Emergence of topological superconductivity in doped topological Dirac semimetals under symmetry-lowering lattice distortions

Recently, unconventional superconductivity having a zero-bias conductance peak is reported in doped topological Dirac semimetal (DSM) with lattice distortion. Motivated by the experiments, we theoretically study the possible symmetry-lowering lattice distortions and their effects on the emergence of unconventional superconductivity in doped topological DSM. We find four types of symmetry-lowering lattice distortions that reproduce the crystal symmetries relevant to experiments from the group-theoretical analysis. Considering inter-orbital and intra-orbital electron density-density interactions, we calculate superconducting phase diagrams. We find that the lattice distortions can induce unconventional superconductivity hosting gapless surface Andreev bound states (SABS). Depending on the lattice distortions and superconducting pairing interactions, the unconventional inversion-odd-parity superconductivity can be either topological nodal superconductivity hosting a flat SABS or topological crystalline superconductivity hosting a gapless SABS. Remarkably, the lattice distortions increase the superconducting critical temperature, which is consistent with the experiments. Our work opens a pathway to explore and control pressure-induced topological superconductivity in doped topological semimetals.

Recently, the lattice-distortion induced superconductivity in DSMs of Cd 3 As 2 [27][28][29] and Au 2 Pb [30][31][32][33][34] is reported.For Cd 3 As 2 , it does not show any superconductivity at the ambient pressure until 1.8 K [27][28][29] .The structural phase transition occurs near 2.6 GPa from a tetragonal lattice with D 4h point group symmetry (I4 1 /acd) to a monoclinic lattice with C 2h point group symmetry (P2 1 /c).Then, superconductivity emerges at T c ≈ 1.8 K under pressure higher than 8.5 GPa.When the pressure increases further, T c keeps increasing from 1.8 K to 4.0 K in the hydrostatic pressure experiment 28 .Similarly, Au 2 Pb shows superconductivity at T c ≈ 1.2 K after a structural phase transition from the cubic with O h symmetry (Fd3m) to the orthorhombic lattice with D 2h symmetry (Pbnc) 30,32,34 .T c increases up to 4 K at 5 GPa, then decreases with further compression 34 .For both materials, the point-contact measurements reported that measured T c using a hard contact tip is much higher than the measured T c using a soft tip 27,29,32 .The point-contact measurements for Cd 3 As 2 showed the zero-bias conductance peak (ZBCP) and double conductance peaks symmetric around zero bias, which was interpreted as a signal of a topological Majorana surface superconductivity is reported in Cd 3 As 2 [27][28][29] and Au 2 Pb [30][31][32][33][34] .Both materials have Dirac points protected by TRS, IS, and C 4 rotational symmetry and share the tetragonal crystal system with D 4h point group symmetry.For this reason, we consider the undistorted topological DSM having a D 4h point group symmetry as a representative model system.

Model and symmetry
The general 4 × 4 Hamiltonian representation is The coefficient function a i (k) are real functions and Γ i = s j σ k are 4 × 4 gamma matrices where s j and σ k are Pauli matrices for spin and orbital degrees of freedom in the spin (↑, ↓) and the orbital (1, 2) spaces, respectively.The symmetry constraints can simplify the Hamiltonian's form in Eq. (1).Due to TRS and IS, the Hamiltonian satisfies the following equations: where T = is y K is the time-reversal operator ( K is the complex conjugation operator) and P is the inversion operator.Because the inversion does not flip the spin, the inversion operator has orbital dependency only, and it can be chosen as P = −σ z for topological DSM without loss of generality 39,45 .Then, due to TRS and IS, among sixteen Γ i matrices, only six Γ i matrices are allowed.They are Γ 0 = 1 4×4 , Γ 1 = σ x s z , Γ 2 = σ y s 0 , Γ 3 = σ x s x , Γ 4 = σ x s y , and Γ 5 = σ z s 0 .We set a 0 (k) = 0 since it does not contribute to the formation of Dirac points 39,45 .The D 4h point group symmetry imposes more constraints on the Hamiltonian's form in Eq. (1).The generators of D 4h point group can be chosen as inversion P, fourfold rotation about the z axis C 4z , and twofold rotation about the x axis C 2x .Their matrix representations are chosen as where we adopt the following basis set known to describe the low-energy effective Hamiltonian of Cd 2 As 3 45 .
where J is the total angular momentum.Other rotation and mirror symmetries are given by C 2z = iσ z s z , M xy = −is z , M yz = −is x , M zx = −iσ z s y , M (110) = i(σ z s x − s y )/ √ 2, and M (1 10) = i(σ z s x + s y )/ √ 2. The subscript in each mirror operator represents the corresponding mirror plane by using either Cartesian coordinates or Miller indices.The group elements are derived in Sec.S1 in Supplementary Information.Due to this D 4h symmetry, the Hamiltonian in Eq. (1) satisfy where U and S are transformation matrices for an element of D 4h group in the spin-orbital and momentum spaces, respectively.For the group generators, the Hamiltonian in Eq. (1) satisfies where R 4z k = (−k y , k x , k z ) and R 2x k = (k x , −k y , −k z ).Because the transformation properties of gamma matrices are given by Table 1, Eq. ( 5) imposes constraints to each coefficient functions a i (k), which is summarized in Table 2. Therefore, the general form of the Hamiltonian of DSM having D 4h point group symmetry is obtained.
Table 1.Transformation properties of gamma matrices under symmetry operations.Under an operation O, each gamma matrices satisfies the relation of OΓ i O −1 = ±Γ j .In each entry, if i = j, the overall sign is written, otherwise the explicit form is given.The gamma matrices are classified according to the irreducible representation (IR) of D 4h point group.Γ 0 , Γ 5 , Γ 4 , and Γ 3 belong to the A 1g , A 1g , B 1u , and B 2u irreducible representations, respectively.Γ 1 and Γ 2 belong to the two-dimensional E u irreducible representation.

Lattice model
For concreteness, we construct an explicit lattice model that describes a class of Dirac semimetals such as Cd 3 As 2 and Au 2 Pb.The coefficient functions of Hamiltonian in Eq. ( 1) are given by 39,45 where M , t xy , t z , v, β , and γ are material-dependent parameters.The energy eigenvalues are given by If t z > (M − 2t xy ) > 0, the Hamiltonian hosts a pair of Dirac points at (0, 0, ±k 0 ) as shown in Fig. 1(a).Here, k 0 is determined by M − 2t xy − t z cos k 0 = 0.These Dirac points are protected by the C 4z symmetry 39 .Due to the C 4z , the four bands on the k z axis can have different C 4z eigenvalues, which lead to fourfold degenerate Dirac points.

Low-energy effective Hamiltonian
Near the Dirac points (0, 0, ±k 0 ), the low-energy effective Hamiltonian takes the form of Dirac Hamiltonian, which is given by where v z = t z k 0 .The energy spectrum shows anisotropic energy-momentum dispersion, which is given by

Symmetry-lowering distortions
In the absence of lattice distortions, Cd 3 As 2 [27][28][29] and Au 2 Pb [32][33][34] share the same D 4h point group symmetry and show no superconductivity.However, both materials showed superconductivity after the structural phase transition under pressure or cooling, and the superconducting critical temperature increases with the pressure 28,34 .At the high pressure, Cd 3 As 2 becomes a monoclinic lattice having C 2h point group symmetry 28 and Au 2 Pb becomes an orthorhombic lattice having D 2h point group symmetry 32 .Thus, IS is preserved even under lattice distortions.In addition, the superconductivity appears under the small lattice distortions in the hydrostatic experiments 28,34 .Therefore, we assume that both TRS and IS are preserved under lattice distortions and the effect of the lattice distortion can be implemented as a perturbation 54 .
We now classify the possible symmetry-lowering lattice distortions.The form of the perturbation Hamiltonian for the lattice distortions is given by where d i (k) is a real-valued function of momentum and Γ i is the gamma matrix.Because Γ 1 , Γ 2 , Γ 3 , and Γ 4 are odd under T and P, the coefficient functions d 1 (k), d 2 (k), d 3 (k), and d 4 (k) are odd functions with respect to k.Similarly, the coefficient a 0 (Sk), a 5 (Sk) Symmetry constraints on a i (k).They are determined by Eq. (5).If the coefficient function is proportional to itself, a i (Sk) = ±a i (k), the overall sign is denoted.If not, the explicit form is denoted.functions d 0 (k) and d 5 (k) are even functions with respect to k.Thus, the allowed lattice distortion terms can be either k odd Γ 1,2,3,4 or k even Γ 0,5 types.
Because we assume TRS and IS to remain under lattice distortions, the Hamiltonians for distorted and undistorted DSM have the same form of H = ∑ i a i (k)Γ i .The only difference between the two Hamiltonians is the transformation properties of the coefficient function a i (k).In the absence of lattice distortions, a i (k) needs to satisfy all transformation properties under all symmetry operations of D 4h point group in Table 2.However, in the presence of lattice distortion, a i (k) only needs to satisfy the transformation properties under the remaining symmetry operations, so a i (k) is less constrained.

Lattice Hamiltonian with lattice distortions
To discuss the effect of lattice distortions explicitly, we introduce the possible symmetry-lowering lattice distortions in the lattice model in Eqs.(7-11).For weak lattice distortions, the lattice distortions are approximately proportional to sin k i and cos k i as only nearest neighbor hoppings are relevant.Because we are interested in the Dirac physics near the Dirac points (0, 0, ±k 0 ), we assume that k x /k z 0 and k y /k z 0, which implies that sin k x and sin k y are smaller than sin k z and cos k i .Hence, sin k z and cos k i are dominant momentum dependent terms in the leading order, and the allowed lattice distortions are either sin k z Γ 1,2,3,4 or cos k i Γ 0,5 types.Because cos k i Γ 0,5 types are included in the trivial A 1g class of D 4h point group, they do no break any symmetry.On the other hand, sin k z Γ 1,2,3,4 types are included in B 1g , B 2g , and E g , and they break the crystal symmetry properly, which are summarized in Table 3.Therefore, in the leading order, there are four types of symmetry-lowering lattice distortion, which are given by where n i is the strength of each lattice distortion.For convenience, each lattice distortion is denoted as n i type lattice distortions in this work.From now on, we will consider these four types of symmetry-lowering lattice distortions, and the possible higher-order terms are discussed in Sec.S2 in Supplementary Information.Therefore, the coefficient functions in Eq. ( 1) are given by Under lattice distortion, the fourfold rotation symmetry is broken.Thus, the Dirac point is gapped, which can be seen from the energy eigenvalues on the k z axis, Thus, the Dirac point is gapped unless As a result of the gap-opening, the DSM becomes a 3D topological insulator because of the band inversion at the Γ point 36,39 .Counting all the parity eigenvalues for the time-reversal-invariant momenta (TRIM) points of the bulk Brillouin zone (BZ) 1,55 gives a nontrivial Z 2 invariant.

