can no longer be regarded as unimportant. We need to go beyond the original AG approximation, and incorporate the vertex correction explicitly in the selfconsistent gap equations.
The correction to the fermion-disorder coupling vertex is given by which can be further written as Using the transformation k + q → k, and employing the Feynman parameterization we convert the vertex correction into the form We further make the replacement k + x (p − q) → k, and then obtain One can verify that Ξ(ω, p, q) is the function of |p − q|, which means Ξ(ω, p, q) ≡ Ξ(ω, |p − q|). Performing the integrations over k and x leads to the following vertex correction: where γ is defined in Eq. (24), and After including the vortex correction, the equations for A 1 and A 3 now become We now employ the re-scaling transformations k Λ → k, ω vΛ → ω, and ∆ vΛ → ∆, and, making use of the relation where the gap ∆ is still given by Eq. (23).
In the limit of ∆ = 0, the system stays in the nonsuperconducting phase. We present the ω-dependence of A and ωA in Fig. 1(c) and Fig. 1(d) respectively. Comparing them to Fig. 1(a) and Fig. 1(b), we can observe that the disorder scattering rate Γ is made larger by the inclusion of the vertex correction. Katanin [112] performed a functional RG analysis of this problem, and also found a larger scattering rate Γ and a larger ρ(0) comparing to those obtained by adopting the SCBA. Now consider the superconducting phase where ∆ = 0. The g-dependence of gap ∆ is displayed in Fig. 2 (c) and (d). A clear result is that the zero-energy gap ∆ obtained after including the vertex correction is larger by several orders than that obtained without the vertex correction if the attraction and disorder are both weak. It is thus apparent that, although the vertex correction does not change the qualitative conclusion reached by the original AG analysis, it plays a very important role in the precise determination of the superconducting gap, particularly in the case of weak pairing interaction and weak disorder. The gap and accordingly the critical temperature T c would be severely underestimated if the vertex correction is not taken into account. Dependence of ∆ on γ with different values of g is presented in Fig. 3(c) and (d). We can find that the qualitative property of the disorder effect on superconductivity remains nearly the same after including the vertex correction.
In the case of ∆ = 0, we show the dependence of A(ω) on ω in the presence of vertex correction in Fig. 4(b). It is obvious that A(ω) approaches to a smaller constant comparing to the one shown in Fig. 4(a). The reason for this behavior is that the gap ∆ becomes larger after including the vertex correction and thus leads to a stronger suppression effect for the disorder scattering rate.
The energy and momenta dependence of the vertex function Ξ(ω, k) are shown in Fig. 5. An apparent fact is that the vertex correction is important at low energies and small momenta, and can be nearly ignored only when the energy and momenta are sufficiently large. As the pairing interaction gets stronger, the vertex correction becomes less important [84].
The semimetal-superconducting QCP exists in a clean 2D DSM, but is eliminated when the system contains weak random chemical potential. Since there is always certain amount of impurity in the material, it seems extremely difficult to realize and probe the predicted quantum critical phenomena at such a QCP.

III. SUPERCONDUCTIVITY IN 3D DIRAC SEMIMETAL
In this section, we will investigate the fate of superconductivity formed by Cooper pairing of 3D Dirac fermions, which could emerge at low energies at the QCP between a normal insulator and a 3D topological insulator. This type of 3D DSM has been observed in TiBiSe 2−x S x [119,120] and Bi 2−x In x Se 3 [121,122] by fine tuning the doping level. Theoretical works [123,124] predicted that a crystal-symmetry protected 3D DSM might be realized in such materials as A 3 Bi (A=Na, K, Rb) and Cd 3 As 2 . Shortly after this prediction, ARPES and quantum transport measurements have reported evidence of 3D DSM state in Na 3 Bi and Cd 3 As 2 [125][126][127][128][129].
Recent RG analysis revealed that weak attraction is irrelevant in 3D DSM, and that only sufficiently strong attraction can induce superconductivity [93] and there is also a QCP separating the semimetallic and superconducting phases. In the non-superconducting phase, the physical effect caused by the random chemical potential is a subject of considerable interest [113-118, 130, 131]. Recent studies based on self-consistent Born approximation (SCBA), RG analysis, and exact numerical simulation all found that there is a QCP between the semimetal and CDM phases by adjusting the strength of random chemical potential [113][114][115][116][117][118]. If the rare region effect is considered, it was found that arbitrary weak disorder drives the system to enter into the CDM phase [130,131]. In this paper, we do not consider the rare region effect, and focus on the fate of s-wave superconductivity.
Similar to 2D DSM, the 3D DSM hosts only discrete Fermi points, and thus the original AG approximation is also inapplicable. In addition, the zero-energy DOS vanishes in both cases, and might become finite if the system is turned by random chemical potential into a CDM. We now parallel the analysis performed in the last section, and include explicitly the vertex correction in the self-consistent equations for A and ∆. The disorder effect on superconductivity can be analogously analyzed.