The effect of lattice distortions
The four types of symmetry-lowering lattice distortions in Eq. ( 16) are classified according to the irreducible representation of D 4h group.The symmetry-lowering lattice distortions break D 4h point group symmetry into its subgroup symmetry, which is summarized in Table 3.The n 1 and n 2 type lattice distortions are included in the one-dimensional class B 1g and B 2g , and break D 4h point group symmetry into D 2h and D 2h , respectively.The n 3 and n 4 type lattice distortions are included in the two-dimensional class E u and break D 4h point group symmetry into C 2h .Note that n 2 type lattice distortion is related to the n 1 type lattice distortion via π/4 rotation, while n 4 type lattice distortion is related to the n 3 type lattice distortion via π/2 rotation.
We investigate the explicit effects of the lattice distortions on the crystal systems and the Fermi surfaces using the lattice model in Eq. (17). Figure 1 shows the crystal structures, the 3D band structures, and Fermi surfaces under various lattice distortions.Under n 1 type lattice distortion, the crystal system and Fermi surface are elongated along x or y direction, C 4z symmetry is broken, the Dirac point is gapped, and the crystal system becomes orthorhombic [Fig.1(b,g)].Similarly, under the n 2 type lattice distortion, the crystal system and Fermi surface are elongated along diagonal lines either x = y or x = −y, C 4z symmetry is broken, the Dirac point is gapped, and the crystal system becomes orthorhombic [Fig.1(c,h)].We denote the symmetry point group of this right rhombic prism as D 2h .Under n 3 type lattice distortion, the crystal structure undergoes structural phase transition from tetragonal to monoclinic [Fig.1(d)].Two Dirac points in the band structure are shifted oppositely along k y direction and the centers of each Fermi surfaces are also oppositely shifted along the same k y direction [Fig.1(h)].Similar effects occur under n 4 type lattice distortion [Fig.1(e,j)] because n 4 type lattice distortion are related with the n 3 type lattice distortion via π/2 rotation.The point groups of these distorted systems under n 3 and n 4 type lattice distortions are denoted as C 2h(x) and C 2h(y) , respectively.Therefore, the four types of symmetry-lowering lattice distortions explain the lattice distortions of Cd 3 As 2 and Au 2 Pb under pressure.

Low-energy effective Dirac Hamiltonian under lattice distortions
Near the Dirac points (0, 0, ±k 0 ), the coefficient functions of the low-energy effective Hamiltonian can be approximated as With this low-energy effective Hamiltonian, we show that the lattice distortion acts as a Dirac mass term and increases DOS at Fermi surface.We assume that the Fermi level is slightly above the Dirac points in undistorted lattice, or near the bottom of the conduction band minima after gap-opening at the Dirac points.
For n 1 and n 2 type lattice distortions, the low-energy effective Hamiltonian is given by So, n 1 and n 2 type lattice distortion terms act as Dirac mass terms.The energy eigenvalue is given by where By the assumption of the total electron number conservation under a weak lattice distortion, the lattice distortion dependent DOS at the Fermi surface is given by which indicates that DOS at the Fermi level is enhanced under the lattice distortion.Here, µ 0 indicates the chemical potential of the undistorted lattice.See the detailed derivations in Sec.S2.4 in Supplementary Information.Next, we consider the n 3 type lattice distortion.The n 3 type lattice distortion shifts the gap minima along the k y direction from (0, 0, ±k 0 ) to (0, ±k is the Dirac mass term.The energy eigenvalue is given by Similar to n 1 and n 2 type lattice distortions, DOS at the Fermi surface are given by DOS(n 3 ) = 1 which means that the DOS at the Fermi level is enhanced under n 3 type lattice distortion.Similarly, for n 4 type lattice distortion, the low-energy effective Hamiltonian and DOS are easily calculated because n 3 and n 4 type lattice distortions are related via π/2 rotation.

Multiple symmetry-lowering lattice distortions
So far, we have considered only one type of lattice distortions.However, more than two types of lattice distortions can be turned on simultaneously.In this case, the final subgroup symmetry determines the crystal system and its physical properties.When both n 1 and n 3 types lattice distortions are turned on, the remaining subgroup has P, C 2x , M yz symmetries.This subgroup is the same point group of the distorted Dirac semimetal under single n 3 type lattice distortion.In other words, under n 3 type lattice distortion, the addition of n 1 type lattice distortion is also allowed.A similar argument can be applied to n 2 and n 4 types lattice distortions.When both n 1 and n 2 type lattice distortions are turned on, the remaining symmetries are P, C 2z , M xy symmetries.We denote this point subgroup as C 2h(z) , and we will not consider this case seriously because there is no real material that corresponds to this case.Similarly, the other combinations such as (n 2 , n 3 ), (n 1 , n 4 ), (n 3 , n 4 ), (n 1 , n 2 , n 3 ), (n 1 , n 2 , n 4 ) break all crystal symmetries except the inversion, and hence these cases are not interested in this work.

BdG Hamiltonian
To discuss the effects of lattice distortions on the superconductivity in doped DSM, we construct the Bogoliubov-de Gennes (BdG) Hamiltonian within mean-field approximation while keeping TRS and the crystal symmetry 56,57 .The BdG Hamiltonian is given by where τ i is the Pauli matrices in the Nambu space.∆(k) and µ are a pairing potential and a chemical potential, respectively.H(k) is the normal state Hamiltonian in Eq. ( 1).The basis is taken as While the pairing mechanism of doped DSM is not known yet, we assume the following onsite density-density interaction as a superconducting pairing interaction 36,37,58,59 : where n i (x) is the electron density operators for ith orbital (i = 1, 2).U and V are intra-orbital and inter-orbital interaction strengths, respectively, and we assume that at least one of them is attractive and responsible for superconductivity.Because the pairing interaction depends on the orbital and is local in x, the mean-field pairing potential is orbital dependent but momentum independent: ∆(k) = ∆.

Symmetry of BdG Hamiltonian
The BdG Hamiltonian in Eq. ( 23) has time-reversal symmetry T , particle-hole symmetry C, and chiral symmetry Γ: where T = is y σ 0 τ 0 K and C = is y σ 0 τ y K are time-reversal and particle-hole symmetry operators, respectively, and Γ = TC = s 0 σ 0 τ y is the chiral operator.K is the complex conjugation operator.Therefore, the BdG Hamiltonian belongs to in DIII class according to the classification table of topological insulator and superconductor 60 .
If the pairing potential satisfies P∆(k)P −1 = η P ∆(−k), the BdG Hamiltonian has the inversion symmetry: PH (k) P−1 = H (−k), with P = diag(P, η P P), where P and P are the inversion operators for the DSM and BdG Hamiltonians, respectively, and η P is the inversion parity.
If η P = 1 (η P = −1), the superconducting phase is an inversion-even-parity (inversion-odd-parity) superconductor.For a single-orbital superconductor, P is the identity operator, and an inversion-odd-parity (inversion-even-parity) pairing is equivalent to the spin-triplet (spin-singlet) pairing.However, because of the spin-orbit coupling and multi-orbital band structure, the pairings are more complex in our case.
From now on, we consider momentum independent pairing potentials, ∆(k) = ∆, because we assume onsite pairing interaction as discussed in Eq. (26).In the absence of lattice distortions, the BdG Hamiltonian has D 4h point group symmetry 37 .If a pairing potential satisfies the transformation property of U∆s y U T s y = η U ∆ under a symmetry operation of D 4h point symmetry group, the BdG Hamiltonian satisfies the corresponding symmetry: where U is the symmetry operator in spin and orbital spaces, η U is a phase factor, and Ũ = diag(U, η U s y U * s y ) is the extended symmetry operator in the Nambu space.
For the generators of D 4h point group, if the pairing potential satisfies C 4z ∆s y C T 4z s y = η C 4z ∆ with η C 4z = e iπr/2 (r = 0, . . ., 3) and C 2x ∆s y C T 2x s y = η C 2x ∆ with η C 2x = ±1, then the BdG Hamiltonian satisfies the corresponding rotation symmetry: Pairing fermion bilinear matrix form where M = diag (M, η M s y M * s y ) is a mirror operator for BdG Hamiltonian and k (k ⊥ ) is the momentum vector parallel (perpendicular) to the mirror plane.The η M is the mirror parity of the pairing potential under the mirror operation M.
In Table 4, the transformation properties of all possible pairing potentials under the rotation and mirror operators are summarized.The details of each pairing potential will be discussed below.

Pairing potentials
We now investigate the possible superconducting pairing potentials in the presence of lattice distortions.Since we are considering multi-orbital superconductivity in the basis of two spins and two orbitals, pairing potentials can be represented as a product of spin Pauli matrices and orbital Pauli matrices, which leads to sixteen matrices.Among them, only six matrices are allowed because of the fermion statistics (∆s y = s y ∆ T ).We denote them as ∆ 1 , ∆ 1 , ∆ 2 , ∆ 3 , ∆ 41 , and ∆ 42 , whose forms and properties are listed in Table 4. Due to Pauli's exclusion principle, the fermion bilinear form of each pairing potential shows antisymmetric property under the particle exchange.Because the pairing potential is momentum independent, the spatial part is symmetric, while the spin-orbital part is antisymmetric under the particle exchange.Thus, if the spin part is singlet, the orbital part is triplet, and vice versa.Therefore, ∆ 1 's and ∆ 41 are the spin-singlet orbital-triplet pairings and ∆ 2 , ∆ 3 , and ∆ 42 are the spin-triplet orbital-singlet pairings as shown in the bilinear form in Table 4.
Six pairing potentials can be classified according to the irreducible representations of the unbroken point group, and the superconducting critical temperatures for the pairing potentials in the different classes are independent 36,37,[57][58][59] .In the absence of lattice distortions, the pairing potentials are classified according to the D 4h group: ∆ 1 's, ∆ 2 , ∆ 3 and ∆ 4 's belong to A 1g , B 1u , B 2u and E u irreducible representations, respectively, which are summarized in Table 4.
The pairing potential belonging to a specific irreducible representation of the D 4h group can be decomposed into a combination of different irreducible representations depending on the symmetry of the distorted lattice.Some pairing potentials in the D 4h group's individual representations can be included in the same representation and vice versa.As an example, in the   5.

Superconducting nodal structure
In this subsection, we classify the superconducting nodal structures under lattice distortions and study the symmetry and topology that guarantee the classified nodal structures.
Figure 2 shows the typical nodal structures of superconducting phases of the doped DSM under lattice distortions.There are three types of nodal structures: Full gap, point nodal, and line nodal structures, which are summarized in Table 6.For ∆ 1 and ∆ 1 superconducting phases, ∆ 1 phase is fully gapped and ∆ 1 phase has two nodal rings regardless of lattice distortions [Fig.2(a-e)].For ∆ 2 and ∆ 3 phases, nodal points exist at the intersections between the k z axis and the Fermi surfaces in the absence of lattice distortions [Fig.2(a)].These points are known to be protected by C 4z symmetry 36,37 .Even under lattice distortions, if there is an unbroken mirror symmetry, the topologically protected nodal points can exist and they are protected by the corresponding mirror symmetry [Fig.2(b-e)].For ∆ 41 and ∆ 42 phases, there are accidental nodal points at the intersections between the k z axis and the Fermi surfaces in the absence of lattice distortions [Fig.2(a)].However, in the presence of lattice distortions, if there is an unbroken mirror symmetry, there can exist the topologically protected nodal points in the corresponding mirror plane [Fig.2(b-e)].Note that all nodal points under lattice distortions in Fig. 2(b-e) are protected by the topological mirror winding numbers, as discussed later.
We now analytically investigate the condition of nodal points in each superconducting phase.Usually, nodal points can exist where the quasi-particle energy spectrum vanishes E (k) = 0, which gives a set of equations for the momentum variables (k x , k y , k z ).If the number of variables N V is greater than or equal to the number of independent equations N E , then nodal structures can exist.That is, N E ≤ N V = 3 is the necessary condition for the existence of the nodes.Moreover, if there is mirror symmetry, the necessary condition changes because the number of independent variables is reduced in the corresponding mirror plane.That is, the necessary condition becomes N E ≤ N V = 2.If there is additional mirror symmetry, the necessary condition can be further reduced to N E ≤ N V = 1 on the intersection of two mirror planes.
First, we consider ∆ 1 and ∆ 1 superconducting phases.The full gap structure of ∆ 1 phase is directly seen from the energy eigenvalues of where |a| = ∑ 5 i=1 a i (k) 2 .Unless ∆ 1 = 0, ∆ 1 phase is fully gapped.For ∆ 1 phase, the energy eigenvalues are given by From E (k) = 0, one can obtain the following equations: Because the number of variable (N V = 3) is larger than the number of equation (N E = 2), a one-dimensional solution can exist, which leads to the nodal lines.Because this argument works regardless of the lattice distortions, the nodal rings can exist for all cases in Fig. 2. On the other hand, under some lattice distortions, a mixture of ∆ 1 and ∆ 1 phases is allowed when ∆ 1 and ∆ 1 are in the same representation as shown in Table 5.In such case, the gap structures have full gap (nodal lines) when See the detailed calculation in Sec.S3 in Supplementary Information.Next, consider the ∆ 2 and ∆ 3 superconducting phases.In the absence of lattice distortions, the nodal points in ∆ 2 and ∆ 3 phases are protected by C 4z symmetry 36,37 .On the other hand, under lattice distortions, a mirror symmetry can protect the nodal points that appear in Fig. 2(b,c,e).For ∆ 3 phase, the energy eigenvalues are given by From E (k) = 0, we get the following equations: Because N E = 4 is larger than N V = 3, there seems to be no allowed nodal point.However, mirror symmetries can allow nodal points.For example, consider D 2h point group with M yz and M xz mirror symmetries.Under the M yz mirror operation, a 1 (k) and a 4 (k) are odd according to Table 2, which gives a 1 (k) = a 4 (k) = 0 at the mirror plane (0, k y , k z ).Similarly, M xz mirror symmetry gives a 2 (k) = a 4 (k) = 0 at the mirror plane (k x , 0, k z ).Thus, along the k z axis, a 1 (k) = a 2 (k) = a 4 (k) = 0 and Eq. ( 39) is reduced to Because N E = 1 is equal to N V = 1, nodal points can exist as shown in Fig. 2(b).However, when M yz and M xz mirror symmetries are broken, the nodal points for the ∆ 3 phase are not protected as shown in Fig. 2(c,d).
Similarly, the nodal points in ∆ 2 phase can be understood using M 110 and M 1 10 mirror symmetries.These mirror symmetries allow nodal points on the k z axis in Fig. 2(c).On the other hand, when M 110 and M 1 10 mirror symmetries are broken, the nodal points disappear as shown in Fig. 2(b,d).For the C 2h(z) case, a mixture of ∆ 2 and ∆ 3 phases is possible because ∆ 2 and ∆ 3 are included in the same A u representation.However, there is no allowed nodal point as shown in Fig. 2(d) because there is no helpful mirror symmetry.See the details in Sec.S3 in Supplementary Information.
Finally, consider ∆ 41 and ∆ 42 phases.Without lattice distortions, there are accidental nodal points on the k z axis [Fig.2(a)].The existence of such nodal point is easily seen using four mirror symmetries M xz , M yz , M 110 , and M 1 10 .These mirror symmetries force a i (k) = 0 for i = 1, • • • , 4 on the k z axis according to Table 2.Then, the equations for nodal points are given by Because N E = N V = 1, the nodal points exist.Because the ∆ 41 and ∆ 42 pairing potentials included in E u representation of D 4h point symmetry group, they break the D 4h symmetry spontaneously to D 2h .Hence, some of non-zero a i (k are spontaneously generated and the corresponding conditions are introduced, which makes the nodal points vanish.Thus, these nodal points are accidental.However, under lattice distortions, the nodal points can be protected by the unbroken mirror symmetry.For example, when the point group is D 2h under the n 1 type lattice distortion, ∆ 41 and ∆ 42 are included in the different representations and thus we can consider each phase separately.For ∆ 41 phase, a 1 (k) = a 4 (k) = 0 on the mirror plane (0, k y , k z ) due to M yz symmetry.Then, the equations for nodes are given by Because

Stability of nodal structures
There are two types of nodes in Table 6, which are symmetry-protected node and topologically-protected node.In this subsection, we investigate the stability of them.

Chiral winding number
Because of the chiral symmetry of the BdG Hamiltonian, the nodal lines can be protected by a chiral winding number 4,60,61 .
The chiral winding number is defined along a path C enclosing a singular point in the Brillouin zone as shown in Fig. 3(a): where Γ is the chiral operator.As shown in Sec.S4 in Supplementary Information, the transformation property of the winding number under PT symmetry is given by where the parity η A,B = ±1 is determined by the relation AB = η A,B BA.For the inversion-even-parity (inversion-odd-parity) pairing potential, η Γ, PT is −1 (+1).Thus, the chiral winding number is zero for the inversion-odd-parity superconductor and only the inversion-even-parity superconducting phases having ∆ 1 and ∆ 1 pairing potentials can have a nontrivial chiral winding number.
Because ∆ 1 phase is fully gapped, the chiral winding number is zero.On the other hand, two nodal rings in ∆ 1 phase are topologically protected by the chiral winding numbers.The calculated chiral winding numbers around the nodal rings are W = ±2 [Fig.3].These chiral winding numbers do not change even under the lattice distortions because chiral winding number depends only on T , C, P, and Γ symmetries.Thus, the topologically-protected nodal rings in ∆ 1 phase exist regardless of the lattice distortion [Fig.2].

Mirror chiral winding number
If there is mirror symmetry, the BdG Hamiltonian commutes with the mirror symmetry operator in the mirror plane: where M is a mirror operator and k M is the momentum vector located in the mirror plane.Then, the BdG Hamiltonian can be block-diagonalized according to the mirror eigenvalues λ = ±i.Besides, if the mirror operator commutes with the chiral operator, the chiral operator also can be block diagonalized according to the same mirror eigenvalue.Then, the winding number W λ in each mirror eigenvalue sector can be defined.The condition in Eq. ( 46) is satisfied only when the pairing potential is mirror even.The reason is as follows: In our convention, the mirror operator for BdG Hamiltonian is defined as where M and s y M * s y are mirror operators for electron part and hole part, respectively.η M = ±1 is the mirror parity of a pairing potential, which is given in Table 4.Because the mirror operator commutes with the time-reversal operator [T, M] = 0, all the mirror operator satisfies s y M * s y = M.Then, M = Mτ 0 ( M = Mτ z ) for the mirror-even-parity (mirror-odd-parity) pairing potential.Thus, only the mirror-even-parity superconducting phase satisfies the condition of Eq. ( 46).Furthermore, the mirror chiral winding number can be defined as W M = W i −W −i , where W λ is the chiral winding number for each block having a mirror eigenvalue λ .The mirror chiral winding number W M can also be defined for a path C that encloses the Dirac point in the mirror plane as shown in Fig. 3(b).When the path C is parametrized by θ ∈ [0, 2π), the mirror chiral winding number is given by 37,62 ∆ 2 and ∆ 3 phases In the absence of lattice distortions, the C 4z symmetry protects the nodal points by assigning different eigenvalues 36,37 .The same nodal points are also topologically protected by the mirror chiral winding number in Eq.( 47) because the ∆ 2 and ∆ 3 pairing potentials are mirror-even.For ∆ 3 pairing potential, which is mirror-even under M xz and M yz , the calculated mirror chiral winding numbers around the nodal points are ±2 [Fig.3(c)].Similarly, the nodal points in the ∆ 2 phase are topologically protected by M 110 and M 1 10 mirror chiral winding numbers.Even though C 4z symmetry is broken under lattice distortions, the mirror chiral winding number topologically protects the nodal points if the corresponding mirror symmetry is unbroken.For example, consider D 2h point group which has M xz and M yz mirror symmetries.Among ∆ 2 and ∆ 3 pairings, ∆ 3 pairing is mirror even under M xz and M yz .Thus, the nodal points in the ∆ 3 phase are topologically protected by the corresponding mirror chiral winding numbers [Fig.3(c)].
Furthermore, the nodal points are positioned on the k z axis because C 2z symmetry gives an additional constraint as follows: Let W M (k) denote a mirror chiral winding number at k.Then, the mirror chiral winding number at C 2z k is related with that at k by where η C 2z is the parity of the pairing potential under C 2z transformation.The detail derivation is in Sec.S4 in Supplementary Information.Since η C 2z = 1 for ∆ 2 and ∆ 3 , W M (k) = W M (C 2z k), which means that the mirror chiral winding numbers are the same for the two nodal points that are related by C 2z rotation.Now, let us assume that a nodal point on the k z axis in the absence of lattice distortions deviates from the k z axis under the n 2 type lattice distortion.Due to the C 2z symmetry, there exists another nodal point having the same mirror chiral winding number.Thus, the total mirror winding number under lattice distortion becomes twice the original winding number, which is a contraction with the topological charge conservation.Therefore, the nodal points should be located on the k z axis under the n 2 type lattice distortion.
A similar argument can be applied to the D 2h case having M 110 and M 1 10 mirror symmetries.The nodal points in the ∆ 2 phase is topologically protected by the M 110 and M 1 10 mirror chiral winding numbers and the nodal points are located on the k z axis due to the C 2z symmetry [Figs.2(c) and 3(b)].For C 2h(x) case, M yz is unbroken while C 2z is broken.Thus, nodal points on M yz plane in ∆ 3 phase are protected by the M yz mirror chiral winding number and can be deviated from k z axis due to the C 2z symmetry breaking [Fig.2(e)].In the absence of lattice distortions, the nodal points in each ∆ 41 and ∆ 42 phases [Fig.2(a)] are accidental nodal points because a single phase, either ∆ 41 or ∆ 42 phase, would break the D 4h point group symmetry spontaneously.Only if we neglect such lattice symmetry breaking, the accidental nodal points can be understood to be protected by the different eigenvalues of C 2z and s z symmetry operators (see the details in Sec. 3 in Supplementary Information).Note that the existence of the accidental point nodes also can be verified via C 4z symmetry 36,37 .In the viewpoint of topological winding numbers, the mirror chiral winding numbers are zero in the absence of lattice distortions [Fig.3(d)] .Due to the C 2z symmetry, Eq. (48) gives which implies that W M = 0 on the k z axis.Here, η C 2z = −1 is used for ∆ 41 and ∆ 42 .Thus, the nodal points are not topologically protected for D 4h case.However, under lattice distortions, nodal points can be topologically protected by the mirror chiral winding number.Let us consider the D 2h point group under the n 1 type lattice distortion.Since ∆ 41 and ∆ 42 pairings are mirror-even under M yz and M xz operations, the corresponding mirror chiral winding number protects nodal points in each mirror plane [Fig.2(b)].The calculated mirror chiral winding numbers are W M = ±2 [Fig.3(d)].Note that the nodal points are off the k z axis and the  Finally, we discuss a gap structure change of ∆ 42 phase under n 1 type lattice distortion [Fig.3(d)].When n 1 = 0, each nodal points has W M = 0 and a quadratic energy-momentum dispersion relation along the k x .With the increasing lattice distortion, nodal points with W M = ±2 are created pairwise from a nodal point with W M = 0, and linear energy-momentum dispersion relation for all three momentum directions appears.Similar gap structure changes occur under the other lattice distortions.