A. Clean case
The mean-field Hamiltonian for 3D DSM is formally similar to that of 2D DSM, and will be not explicitly given here. We directly write down the the gap equation obtained in the clean limit: where g is the strength parameter for pairing interaction. Integrating over momenta results in where Λ is the momentum cutoff. The critical attraction strength can be determined by taking ∆ = 0, satisfying the following equation: The critical value is g c0 = 4π 2 v/Λ 2 , which defines the superconducting QCP.

B. Analysis without vertex correction
Under the original AG approximation, the selfconsistent equations for A and ∆ are given by where In the derivation, we have employed the transformations: k Λ → k, ω vΛ → ω, and ∆ vΛ → ∆. Before analyzing the disorder effect on superconductivity, it is necessary to first consider the impact of weak disorder on the low-energy behavior of Dirac fermions in the non-superconducting phase. This is an interesting problem and has been studied recently by means of several different approaches [113-118, 130, 131].
In the limit of ∆ = 0, the equation for A has the form The solutions for this equation are shown in Fig. 6 (a) and (b). We find that A(ω) approaches a finite value in the limit ω → 0 if γ is smaller than a critical value γ c . In contrast, if γ > γ c , A(ω) is divergent in the limit ω → 0, yet satisfying where Γ takes a finite value. The constant Γ should be identified as the disorder scattering rate. A finite DOS is generated at the Fermi level, which tends to favor superconductivity. The dependence of Γ on γ is depicted in Fig. 7(a), which clearly shows that γ c = 1. Making use of the approximation ωA(ω) → Γ, we rewrite Eq. (40) in the form By setting Γ = 0, we find that γ c = 1. According to the results presented in Figs. 6(a) and (b), and Figs. 7(a) and (b), the system undergoes a quantum phase transition from the semimetallic phase to a CDM phase at γ = γ c . This result is consistent with previous studies based on perturbative RG [113,114], functional RG [117], and direct numerical calculation [116]. We point out here that we did not consider the effects of rare region for simplicity [84,130,131]. We then turn to solve the coupled equations (37) and (38), which will be used to analyze the impact of disorder on superconductivity. As can be seen from Fig. 7(b), the critical value g c decreases as the disorder parameter γ grows, indicating that superconductivity is promoted. More concretely, in the range 0 < γ < 1, g c is made smaller than g c0 but remains finite. Thus, there is still a superconducting QCP and it is possible to observe the corresponding quantum critical phenomena. If γ > 1, we take the limit ∆ → 0 for Eq. (38) and then obtain an equation for g c : At low energies, A(ω) behaves as A(ω) ∼ Γ |ω| . Accordingly, the integral in Eq. (43) is divergent, indicating that the critical value vanishes, i.e., g c → 0. It turns out that even arbitrarily weak attraction suffices to form Cooper pairs. The dependence of ∆ on g at different values of γ can be found in Fig. 8(a) and (b). The dependence of ∆ on γ at different values of g is shown in Fig. 9(a) and (b). We observe that the gap ∆ displays a non-monotonic dependence on γ if g is relatively small: ∆ is enhanced by weak disorder but gets suppressed by sufficiently strong disorder. However, the gap is always suppressed when g becomes relatively large. This behavior is qualitatively similar to 2D DSM.
The asymptotic behavior of A(ω) for different values of g and γ is shown in Fig. 10(a). We can find that A(ω) generally approach to some finite value determined by g and γ. In the semimetal phase, ∆ is vanishing, and A(ω) is saturated to finite value and ωA(ω) vanishes in the lowest energy limit. In the superconducting phase, the nonzero gap ∆ serves as a cutoff and prevents A(ω) from being divergent in the lowest energy limit.