Surface spectrum
Surface Andreev bound state (SABS) in superconducting phases of the topological DSM have been studied in the absence of lattice distortion. 37.In this subsection, we systematically investigate SABS in superconducting phases under lattice distortions.There are four types of gapless surface Majorana states under lattice distortions.Three types are topologically protected by mirror chiral winding, mirror Chern, and zero-dimensional winding numbers.The fourth type is protected by mirror symmetry and corresponding eigenvalues.
Using the Möbius transformation based method 63 , we calculate the surface band structures.Figure 4 shows the numerically obtained surface spectra for (010) surface in various superconducting phases under lattice distortions.For ∆ 1 and ∆ 1 phases, there is no SABS; ∆ 1 phase is fully gapped and topologically trivial, and ∆ 1 phase has two nodal lines having opposite chiral winding numbers as shown in Fig. 3(a), which does not have protected SABS because of the positions and shapes of two nodes in momentum space.On the other hand, ∆ 2 , ∆ 3 , ∆ 41 , and ∆ 42 have various types of SABS [Fig.4], which are summarized in Table 7.
Without loss of generality, we will focus on the (010) surface and the surface Brillouin zone (k x , k z ).A similar analysis for the (010) surface can be easily applied to the other surfaces such as (100), (110) planes, because the results for the other plane only depend on the mirror symmetries and the transformation properties of the pairing potentials under the unbroken symmetries.For convenience, we consider the surface states in the three regions: Region I, II, and III, which are (0, k 1 )-(0, k 2 ), (k 2 , 0)-(0, 0), (0, 0)-(π/2, 0), respectively.Here, k 1 and k 2 (k 1 > k 2 > 0) indicate two intersecting points between the upper Fermi surface and the k z axis.
First, we consider the flat SABS in the Region I, which is topologically protected by the nontrivial mirror chiral winding number in Eq. (47).For example, let us consider ∆ 3 phase and M yz mirror symmetry.For D 4h , D 2h , and C 2h(x) cases, M yz mirror is unbroken and ∆ 3 has odd parity under M yz , which leads to the opposite mirror chiral winding numbers (W M = ±2) for two nodal points near the upper Fermi sphere as shown in Fig. 3(c).Then, there exists a flat SABS on (010) surface as shown in Fig. 4(b,f,j).To understand such SABS on (010) surface, the mirror winding number W M (k z ) along the mirror invariant k z axis is defined as 37 which is nontrivial between nodal points.Therefore, between the nodal points, there exists a flat SABS.Similarly, for ∆ 41 phase, M yz mirror symmetry gives nontrivial mirror chiral winding numbers, which guarantees the existence of the zero-energy flat SABS in the Region I on (010) surface [Fig.4(k)].Note that, under the n 3 type lattice distortion, the mixture of ∆ 3 and ∆ 41 phases are allowed.But the flat SABS is still present due to the M yz mirror chiral winding number.Second, we consider the gapless SABS protected by the mirror Chern number C M .The topological mirror superconducting phases 36,64 are allowed for ∆ 2 and ∆ 3 phases because ∆ 2 and ∆ 3 pairing potentials are M xy mirror-odd and the corresponding mirror Chern numbers for each mirror eigenvalue block are nontrivial.Under the n 1 (n 2 ) type lattice distortion, ∆ 2 (∆ 3 ) phase is fully gapped, and the mirror Chern number defined in M xy plane is nontrivial (C M = ±2), which leads to a topologically-protected Majorana states on M xy plane.For example, see the surface spectra in the Region III in Fig. 4(e).
Third, we consider the gapless SABS protected by the zero-dimensional topological number.Since ∆ 2 and ∆ 42 pairings are odd under M yz , a zero-dimensional topological number ρ(k x ) can be defined using ΓM yz 36, 37 .Then, the zero-dimensional topological number protects the gapless state in the Region III.See the surface spectra at the Region III in Fig. 4(d, h, i, l) and Table 7.Similarly, ∆ 2 and ∆ 3 pairings are odd under M yz , a zero-dimensional topological number ρ(k z ) is defined using CM yz 36, 37 , which protects the gapless states in the Region II for D 4h and D 2h cases.See the surface spectra at the Region II in Fig. 4(a, b, e, f) and Table 7.
Fourth, we consider the gapless SABS protected by mirror eigenvalues.If the pairing potential has an odd parity under the mirror operation, the mirror eigenvalues for the electron and hole bands are different, which protects the band crossing of surface states 36,37 .For example, consider ∆ 2 phase and M yz symmetry.Because (k x , k y , k z ) → (−k x , k y , k z ) under M yz , the mirror eigenvalues are properly defined on the k z axis.Moreover, ∆ 2 pairing has odd parity under M yz symmetry.Hence, the different mirror eigenvalues protect the gapless states in the Region I. See Fig. 4(a, e, i).Similarly, ∆ 42 phases has odd parity under M yz , which protects the gapless states in the Region I and II.See Fig. 4(d, h, l).
In summary, we find the various types of surface states depending on the pairing potentials and lattice distortions.Even under the lattice distortions, most of the inversion-odd-parity superconducting phases have gapless SABS, which may be observed as zero bias conductance peak (ZBCP) in experiments.

Superconducting critical temperature and phase diagram
In this subsection, we study superconducting critical temperatures and their enhancements under lattice distortions.We also investigate the phase diagram for the various superconducting phases under lattice distortion.
In the weak-coupling limit, the superconducting critical temperature T c can be calculated by solving the linearized gap equation and a phase diagram for various pairing potentials is obtained by comparing the critical temperatures 37,[56][57][58][59] .The linearized gap equation can be expressed using the pairing susceptibility 37,[56][57][58][59] .The pairing susceptibility χ i for each pairing potential ∆ i is given by Here, β = 1/(k B T ) is the inverse temperature, k B is the Boltzmann constant, ω n is the Matsubara frequency, and ∆ i is the matrix representation of a pairing potential listed in Table 4. G 0 (k) = P k iω n −ε k is the single-particle Green's function of the normal state and P k ≡ ∑ m=1,2 |φ m,k φ m,k | is the projection operator onto the two degenerate Bloch states in the conduction bands.Here, Then, the superconducting susceptibility has the following generic form: where f i (k) is momentum dependent form factor.The explicit expressions for form factors f i (k) are given in Sec.S5 in Supplementary Information.With these susceptibilities, we now solve the linearized gap equation.The linearized gap equations are obtained by minimizing the mean-field free energy in the weak coupling limit.Since superconducting critical temperatures with pairing potentials in the same classes are not independent, the ∆ i 's in the same class can appear in the same linearized gap equation.
First, consider the gap equation in the absence of lattice distortions.According to the irreducible representation of D 4h , ∆ 1 's, ∆ 2 , ∆ 3 and ∆ 4 's belong to A 1g , B 1u , B 2u and E u irreducible representations (see Table 5).Then, the gap equations are given by where χ i, j is the generalized superconducting susceptibility for mixed pairings ∆ i and ∆ j by replacing the second ∆ i with ∆ j in Eq. ( 51).Using the low-energy effective Hamiltonian in Eq. ( 13), the superconducting susceptibility can be further simplified and hence one can solve the gap equation analytically.Using an ellipsoidal coordinate, the superconducting susceptibility can be represented as a product of two independent integrals (see more details in Sec.S5 in Supplementary Information):

17/42
Here, the radial integral part R(β c ) is given by where E is an integration variable and ω D is the energy cutoff of the pairing potential.The angular integral part Ω i (µ) is given by where the form factor f i (k) is represented as a function of r, θ and φ in the ellipsoidal coordinates.After the integration over θ and φ , the susceptibilities can be obtained as follows: where . Then, the linearized gap equations are given by If we denote the critical temperature c for a pairing potential ∆ i , then the gap equations are given by R(T Because R(x) is a monotonically decreasing function with respect to x, T (2) . Thus, the highest T c is determined among T When the chemical doping is low, the superconducting phase diagram for undistorted Dirac semimetal is shown in Fig. 5(a).When the intra-orbital interaction U is strong, the conventional s-wave superconductivity with pairing potential ∆ 1 is the dominant phase.However, with the increasing inter-orbital interaction V , the unconventional superconducting phase with inter-orbital pairing potential ∆ 2 or ∆ 3 can emerge.Figure 5(b) shows the numerically obtained critical value of U/V ratio using the lattice Hamiltonian.Thus, by controlling the U/V ratio, both conventional and unconventional superconductivity can emerge for for the large range of chemical doping.The calculated value of U/V ratio is similar with 2/3 using the low-energy effective Hamiltonian, which means that ∆ 2 or ∆ 3 phase can emerge for the large range of chemical doping.
Next, consider the effect of n 1 and n 2 types of lattice distortions on the superconducting temperatures and the phase diagrams.When n 1 type lattice distortion is turned on, the point group becomes D 2h .In this case, only ∆ 1 and ∆ 1 belong to the same A g class, and the others are belong to different classes (see Table 5).So the linearized gap equation is given by Similar to D 4h case, the susceptibility can be analytically calculated when the chemical doping level is small.The relevant gap equations that determine the phase map are given by Thus, the phase boundary is given by , which corresponds to the black arrows in (c,e).T 0 is the critical temperature of the ∆ 1 phase in the absence of the lattice distortions.
Similarly, the other cases can be calculated.See the details in Supplementary Information.Figure 5 shows the numerically calculated phase maps under the n 1 and n 2 types of lattice distortions using the low-energy effective Hamiltonian.The phase diagrams are plotted in the plane of the U/V ratio versus strength of n 1 or n 2 type lattice distortion.In each diagram, the dominant phases are conventional spin-singlet ∆ 1 phase and unconventional spin-triplet ∆ 2 or ∆ 3 phase depending on the parameters.When U/V is small (large) enough, ∆ 2 or ∆ 3 (∆ 1 ) phase emerges.Remarkably, the unconventional superconductivity can emerge with increasing lattice distortions.As an example, near the phase boundary of U/V ≈ 0.7, there is a phase transition between conventional superconducting ∆ 1 and unconventional superconducting ∆ 2 phases when n 1 increases [see the black arrow in Fig. 5(c)].To see this phase transition more clearly, we plot the normalized superconducting critical temperatures along the black arrow [Fig.5(d)].When n 1 = 0, the ∆ 1 phase is dominant.With increasing n 1 , the superconducting critical temperatures for ∆ 2 are increasing, which leads to the ∆ 2 superconducting phase under enough lattice distortion.Note that T c 's for ∆ 1 , ∆ 2 , ∆ 41 , and ∆ 42 increase while T c for ∆ 3 decreases with the increasing n 1 [Fig.5(d)].This can be explained by the expectation values of the Cooper pairings and spin-orbital texture at the Fermi surface, which will be discussed later.Because n 1 and n 2 type lattice distortions are related with π/4 rotation, similar features are observed except for the exchange of ∆ 2 and ∆ 3 phases [Fig.5(e,f)].
For the n 3 type lattice distortion, similar features can be observed in Fig. 6.Under the n 3 type lattice distortion, n 1 type lattice distortions also can be involved as discussed before.Thus, we plot three representative phase diagrams for n 1 = 0.0, 0.05, and 0.1.Surprisingly, when n 1 = 0, ∆ 2 and ∆ 3 phases are degenerate, and they are dominant unconventional phases as shown in Fig. 6(a,d).With increasing n 1 , the region of the unconventional phase ∆ 2 increases [Fig.6(a-c)] and the degenerate ∆ 2 and ∆ 3 phases become distinguishable.
Under n 1 , n 2 , and n 3 lattice distortions, the T c 's of ∆ 2 , ∆ 3 , ∆ 41 , and ∆ 42 increases much more than that of ∆ 1 [Figs.5(d,f), 6(d-f)], and hence the unconventional superconducting phases emerge.The mechanism of this will be discussed below.

Mechanism for T c enhancement of unconventional superconductivity
The T c enhancement of unconventional superconductivity under lattice distortions can be understood by the enhancement of DOS at Fermi surface and the enhancement of the expectation values of unconventional pairings at Fermi surfaces due to the 19/42 (b) 0.0 0.3 0.6 0.9 1.2 0.0 0.3 0.6 0.9 1.2 .0 0.3 0.6 0.9 1.2  -c).Here, U/V = 0.7 and T 0 is the critical temperature of the ∆ 1 phase in the absence of the lattice distortions.In (d), the red and orange lines for ∆ 2 and ∆ 3 overlap.
unique spin-orbital texture.First, we consider the increment of DOS at the Fermi surface.Under the lattice distortions, the DOS's at the Fermi surface increase as shown in Eqs. ( 20) and (22).Then, the superconducting critical temperature increases under lattice distortions because T c ∝ e − 1 gN(0) .Here, g is the strength of the pairing potential in the standard BCS theory and N(0) is the DOS at Fermi surface.Due to this enhancement of DOS, most of the superconducting temperatures increase under the lattice distortions [see Figs.5(d,f) and 6(d,e,f)].However, some unconventional superconducting temperatures decrease while some unconventional superconducting temperatures increase under lattice distortions.To understand this, we investigate the pairing expectation values for each superconducting pairing potentials.
As a representative example, we calculate the normalized expectation values for the ∆ 1 , ∆ 2 , and ∆ 3 pairings at the Fermi surface with and without the n 1 type lattice distortion [Figure 7(a,b)].For a clear comparison, the differences are calculated [Fig.7(c)].Without lattice distortions, ∆ 1 is uniform while ∆ 2 and ∆ 3 show zeros on the k z axis.With the n 1 type lattice distortion, ∆ 2 increases while ∆ 3 decreases, which leads to ∆ diff 2 > 0 and ∆ diff 3 < 0 [Fig.7(c,d)].On the other hand, ∆ diff 1 = 0.These behaviors of the expectation values of ∆ i explains that the tendency of T c under lattice distortions.T c of ∆ 2 phase increase greater than that of ∆ 1 phase while T c of ∆ 3 phase decreases under n 1 type lattice distortion [Fig.5(a)].Similarly, the effect of the other types of lattice distortions on T c can be understood by the expectation value change of the pairing potentials.
Microscopically, we can understand the emergence of unconventional superconducting phases under lattice distortions as a result of the enhancement of inter-orbital pairing at the Fermi surface.Even though our argument can be applied to all distortions, we discuss the effect of n 1 type lattice distortion for convenience.We consider two Fermi surfaces encapsulating Dirac points (0, 0, ±k 0 ) which are related by time-reversal and inversion.On the upper Fermi surface near the Dirac point (0, 0, +k 0 ), the Dirac Hamiltonian in Eq.( 18) in the k x -k z plane is given by The spin and orbital parts can be diagonalized separately and the total wavefunction can be represented by the product of spin and orbital wavefunctions 37 : Let us diagonalize the spin part.The spin part of Hamiltonian is given by , are plotted with respect to n 1 .Note that the upper Fermi surfaces encloses the Dirac point (0, 0, k 0 ) as shown in Fig. 1(k-o).
where h = (n 1 sin k 0 , 0, vk x ).Since this Hamiltonian is a product of momentum and spin operators, the spin wavefunction can be represented in the helicity basis |λ spin with λ = ±1: Next, we diagonalize the remaining orbital part.Depending on the spin helicity λ , the Hamiltonian in Eq. ( 68) can be written as follows: where The orbital wavefunction can be represented by the pseudo-spin along dλ .For each spin helicity λ , there are two orbital wavefunctions κ dλ orbital with κ = ±1 that satisfy the following equations: where When the chemical potential is positive, two degenerate wavefunctions located in conduction bands participate in the superconducting pairing.These wavefunctions are given by which form a Kramer's pair due to the PT symmetry regardless of lattice distortions: PT operation conserves the momentum while it flips helicity and the x-component of the orbital because T = is y K and P = −σ z .Since we have obtained the spin and orbital texture in one Fermi surface, we can obtain the spin and orbital texture of the other Fermi surface by applying either time-reversal or inversion operator.Let Ψ(k) be a wavefunction on the Fermi surface.Because there is no σ y in the Hamiltonian Eq. ( 68), the time-reversal partner T Ψ(k) has the same orbital direction and the opposite spin direction regardless of lattice distortions comparing with Ψ(k).On the other hand, since P = −σ z , the inversion partner PΨ(k) has the opposite d x while keeping d z and spin direction comparing with Ψ(k). Figure 8 shows the numerically calculated spin and orbital textures using the lattice model.The P and T symmetry operators connects spin and orbital wavefunctions in Fig. 8.The red and green arrows indicate time-reversal and inversion pairs, respectively.
Using these spin and orbital textures, let us investigate how the lattice distortions promote the unconventional pairings.The conventional ∆ 1 pairing is not affected by the lattice distortion.The expectation value of ∆ 1 is constant over the entire Fermi surface regardless of lattice distortion as shown in Fig. 7(a,b).Because that are related by time-reversal, the expectation value of ∆ 1 is constant due to TRS.In other words, because ∆ 1 is represented by the identity matrix 1 4×4 , the expectation value of the ∆ 1 over the Fermi surface is constant even under the lattice distortions.
On the other hand, n 1 type lattice distortion can increase the expectation values of the inter-orbital pairing ∆ 2 .For example, let us consider two wavefunctions located at the south pole of the upper Fermi surface (k z = +k s with k s < k 0 ) and the north pole of the lower Fermi surface (k z = −k s ).Two wavefunctions are indicated by the orange and cyan arrows in Fig. 8(c,d).At k z = ±k s , the Dirac Hamiltonian in Eq.( 18) is given by Dirac and H (−) Dirac correspond k z = k s and k z = −k s , respectively.When n 1 = 0, wave functions on the conduction bands at k z = ±k s are given by where |1 orbital and |2 orbital indicate the orbital basis for σ matrix as defined before.|↑ x spin and |↓ x spin indicate the spin up and down along x-direction.Thus, the expectation value of inter-orbital pairing is zero for these wavefunctions because the orbital states of the wavefunctions in Eqs.(76) and, (77) are same.On the other hand, when n 1 = 0, the x-component of the orbital pseudo-spin is generated [indicated in the large cyan arrows in Fig. 8(d)].The wave functions at k z = ±k s are given by where d ± = (±n 1 sin k 0 , 0, −v z (k 0 − k s )).Therefore, under the lattice distortion, the expectation value of the inter-orbital pairing is allowed and ∆ 2 pairing is enhanced.This mechanism for the enhancement of unconventional pairings can be applied to the other cases.In summary, the emergence of unconventional superconductivity under lattice distortion can be understood due to the enhancement of inter-orbital pairings and DOS at Fermi surfaces.8. Possible topological superconductivity in doped DSM under lattice distortions.SC, FG, LN, and PN denote superconductor, full gap, line node, and point node, respectively.P odd and P even represent the inversion-odd and inversion-even parity superconductors.M odd and M even represent the mirror-odd and mirror-even parity superconductors.C M is the mirror Chern number.W is chiral winding number defined by Eq. ( 43).W M is the mirror chiral winding number defined by Eq. ( 47).
Here, the 2Z indicates the even number of the corresponding surface Andreev bound state (SABS).

Topological superconductivity of doped Dirac semimetal
As summarized in Table 8, we characterize possible superconducting states in doped Dirac semimetal by the gap structures, topological winding numbers, and surface spectra.First, the conventional superconducting phase having ∆ 1 pairing potential can emerge.Because T c of the ∆ 1 phase increases under lattice distortions as shown in Figs. 5 and 6, conventional fully-gapped s-wave superconductivity can emerge.
Second, we consider the inversion-odd-parity superconductor.The BdG Hamiltonian in Eq. ( 23) are included in the DIII class according to 10-fold Altland-Zirnbauer classes 4, 60 because T 2 = −1,C 2 = +1, and Γ 2 = +1.With the additional inversion symmetry, the DIII class superconductor can be an inversion-odd-parity topological superconductor 58 classified by Z 2 invariants (−1) w DIII , where Here, Γ is the chiral operator, and Q is the so-called Q-matrix 4, 60 (or projection matrix).The sufficient condition for realizing the inversion-odd-parity topological superconductor is that it has an inversion-odd-parity pairing with a full gap and its Fermi surface encloses an odd number of time-reversal-invariant momenta.In the absence of lattice distortions, the inversion-oddparity pairings, ∆ 2 , ∆ 3 , ∆ 41 , and ∆ 42 , are not fully gapped [Fig.2(a)] and cannot be such a topological superconductor.However, under the lattice distortions, these inversion-odd-parity phases can be fully gapped, and the sufficient condition above can be satisfied for the large chemical potential (µ > M 0 ) because the Fermi surface can enclose only (0, 0, 0) in BZ.However, when the chemical potential is large with a lattice distortion, the band structure near the Fermi energy is far from that of DSM.Because we are discussing the Dirac physics, we do not consider such a superconducting phase in this work.Third, topological mirror superconducting phases 36,64 can exist under lattice distortions.Topological DSM has a nontrivial mirror Chern number defined in the M xy plane and the corresponding surface states on the mirror-symmetric boundary 36,39 .Similarly, topological mirror superconductivity for ∆ 2 and ∆ 3 phases can exist under lattice distortions.Under the n 1 (n 2 ) type lattice distortion, ∆ 2 (∆ 3 ) phase is fully gapped, the ∆ 2 (∆ 3 ) pairing potential is mirror-odd under the M xy symmetry, and the mirror Chern number defined in M xy plane is nontrivial (C M = ±2), which leads to topological mirror superconductivity with a topologically-protected Majorana states on the mirror symmetric boundary.For example, see the gapless surface spectra of ∆ 2 and ∆ 3 phases in Region III in Fig. 4(a,b,e,f).Due to the TRS and IS, this topological mirror superconductor is classified as 2Z.
Fourth, topological line nodal superconducting phases can exist under lattice distortions.As discussed in Fig 3(b,c), the inversion-even-parity ∆ 1 pairing allows a topologically-protected nodal lines protected by the chiral winding number in Eq. ( 43).According to this chiral winding number, in general, the topological line nodal superconductor in doped topological DSM is classified as 2Z.The reason is as follows.Since there are PT and PC, the nodal points are fourfold degenerate, which means that there are even number of winding source at the same points.Therefore, our generic model has a topological winding number of even integers.Note that the topological class of a line node in 3D DIII superconductor using Clifford algebra 62 is 2Z, which is consistent with our result.However, there is no surface state because ∆ 1 phase has two nodal lines having opposite chiral winding numbers [Fig.3(a)].
Fourth, topological point nodal superconducting phases can exist under lattice distortions.For an inversion-odd-parity and mirror-even-parity pairing potential, we have a topological point nodal superconductor of which nodal points are protected by the mirror chiral winding number in Eq. (47).Because the chiral winding number is zero for inversion-odd-parity superconductor (W = W λ =i +W λ =−i = 0), the mirror chiral winding number is given by W M = W λ =i −W λ =−i = 2W λ =i .From this mirror chiral winding number, this topological point nodal superconductor is classified as 2Z.Note that the classification of a point node using Clifford algebra 4,62 is MZ considering one mirror sector, which is consistent with our results.