C. Beyond AG approximation
Paralleling the analysis carried out in the case of 2D DSM, we now examine the role played by the vertex correction. After including the vertex correction into the equations of A and ∆, Eq. (37) becomes but the gap equation Eq.(38) is not changed. Straightforward calculations lead us to the following expression for the vertex function Ξ(ω, k): where First, we assume g = 0 and analyze the influence of disorder in the normal phase of 3D DSM. The dependence of A(ω) and ωA(ω) on ω are displayed in Fig. 6(c) and (d). We can find A(ω) is saturated to a finite value if γ is enough. Whereas, A(ω) becomes divergent and ωA(ω) approaches to a finite value provided γ is large enough. These results are qualitatively the same as those obtained by ignoring the vertex correction presented in Fig. 6(a) and (b). However, the magnitude of γ c becomes smaller as the vertex correction is taken into account, which can be easily observed from Fig. 7. We notice that, a recent functional RG analysis [117] incorporated the vertex correction and argued that the critical value of disorder strength is smaller than that obtained by using the SCBA. Ominato and Koshino [118] also emphasized the importance of the vertex correction in the estimate of disorder scattering rate.
Through numerical calculations of the Eqs. (44), (45), and (38), we obtain the phase diagram shown in Fig. 7(d). According to this phase diagram, g c becomes smaller as γ increases. Comparing Fig. 7(b) to Fig. 7(d), we observe that the area of the superconducting phase is broadened when the vertex correction is incorporated.
The dependence of ∆ on g is given in Fig. 8(c) and (d), and the dependence of ∆ on γ in Fig. 9(c) and (d). The gap ∆ is enhanced by weak disorder and suppressed by sufficiently strong disorder when g is small, but is always suppressed by disorder if g takes a large value. These results are qualitatively the same as those obtained by ignoring the vertex correction. Comparing Fig. 8(a) and (b) with (c) and (d), we can find that the enhancement effect of the superconducting gap induced by weak disorder for small g is even more significant when the vertex correction is considered. According to Fig. 10, A(ω) is still saturated to certain finite constant. This is qualitatively the same as the case of neglecting the vertex correction. Quantitatively, A(ω) does acquire certain amount of modification after including the vertex correction.
The behavior of Ξ(ω, |k|) is presented in Fig. 11. In Figs. 11(a) and (b), we take ∆ = 0 by assuming that g = 0. According to Figs. 11(a) and (b), Ξ is amplified when γ becomes larger. From Figs. 11(c) and (d), we see that in the superconducting phase, Ξ decreases with growing g, indicating that the vertex correction becomes less important once a finite gap is opened.

IV. SUPERCONDUCTIVITY IN 2D SEMI-DIRAC SEMIMETAL
The dispersion for 2D semi-Dirac fermions is given by which is linear along one momentum component (k y ) but quadratic along the other one (k x ). The corresponding materials is usually called 2D semi-DSM. Such type of fermions might emerge at the QCP between a 2D DSM and a band insulator upon merging two separate Dirac points to a single one. Generation of semi-Dirac fermions through merging pairs of Dirac points is theoretically predicted to exist in deformed graphene [17][18][19][20], pressured organic compound α-(BEDT-TTF) 2 I 3 [19][20][21], few-layer black phosphorus that is subject to a perpendicular electric field [22,23] or doping [24], and also some artificial optical lattices [25,26]. Experimentally, the merging of Dirac points and the appearance of semi-Dirac fermions were observed in ultracold Fermi gas of 40 K atoms arranged on a honeycomb lattice [27] and microwave cavities with graphene-like structure [28]. Kim et al. [29] realized semi-DSMs in few-layer black phosphorus at critical surface doping with potassium. Moreover, robust semi-DSM state was predicted to emerge in TiO 2 /VO 2 nanostructure under suitable conditions [30,31]. It was suggested by first-principle calculations that semi-Dirac fermions are the low-energy excitations of the strained puckered arsenene [33,34]. In addition, semi-DSM state may also be realized [40] at the QCP between normal insulator and topological insulator, and the QCP between normal insulator and 2D DSM in time-reversal invariant 2D noncentrosymmetric system.
Recently, Uchoa and Seo [41] studied the possibility of Cooper pairing in 2D semi-DSM by making a mean field analysis, and argued that s-wave superconductivity is favored. Close to the QCP between semimetallic and superconducting phases, they found that the anisotropy of quasiparticles leads to a novel smectic state of superconducting stripes. Roy and Foster [54] investigated the influence of various short-range interactions on the low-energy behavior of 2D semi-DSM by making a RG analysis, and also discovered an s-wave superconductivity. In the non-interacting limit, recent SCBA [43] and RG [43,44] studies showed that arbitrarily weak random chemical potential turns the 2D semi-DSM into a CDM state, analogous to what happens in 2D DSM.
Like 2D and 3D DSM, the 2D semi-DSM also has a vanishing zero-energy DOS if the chemical potential is tuned exactly at the Dirac point. Consequently, the original AG approximation is no longer valid. In what follows, we will examine how weak random chemical potential affects the s-wave superconductivity in a 2D semi-DSM by means of the AG approach and its generalization. Once again, we will not give the mean-field Hamiltonian, and starts our discussion directly from the gap equation.