Discussion
Now, we compare our results with experimental works in doped DSM of Au 2 Pb [30][31][32][33][34] and Cd 3 As 2 [27][28][29] .Au 2 Pb shows superconductivity at T c ≈ 1.2K with D 2h symmetry at the ambient pressure 30,32,34 .This structural transition corresponds to the n 1 or n 2 type lattice distortion.T c increases to 4 K until 5 GPa under compression 34 .The point-contact measurements also reported that T c ≈ 2.1 K using a hard contact tip is higher than the measured T c ≈ 1.13 K using a soft tip.Assuming that the hard tip induces higher pressure than the soft tip, the experimental results are consistent with our result that T c is enhanced with increasing n 1 or n 2 lattice distortion [Fig.5].The experiments reported that the superconductivity is either conventional 33,34 or unconventional 32 depending on the physical situations.From our analysis, the superconducting phase of Au 2 Pb is expected to be either a conventional fully gapped or unconventional topological mirror superconductor with a gapless SABS depending on physical parameters.
Similarly, in Cd 3 As 2 , the structural phase transition occurs near 2.6 GPa, resulting in a monoclinic lattice C 2h .Then, a superconductivity emerges at T c ≈ 1.8 K under pressure higher than 8.5 GPa.This structural transition corresponds to n 3 or n 4 type lattice distortion.When the pressure increases further, T c keeps increasing from 1.8 K (8.5 GPa) to 4.0K (21.3 GPa), which is consistent with the enhancement of T c under lattice distortions [Fig.6].In this case, n 1 or n 2 can also be added without breaking the symmetry further.From our analysis, the superconducting phases of Cd 3 As 2 are expected to be either a conventional or topological mirror superconductor with a gapless SABS.We emphasize that the topological nodal superconductor having a flat SABS can appear only if either n 3 or n 4 lattice distortion is turned on.The point-contact measurements for Cd 3 As 2 showed the zero-bias conductance peak (ZBCP) and double conductance peaks symmetric around zero bias, which was interpreted as a signal of a Majorana surface states [27][28][29] .Even though our result cannot directly explain the result of the point-contact measurement, the unconventional superconductivity having gapless Majorana fermion can emerge regardless of the lattice distortions according to the surface spectra [see Fig. 4], which seems to support the measured conductance peaks.Further experimental studies that reveal the nature of superconductivity are necessary, and our theoretical results will be a helpful guideline to interpret the experimental result and search for the possible topological superconductivity in DSM.

Summary
In this work, we have studied the possible symmetry-lowering lattice distortions and their effects on the emergence of unconventional superconductivity in doped topological DSM.From the group theoretical analysis, four types of symmetrylowering lattice distortions that reproduce the crystal systems present in experiments are identified.We investigated the possible superconductivity under such symmetry-lowering lattice distortions considering inter-orbital and intra-orbital electron densitydensity interactions.We found that both conventional and unconventional superconductivity can emerge depending on the lattice distortion and electron density-density interaction.Remarkably, the unconventional inversion-odd-parity superconductivity hosts gapless surface Andreev bound states (SABS) even under lattice distortions.We found that the lattice distortion enhances the superconducting critical temperature.Therefore, our work is consistent with the observed structural phase transition and the enhancement of superconductivity in Cd 3 As 2 and Au 2 Pb under pressure.We also suggest that enhanced conventional and unconventional superconductivity in doped topological DSM can be controlled by physical parameters such as the pressure and strength of the superconducting pairing interaction.Thus, our work will provide a valuable tool to explore and control the superconductivity in topological materials.

Methods
To study the effects of symmetry-lowering lattice distortions, we assume a minimal 4 × 4 Hamiltonian that describes representative topological Dirac semimetals 39,45 , where the lattice distortions are implemented as a perturbation 54 .To study the superconductivity, we construct the Bogoliubov-de Gennes (BdG) Hamiltonian within the mean-field approximation while keeping TRS and the crystal symmetry 56,57 .The momentum independent pairing potentials are classified using irreducible representations of the unbroken point group 36,37,[57][58][59] .The nodal structures, chiral winding number in Eq. ( 43), and chiral mirror winding number in Eq. ( 47) are calculated using the BdG Hamiltonian.The surface Green's functions are calculated using a Möbius transformation-based method 63 .The superconducting critical temperature T c is calculated by solving the linearized gap equation in the weak-coupling limit 37,[56][57][58][59]  Let us discuss Cd 3 As 2 [3-5].First, it does not show any superconductivity in ambient pressure but becomes superconducting at 1.8 K under pressure (≈ 8.5 GPa) [4].The structural phase transition occurs at 2.4 GPa from a tetragonal lattice with D 4h symmetry to a monoclinic lattice with C 2h symmetry.The space group of the undistorted lattice is I4 1 /acd (No. 142) and the space group of the distorted lattice is P 2 1 /c (No. 14).After the structural phase transition, the beta angle of the original tetragonal structure is shifted from β = 90 • to β = 98 • .Second, the T c increases from 1.8 K to 4.0 K as the pressure increases to 21.3 GPa.Third, the change of volume is not large.Before the structural phase transition, the unit cell of the tetragonal lattice is a reconstructed 2 × 2 × 4 supercell with a total volume V ≈ 4058 Å3 .After the structural phase transition, the total volume of sixteen unit cells in the monoclinic lattice is V ≈ 3636 Å3 , which is 90% of the original volume.Because the volume change is not large, the strength of lattice distortion can be regarded a weak perturbation.Finally, the observed superconductivity is reported as unconventional [3, 4].The transport data under magnetic fields showed anomalous behaviors unexplainable by the usual BCS s-wave superconductivity.The unconventional zero-bias conductance peak (ZBCP) and double conductance peaks symmetric around zero bias shown in the point-contact measurements are interpreted as a possible Majorana surface state.Now, let us discuss Au 2 Pb [6-10].First, it exhibits superconductivity at T c ≈ 1.2 K after a structural phase transition under cooling [6-10] Note that in the reference [6], the axes are differently chosen so that a = 7.90 Å, b = 5.58 Å, and c = 11.19Å.For the consistency with our work, we choose the twofold rotation axis of orthorhombic lattice as z-axis.The space group of undistorted lattice is F d 3m (No. 227) and the space group of distorted lattice is P bcn (No. 60).Second, T c reaches 4 K at 5 GPa, then decreases with further compression [10].Third, the local pressure from the point contact increases the T c [8].The measured T c using a hard tip is higher than the measured T c using a soft tip.Fourth, the change of volume is negligible.The volume of unit cell V ≈ 493 Å3 is nearly same before and after structural phase transition, which means that the strength of the lattice distortion itself is not large.Finally, the superconductivity in Au 2 Pb is also unconventional.The empirical transport data shows usual longitudinal linear magenetoresistivity (MR) behavior, which is not described by the conventional BCS theory [6].Note that the calculated band structure in the low temperature phase for Au 2 Pb has a Fermi surface having electron and hole pockets [6, 8], which seem to provide metallic Fermi surface and conventional BCS pairing [9].However, the interplay between quadratic bands and linear bands from the DSM is not well-understood due to the complexity, which is beyond this work.Instead, we will focus the two Dirac points and influence of the symmetry-lowering lattice distortion.

S2.2. Classification of symmetry-lowering lattice distortions
In the minimal four-band model, the summery-breaking lattice distortion terms can be described by the following 4×4 matrix: where Γ i = s j σ k and d i (k) is a real-valued function of momentum.Here, s j and σ k are Pauli matrices for spin and orbital degrees of freedom in the spin (↑, ↓) and the orbital (1, 2) spaces, respectively.By time-reversal symmetry (TRS) and inversion symmetry (IS), the form of d i and Γ i are restricted.First, as discussed in the main text, TRS and IS allow only six gamma matrices are allowed, which are Γ 0 , Γ 1 , Γ 2 , Γ 3 , Γ 4 , and Γ 5 .Also, the constraint for coefficient function can be easily found.Furthermore, using the group theoretical analysis, possible symmetry-lowering lattice distortions can be classified according to the irreducible representation of D 4h group, which is summarized in Table S2 Because we are interested in the low-energy physics near the Dirac points (0, 0, ±k 0 ), symmetry-lowering lattice distortion terms can be classified order by order with respect to the momentum.Then, the leading terms are 1, k z among 1, k x , k y , k z .Due to T P symmetry, the 1 can be combined with Γ 0 and Γ 5 , and these terms are included in the A 1g class.Because A 1g class does not break any symmetry, we can ignore these terms.On the other hand, k z can be combined with Γ 1 , Γ 2 , Γ 3 , and Γ 4 .In the leading order, these four types of symmetry-lowering lattice distortions are dominant terms as discussed in the main text.Up to the quadratic order of momenta, the possible lattice distortion terms are listed in Table S3.

S2.3. Lattice Hamiltonian under lattice distortions
For definiteness, we consider the explicit perturbation Hamiltonian in the lattice model.Then, we will show that how the lattice distortion terms generate the Dirac mass and shift the Dirac points in the low-energy effective Hamiltonian.
First, consider the n 1 and n 2 type lattice distortions.In this case, the coefficient functions d i (k) are given by where n i,j and n i,j are the constant parameters that represent the strength of the n i type lattice distortions.The second subscript j indicates the power of momentum near Dirac points (0, 0, ±k 0 ) according to Table S3.Near the Dirac point, k z is approximately constant while k x and k y are very small.This is the reason why we only consider the leading terms d 3 (k) = n 1 sin k z and d 4 (k) = n 2 sin k z for n 1 and n 2 type lattice distortions in the main text.If we set n 1 ≡ n 1,0 and n 2 ≡ n 2,0 , then near the Dirac points (0, 0, ±k 0 ) the low energy effective Hamiltonian is given by Thus, the n 1 and n 2 type lattice distortions generate the Dirac mass terms.
Next, consider the n 3 type lattice distortion.Then, the coefficient functions are given by Similar to the n 1 and n 2 case, the n 3,0 sin k z term is dominant near the Dirac point.In this case, however, the gap minimum points are shifted along k y axis from Dirac points (0, 0, ±k 0 ) to (0, ±k y , ±k 0 ), where k (0) y = −n 3,0 sin k 0 /v in the linear order.Then, the low energy effective Hamiltonian is given by where Finally, consider the n 4 type lattice distortion, where the coefficient functions are given by Similar to n 3 case, the Dirac points are shifted along k x axis from (0, 0, ±k 0 ) to (±k where