A. Clean Case
In the clean limit, the equation for the superconducting gap takes the form which can be further written as We employ the transformations which are equivalent to The measures of integration satisfy the relation Adopting the transformations Eqs. (53) and (54), the gap equations becomes where Λ E is a cutoff for the variable E. Taking the limit ∆ → 0, we obtain the following critical value beyond which a nonzero superconducting gap is opened. This g c0 is the QCP that separates the semimetal and superconducting phases.

B. Analysis without vertex correction
Including the disorder scattering, the self-consistent equations obtained under the original AG approximation are found to have the forms where with J 4 = A 2 ω 2 + A 2 ∆ 2 . The scaling transformations ω ΛE → ω and ∆ ΛE → ∆ have been used. By taking ∆ = 0, we obtain the solution of A(ω) in the normal state, and show the results in Figs. 12(a) and (b). We find that A(ω) is divergent in the low energy limit, and ωA(ω) approaches to a constant in 2D semi-DSM, similar to 2D DSM. In the limit ∆ = 0, the integrand in the Eq. (58) satisfies in the low energy regime. Thus, the integration in the Eq. (58) is divergent, which leads to g c /g c0 → 0. Therefore, superconductivity is produced by arbitrarily weak pairing interaction once random chemical potential is considered. Dependence of ∆ on g with different values of γ is presented in Fig. 13, which clearly exhibits the promotion of superconductivity due to weak random chemical potential if g is small.
In the superconducting phase with ∆ = 0, the dependence of ∆ on γ is depicted in Figs. 14(a) and (b). For small values of g, the gap ∆ increases as γ is growing, and then starts to decrease with growing γ when γ becomes sufficiently large. For larger values of g, ∆ decreases monotonously as g increases. We thus can see that the influence of random chemical potential on s-wave superconductivity in 2D semi-DSM is qualitatively the same as 2D DSM.

C. Analysis beyond AG approximation
In this subsection, we move to examine the impact of the vertex correction. After including the vertex correction, the equation for A is of the form where the vertex function Ξ is given by We have used the transformations and also made the assumption that aΛ 2 x = vΛ y = Λ E . The equation for the gap ∆ is still given by Eq. (58).
Including the vertex correction, dependence of ∆ on g with different values of γ is shown in Figs. 13(c) and (d). The relation between ∆ and γ at different values of g is displayed in Figs 14(c) and (d). We observe from Fig. (13) and Fig. (14) that including the vertex correction leads to moderate amplification of the gap ∆. We can also find that vertex correction does not change the qualitative characteristic of influence of random chemical potential on superconductivity in 2D semi-DSM.

V. SUPERCONDUCTIVITY IN BILAYER GRAPHENE
In bilayer graphene with Bernal AB stacking, in which two layers of carbon atoms are rotated by 60 • , the Fermi surface is also composed of discrete points [1,4]. There are also other sorts of configuration of bilayer graphene, such as AA stacking that match the A sublattices of two layers. Here, we focus on bilayer graphene with Bernal configuration, which is the most frequently studied case. In such a system, the fermions display the following dispersion [1,4] where a is a constant. For Bernal bilayer graphene, the Berry phase around the touching points is trivial, in distinct to monolayer graphene that exhibits a non-trivial Berry phase. This difference can be clearly observed in quantum Hall effect experiments [132]. Due to its special dispersion and dimensionality, the bilayer graphene has a finite zero-energy DOS [4]: ρ(0) ∝ 1 a . As a results, an infinitesimal short-range interaction is able to drive some type of phase-transition instability [4,133,134].