S2.4. Enhancement of DOS at Fermi surface
In this subsection, we analytically calculate the DOS at Fermi surface using the low energy effective Dirac Hamiltonian.First, consider the n 1 and n 2 type lattice distortions.Because the energy eigenvalue is given by the Fermi surface is given by the surface of an ellipsoid: Since the ellipsoid (x/a) 2 + (y/b) 2 + (z/c) 2 = 1 has the volume (4π/3)abc, the electron number density N up to energy E is given by In the absence of lattice distortion (|n| = 0), the initial electron number density with the initial chemical potential µ 0 is given by N . The electron number density conservation gives Therefore, the Fermi surface for chemical potential µ(n) is given by where the lattice distortions terms in the right-hand side are canceled.From Eq. ( 12), DOS for an arbitrary energy level E is given by: Using the chemical potential µ(n), the lattice-distortion dependent DOS is given by Therefore, the DOS is enhanced by the lattice distortions.
For n 3 type lattice distortion, one can apply the above calculation simply by substitute the |n| 2 sin 2 k 0 by m 2 using Eq. ( 7).The only difference is the y directional shift of Fermi surface.The result is as follows: The chemical potential is given by where µ 0 is the chemical potential in the absence of the lattice distortion.The lattice-distortion dependent DOS is given by The Fermi surface for the chemical potential µ(n 3 ) is given by R=0.0 R=0.6 R=0.98 R=1.0 R=2.0 R=0.3 R=0.9In this subsection, we discuss the superconducting gap structures for each phases under n 1 and n 2 type lattice distortions.
S3.1.1.∆ 1 and ∆ 1 phases First, we consider the superconducting phases with ∆ 1 and ∆ 1 paring potentials.Because they are included in the same representation regardless of lattice distortions, the mixture of ∆ 1 and ∆ 1 phases is generally allowed.In this case, the energy eigenvalue E(k) is given by By solving E(k) = 0, the equations for nodal points are obtained: In general, for ∆ 1 = 0 and ∆ 1 = 0, the number of momentum variable (N V = 3) is larger than the number of equation (N E = 2) and hence the one-dimensional solution is generally allowed.However, if ∆ 1 > ∆ 1 , Eq. ( 20) can not be satisfied, which leads to a gapped phase.Therefore, the phase transition between full gap and nodal line phases occurs as shown in Fig. S2.
S3.1.2.∆ 2 and ∆ 3 phases Next, we consider the ∆ 2 and ∆ 3 superconducting phases.As discussed in the main text, these two paring potentials can be included in the same or different classes depending on the unbroken point group.In general, the energy eigenvalue is given by By solving E(k) = 0, the following equations for nodal points are obtained: These equations do not seem to allow any nodal point because the number of equation (N E = 4) is larger than the number of momenta (N V = 3).However, if there are mirror symmetries, then the existence of nodal points are guaranteed.Let us consider following cases: (i) For D 4h case, C 4z symmetry protects the nodal points on the k z axis.The C 4z symmetry restricts that a 1 (k) = a 2 (k) = a 3 (k) = a 4 (k) = 0 on the k z axis.Then, Eq. ( 21) on the k z axis becomes (a 5 (k)) 2 = µ 2 + ∆ 2 2 + ∆ 3 2 .
On the k z axis, the number of variables (N V = 1) is equal to the number of equations (N E = 1).Thus, for D 4h case, each ∆ 2 and ∆ 3 phases have nodal points on the k z axis.
(ii) For D 2h case, M yz and M xz symmetries protect the nodal points on the k z axis.a 1 (k) and a 4 (k) are odd-functions under M yz operation, which leads that a 1 (k) = a 4 (k) = 0 on the corresponding mirror plane (0, k y , k z ).Similarly, M xz symmetry gives a 2 (k) = a 4 (k) = 0 on the corresponding mirror plane (k x , 0, k z ).Thus, both M yz and M xz symmetries endow a 1 (k) = a 2 (k) = a 4 (k) = 0 on the k z axis.Then, Eq. ( 21) on the k z axis becomes Since a 3 (k) is generally not zero, there can exist a nodal point only when ∆ 2 = 0. Thus, the ∆ 3 phase can have nodal points in D 2h case.
(iii) For D 2h case, M (110) and M (1 10) symmetries protect the nodal points on the k z axis.Using the transformation properties in Table 2 in the main text, we get under M (110) operation, which leads that a 1 (k) = −a 2 (k) and a 3 (k) = 0 on the k z axis.Similarly, M (1 10) operation gives that a 1 (k) = a 2 (k) on the k z axis.Combining the results of two mirror symmetries, we get a 1 (k) = a 2 (k) = a 3 (k) = 0 on the k z axis.Then, Eq. ( 21) on the k z axis becomes Since a 4 (k) is generally not zero, there can exist a solution for these equations only when ∆ 3 = 0. Thus, the ∆ 2 phase can have nodal points in D 2h case.
(iv) For C 2h(z) case, there is no symmetry that protects nodal points.By solving E(k) = 0, the following set of equations for nodal points is obtained: These equations do not seem to allow any nodal point because the number of equation (N E = 4) is larger than the number of momenta (N V = 3).However, mirror symmetries can guarantee nodal points.Let us consider the following cases: (i) For D 4h case, C 4z symmetry seems to allow the existence of the nodal points on the k z axis.The C 4z symmetry restricts that a 1 (k) = a 2 (k) = a 3 (k) = a 4 (k) = 0 on the k z axis.Then, Eq. ( 22) on the k z axis become On the k z axis, the number of variables (N V = 1) is equal to the number of equations (N E = 1), which seems to allow nodal points.However, these nodal points are accidental nodal points because the D 4h symmetry is spontaneously broken when ∆ 41 or ∆ 42 pairing potentials are present.
(ii) For D 2h case, M yz and M xz symmetries protect the nodal points on mirror plane.a 1 (k) and a 4 (k) are odd-functions under M yz operation, which leads to a 1 (k) = a 4 (k) = 0 on the corresponding mirror plane (0, k y , k z ).Thus, for ∆ 41 phase, Eq. ( 22) becomes Because the number of equation (N E = 2) is equal to the number of momenta (N V = 2) on the mirror plane, which can allow nodal points.Similarly, M xz symmetry gives a 2 (k) = a 4 (k) = 0 on the corresponding mirror plane (k x , 0, k z ).Thus, for ∆ 42 phase, Eq. ( 22) becomes Hence, ∆ 41 and ∆ 42 phases can have nodal points in the mirror planes.
(iii) For D 2h case, M (110) and M (1 10) symmetries allow the nodal points on the corresponding mirror plane.Because there are diagonal mirror symmetries, the following linear combinations of the pairing potentials are meaningful and do not break the D 2h point group symmetry.Because the number of equation (N E = 2) is equal to the number of momenta (N V = 2) on the mirror plane, which can allow nodal points.Similarly, M (1 10) operation gives that a 1 (k) = a 2 (k) on the k z axis.Thus, for ∆ − phase with the condition ∆ + = 0, Eq. ( 22) becomes Hence, ∆ + and ∆ − phases can have nodal points in the mirror planes.
(iv) For C 2h(z) case, there is no symmetry that allows a nodal point.(1) For ∆ 2 and ∆ 42 phases, the energy eigenvalue is given by The nodal equations are given by Because the number of equations (N E = 4) is larger than the number of variables (N V = 3), there is no nodal point.
(2) For ∆ 3 and ∆ 41 phases, the energy eigenvalue is given by The nodal equations are given by obtained.
Using these susceptibilities and the gap equations, we calculated the critical temperatures and phases maps.

Figure 1 .
Figure 1.Crystal systems, band structures, and Fermi surfaces of Dirac semimetal (DSM) under various lattice distortions.(a) Undistorted DSM for comparison.It has a tetragonal lattice.(b-e) Distorted crystal systems under (b) n 1 , (c) n 2 , (d) n 3 , and (e) n 4 type lattice distortions.In (b) and (c), n 1 and n 2 type lattice distortions changes inplane lattice constants, which results in orthorhombic lattices.In (d) and (e), n 3 and n 4 type lattice distortions change the α and β angles, which results in monoclinic lattices.(f-j) The corresponding 3D band structures.In (f-i) [(j)], the band structures are plotted for the k y -k z (k x -k z ) plane and the orange planes are k y = 0 (k x = 0) plane.(k-o) The corresponding Fermi surfaces.In (l-o), all Fermi surfaces are distorted according to types of lattice distortions.In (n) and (o), the Fermi surfaces are shifted as indicated by the black arrows.Each vertical orange line indicates the k z axis.

Figure 2 .
Figure 2. Superconducting nodal structures for pairing potentials under lattice distortions.Nodal structures for (a) D 4h , (b) D 2h , (c) D 2h , (d) C 2h(z) , and (e) C 2h(x) cases.The orange point, line, and plane indicate nodal point and nodal line, and mirror plane (M xz , M yz , M 110 , and M 1 10 ), respectively.In (a-e), the ∆ 1 phases are fully gapped and the ∆ 1 phases have two nodal rings.In(a,b,c,e), nodal points are located in the corresponding mirror planes.In (c), ∆ 42 + ∆ 41 and ∆ 42 − ∆ 41 phases are considered instead of ∆ 41 and ∆ 41 phases.In (d), the system has no mirror symmetries and hence no nodal points.These nodal structures are summarized in Table6.
there can exist nodal points [Fig.2(b)].For ∆ 42 phase, nodal points also can exist due to M xz mirror symmetry [Fig.2(b)].When the point group is D 2h under the n 2 type lattice distortion, nodal points can exist due to M 110 or M 1 10 mirror symmetries [Fig.2(c)].For C 2h(z) , a mixture of ∆ 41 and ∆ 42 phases is possible.However, there is no allowed nodal point due to the lack of mirror symmetry [Fig.2(d)].When the point group is C 2h(x) under the n 3 type lattice distortion, nodal points can exist due to M yz mirror symmetry [Fig.2(e)].See the detailed calculations in Sec.S3 in Supplementary Information.

Figure 3 .
Figure 3. Topologically protected nodal structures and chiral winding numbers.The orange ring, point, plane, and vertical line indicate nodal ring, nodal point, mirror plane, and k z axis, respectively.Each winding number is defined along each blue loop.(a) The chiral winding numbers (W = ±2) protect nodal rings.(b,c) The mirror chiral winding numbers (W M = ±2) protect nodal points on the mirror planes.(d) Evolution of nodal points in ∆ 42 phases and the corresponding mirror chiral winding number under the n 1 type lattice distortion.For clarity, the blue winding loops are omitted.For n 1 = 0, nodal points with W M = 0 are located on k z axis.These are fine-tuned accidental nodal points because D 4h is spontaneously broken into D 2h due to ∆ 42 pairing [see the main text below Eq. (41)].As n 1 increases, the nodal points split into two nodal points with W M = ±2.The bottom plot shows the evolution of the energy dispersion along k x axis.As n 1 increases, the blue (orange) band moves downward (upward), which results in two Dirac points.