A. Clean Case
In the clean limit, the gap equation for s-wave superconductivity in bilayer graphene is given by where g is the strength parameter of pairing interaction. Performing the integrations of momentum and energy, we obtain where Λ is the cutoff of the momentum. In the limit ∆ → 0, g approaches a critical value g c0 that satisfies It is obvious that g c0 → 0, so arbitrarily weak attraction leads to superconductivity in bilayer graphene. This behavior is markedly different from the aforementioned semimetals with vanishing DOS at the Fermi level.

B. Analysis without vertex correction
In the absence of vertex correction, the self-consistent equations for the renormalization factor A(ω) and the superconducting gap ∆ can be written as where γ = n imp u 2 2πa 2 Λ 2 , g 0 = 2π 2 a, and J 4 = A 2 ω 2 + A 2 ∆ 2 . Here, g 0 is chosen as the unit of attraction strength g. Solving the Eqs. (68) and (69), we obtain the dependence of ∆ on g at different values of γ, which is shown in Figs. 15(a) and (b). It is easy to find that ∆ is quantitatively suppressed by random chemical potential. The relation between ∆ and γ at different values of g is displayed in Fig. 16(a). The gap ∆ is suppressed monotonously by increasing γ, but the suppression effect is not significant. Thus, the s-wave superconductivity seems to be robust against weak disorder in bilayer graphene. For bilayer graphene, there is not any evidence of disorder-induced promotion of superconductivity, which occurs in 2D DSM, 3D DSM, and 2D semi-DSM. Such difference originates from the fact that the zero-energy DOS ρ(0) is nonzero in bilayer graphene, but vanishes in the other three types of semimetal.
The gap still satisfy the Eq. (69). Including the vertex correction, the relation between ∆ and g for different values of γ are shown in Figs. 15(c) and (d). The dependence of γ for different values of g is depicted in Fig. 16(b). We can find that vertex correction does not lead to qualitative change of the results. Quantitatively, the suppression effect for the gap by disorder becomes slightly weaker once vertex correction is considered.

VI. SUMMARY AND DISCUSSION
In this paper, we have studied the influence of random chemical potential on s-wave superconductivity in 2D DSM by using the AG diagrammatic approach. It is found that an arbitrarily weak attraction can lead to Cooper pairing instability. When the attraction is weak, the magnitude of the superconducting gap first increases with growing disorder strength parameter and then decreases once the disorder strength exceeds certain critical value. For relatively strong attraction, the gap decreases monotonously as disorder is becoming stronger. To get a more reliable result, we have gone beyond the original AG approximation, and taken into account the correction to the fermion-disorder vertex. After including the vertex correction, the qualitative behavior of superconductivity is not substantially altered. However, if the pairing interaction and disorder are both weak, the magnitude of superconducting gap is significantly enhanced by the vertex correction is considered. Therefore, to determine the gap and the critical temperature precisely, the vertex correction must be carefully incorporated.
We then have applied the generalized AG approach to investigate the fate of s-wave superconductivity in other analogous semimetal materials, including 3D DSM, 2D semi-DSM, and bilayer graphene. For 3D DSM, we have found that the critical pairing interaction strength g c is reduced to a smaller value by weak random chemical potential, and thus there is still a QCP separating the semimetallic and superconducting phases. This QCP provides an ideal platform to study the rich quantum critical phenomena. Nevertheless, when random chemical potential becomes sufficiently strong, the critical value g c vanishes, and superconductivity is achieved no matter how weak the pairing interaction. In both cases, we see that superconductivity is promoted by random chemical potential.
For 2D semi-DSM, the disorder effect on the s-wave superconductivity is nearly the same as that of 2D DSM. In particular, superconductivity can be induced by arbitrarily weak attraction when random chemical potential is present in the system. The vertex correction also plays an important role in the precise determination of the superconducting gap and the critical temperature if both pairing interaction and disorder are weak. However, the vertex correction does not modify the qualitative results obtained by using the original AG approximation.
Comparing to the above three types of semimetals, the bilayer graphene is spectacular since its zero-energy DOS takes a finite value at the Fermi level. As the disorder strength increases, the magnitude of the superconducting gap decreases, yet at a very low speed. Apparently, such behavior is in sharp contrast to that of 2D DSM, 3D DSM, and 2D semi-DSM, where the zero-energy DOS vanishes in the clean limit and acquires a finite value only when the system contains random chemical potential.