Figure 4 .
Figure 4. Surface band structures of superconducting phases under distortions.Surface band structures on the (010) surface for (a-d) D 4h , (e-h) D 2h , (i-l) C 2h(z) and (m-p) C 2h(x) .In each panel, the upper figure indicates the close-up view of the band structure near E = 0 corresponding to the red box in the lower figure.The red vertical arrows indicate the nodal points of the bulk superconducting states.In the insets of (e,h,i,l), the bulk states are gapped.The cyan vertical arrows indicate the gapped surface states.In (b,f,j,k), red horizontal lines show the surface flat bands.The nature of gapless surface state (GSS) is distinguished by the colored circle: Red ones in (a,b,e,f), green ones in(a,b,d,e,f,h,i,l), and black ones in (d,h,i,l) indicate GSS's protected by mirror Chern numbers, zero-dimensional topological numbers, and mirror eigenvalues, respectively.In (j), GSS's are accidental.The details are in Table7 andin the main text.Region I, II, and III are (0, k 1 )-(0, k 2 ), (k 2 , 0)-(0, 0), (0, 0)-(π/2, 0), respectively, where k 1 and k 2 (k 1 > k 2 > 0) indicate two intersecting points between the upper Fermi surface and the k z axis.
the critical temperatures are same at the phase boundary, the phase boundary in Fig.5(a) is determined by the equation R(T which gives the critical value of U/V = 2/3.

Figure 5 .
Figure 5. Phase diagrams for the tetragonal and orthorhombic crystal systems.(a) Superconducting phase diagram in the U and V plane in the absence of lattice distortions.In the orange (blue) region, ∆ 2 or ∆ 3 (∆ 1 ) phase is dominant.The slope of the phase boundary is approximately U/V = 2/3.The white region indicates a non-superconducting phase.(b) The numerically calculated critical value of U/V ratio as a function of the chemical potential in the absence of lattice distortions.Since µ = 0.75t z is the band inversion point, there is a local maximum due to Van Hove singularity near µ = 0.75t z .(c,e) Phase diagrams with respect to (c) n 1 and (e) n 2 type lattice distortions when U = 0.045t z .The corresponding point groups are (c) D 2h and (e) D 2h .Each black arrow indicates the possible phase transition from an inversion-even-parity to inversion-odd-parity superconducting phases.(d,f) The normalized critical temperature T c /T 0 for various pairing potentials with respect to (d) n 1 and (f) n 2 type lattice distortions.In both figures, U/V = 0.75 and U = 0.045t z, which corresponds to the black arrows in (c,e).T 0 is the critical temperature of the ∆ 1 phase in the absence of the lattice distortions.

Figure 6 .
Figure 6.Phase diagrams for the monoclinic crystal system.(a-c) Phase diagrams with respect to U/V ratio and n 3 type lattice distortions for (a) n 1 = 0.0, (b) n 1 = 0.05, and (c) n 1 = 0.1.(d-f) The normalized critical temperature T c /T 0 along the black arrows in(a-c).Here, U/V = 0.7 and T 0 is the critical temperature of the ∆ 1 phase in the absence of the lattice distortions.In (d), the red and orange lines for ∆ 2 and ∆ 3 overlap.

Figure 7 .
Figure 7. Expectation values of pairing potentials at the upper Fermi surface under the n 1 type lattice distortion.(a,b) The normalized expectation values ∆ i of ∆ 1 , ∆ 2 , and ∆ 3 (a) without and (b) with the n 1 type lattice distortion are plotted at the upper Fermi surface of DSM in the k x -k z plane.(c) The differences ∆ diff i ≡ ∆ i n 1 =0 − ∆ i n 1 =0 are plotted.In (a-c), the black arrows indicate the points having zero expectation values.(d) The normalized integrated expectation values of each pairing potentials, FS d 2 k∆ diff i / FS d 2 k ∆ i n 1 =0 , are plotted with respect to n 1 .Note that the upper Fermi surfaces encloses the Dirac point (0, 0, k 0 ) as shown in Fig. 1(k-o).

Figure 8 .
Figure 8. Spin and orbital textures without and with lattice distortion.(a,b) [(c,d)] Numerically calculated spin (orbital) textures at two Fermi surface surfaces.The n 1 type lattice distortion is absent in (a,c) and present in (b,d).In (a,b) [(c,d)], the spin (orbital) textures are represented by the small black (blue) arrows.In (a-d), the textures in left and right panels correspond to the spin helicity up and down wavefunctions, respectively.In (a,b), the red and green arrows indicate time-reversal and inversion pairs, respectively.In (c,d), the orange and blue arrows indicate the possible Cooper pairing between two electrons with opposite momenta.Note that the orbital pseudo-spin vectors connected by orange arrows are parallel regardless of the lattice distortion.On the other hand, the orbital pseudo-spin vectors connected by cyan arrows are parallel in (c) while non-parallel in (d).
. At room temperature, Au 2 Pb is in the cubic Laves phase with cubic symmetry (O h ) with lattice constants a = b = c = 7.90 Å.This cubic symmetry has three fourfold symmetries (C 4x , C 4y , C 4z ) around the high symmetric axes (x, y, z axes), which protect three pairs of Dirac points that are located on the high symmetric axes (k x , k y , and k z axes) in BZ.Under cooling, several structural phase transitions occur at T 1 = 97 K, T 2 = 51 K, and T 3 = 40 K.Even though the intermediate lattice structures are not clearly studied, below T 3 = 40 K the crystal structure becomes orthorhombic with point group symmetry D 2h .The measured the lattice constants are a = 5.58 Å, b = 11.19Å, and c = 7.90 Å.

y
, 0, ±k 0 ), where k (0) z = −n 4,0 sin k 0 /v in the linear order.The low-energy effective Hamiltonian is given by

Fig. S1 .
Fig. S1.Band structures and Fermi surfaces for low-energy effective Dirac Hamiltonians.(a-c) (a), (b), and (c) are band structures for the tetragonal, orthorhombic, and monoclinic crystal systems, respectively.The green lines indicate the chemical potentials.In the insets of (b) and (c), black arrows express the corresponding lattice distortions.Hence, in (b) [(c)], n1 type (n3 type) lattice distortion is turned on.(d-f) (d), (e), and (f) are Fermi surfaces for each crystal system.For the low-energy effective theory, the shape of the Fermi surfaces in (d), (e), and (f) are the same, but the position of the Fermi surface in (f) is shifted along the ky direction.

S3. 2 .
Under n 3 type lattice distortions So far, we have discussed the nodal structure of superconducting phases for the point groups D 4h , D 2h , D 2h , and C 2h(z) under n 1 and n 3 type lattice distortions.Now, we discuss C 2h(x) point group under n 3 type lattice distortion.In this case, the classification of pairing potentials are much different from the previous case: ∆ 2 and ∆ 42 are included in A u representation.∆ 3 and ∆ 41 are included in B u representation.Therefore, mixed phases are generally allowed.The mixed phase composed of ∆ 2 and ∆ 42 is fully gapped [Fig.2(e) in the main text].On the other hand, there are nodal points in mixed phase composed of ∆ 3 and ∆ 41 pairing potentials [Fig.2(e) in the main text].The existence of such nodal points can be shown using the M yz mirror symmetry which is not broken in the C 2h(x) point group.

Table 3 .
Four types of symmetry-lowering lattice distortions are classified according to the irreducible representation of D 4h point group.n 1 and n 2 belong to the B 1g and B 2g irreducible representations of D 4h , respectively, while n 3 and n 4 belong to the two-dimensional E g irreducible representation.For each lattice distortion, the matrix form, remaining essential group elements, and related material are listed.

Table 4 .
The pairing potentials are classified according to the irreducible representation of D 4h point group.∆ 1 , ∆ 1 , Γ 2 , and Γ 3 belong to the A 1g , A 1g , B 1u , and B 2u irreducible representations, respectively.∆41 and Γ 42 belong to the two-dimensional E u irreducible representation.The transformation properties of the pairing potentials are represented by +1 and −1 for even and odd parities.For two-dimensional representation E u , the explicit forms are listed.wherethe extended symmetry operators are given by C4z = diag(C 4z , η C 4z s y C * 4z s y ) and C2x = diag(C 2x , η C 2x s y C * 2x s y ).If the pairing potential satisfies M∆s y M T s y = η M ∆ under a mirror operator M, the BdG Hamiltonian satisfies the corresponding mirror symmetry:

Table 5 .
Pairing potentials classified according to the D 4h point group are reclassified according to the irreducible representation of unbroken subgroup under the lattice distortions.For D 2h group, ∆ 42 + ∆ 41 and ∆ 42 − ∆ 41 pairing potentials belong to in B 3u and B 2u representations, respectively.9/42 k z kx ky

Table 6 .
Nodal structures of superconducting phases under lattice distortions.FG, LN, and PN denote full gap, line node, and point node, respectively.a Topological line node protected by the chiral winding number (W = ±2 for each line node).b Node protected by C 4z symmetry.c Topological point node protected by the mirror chiral winding number (W M = ±2 for each point node).d The nodal point is located on the k z axis.e The nodal point is off the k z axis.f Accidental point node.

Table 7 .
Acc. Acc.W M n/a Acc.M yz M yz ΓM yz Gapless surface Andreev bound state (SABS) on (010) surface.The entry is either a topological number or a symmetry operator which protects corresponding gapless surface states.Region I, II, and III are defined in Fig.4.W M is a mirror chiral winding number that protects the flat SABS between nodal points.C M is a mirror Chern number that protects the gapless SABS in M xy plane.ΓM yz and CM xy indicate the symmetry operators which protect gapless SABS using the corresponding zero-dimensional topological number.M yz and M xy indicate the symmetry operators which protect the gapless SABS protected by the corresponding mirror eigenvalues.Acc.indicates an accidental gapless state.n/a means that there is no gapless state.
, ∆ 3 , ∆ 41 , ∆ 42 . All the details are provided in the main text and Supplementary Information.

Table S2 .
Because the Γ 1 , Γ 2 , Γ 3 , and Γ 4 have odd parities under T and P , the coefficient functions d 1 (k), d 2 (k), d 3 (k), and d 4 (k) should be a function of momenta with only odd powers.Similarly, the coefficient functions d 0 (k) and d 5 (k) should be a function of momenta with only even powers.Therefore, the form of allowed lattice term is either k odd Γ 1,2,3,4 or k even Γ 0,5 types.Group theoretical classification of momenta, gamma matrices, and their products.The T P symmetry allows only two types of lattice distortions, either k odd Γ1,2,3,4 or k even Γ0,5.Thus, resultant perturbation terms are included in the A1g, A2g, B1g, B2g, Eg.

Table S3 .
All possible symmetry-lowering lattice distortions up to quadratic order.Symmetry-lowering lattice distortions are classified according to the irreducible representation (IR) of D 4h point group.n1 and n2 types belong to the B1g and B2g irreducible representations of D 4h , respectively, while n3, n4, n5, and n6 types belong to the same two-dimensional Eg irreducible representation.(110) and (1 10) are the Miller indices for directions.
. k x and k y are in E u class, and k z is in A 2u class.Γ 0 and Γ 5 are in A 1g class, Γ 1 and Γ 2 are in E u class.Γ 3 is in B 2u class and Γ 4 in B 1u class.Hence k odd Γ 1,2,3,4 and k even Γ 0,5 types are included in A 1g , A 2g , B 1g , B 2g , E g .Because the trivial class A 1g does not break any crystal symmetry, the possible symmetry-lowering lattice distortions can be categorized into A 2g , B 1g , B 2g , E g classes.Each lattice distortion breaks D 4h point group symmetry into its subgroup symmetry: A 2g type gives C 4h .B 1g and B 2g types give D 2h and D 2h , respectively.E u types gives C 2h .Because D 2h , D 2h and C 2h are reported in Cd 3 As 2 and Au 2 Pb, we consider these types of lattice distortions as discussed in the main test.