Topological Crystalline Materials of J = 3/2 Electrons:Antiperovskites, Dirac points, and High Winding Topological Superconductivity

Takuto Kawakami,1, ∗ Tetsuya Okamura,1 Shingo Kobayashi,2, 3 and Masatoshi Sato1, †

1Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan2Institute for Advanced Research, Nagoya University, Nagoya 464-8601, Japan3Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan

(Dated: February 28, 2018)

We present a theory of topological crystalline materials of J = 3/2 electrons. The high spintopological phases are expected to be realized in cubic crystals with band inversion such as an-tiperovskite materials, A3BX with A=(Ca, Sr, La), B=(Pb, Sn) and X=(C, N, O). The low energyeffective theory reveals that the system exhibits topological phase transitions with two differentpatterns of Dirac points. For the carrier doped case, it is shown that the system may host uniqueunconventional superconductivity because of its high spin nature as well as an additional orbitaldegrees of freedom intrinsic to topological insulators. The superconducting critical temperature isevaluated by using density-density pairing interactions, and odd-parity Cooper pairs are shown tonaturally realized in the presence of interorbital pairing interaction. It is found that even the sim-plest spin 0 odd-parity pairing state exhibits a novel class of topological superconductivity, dubbedas high winding topological superconductivity. We also discuss experimental signals of the highwinding topological superconductivity in the case of the antiperovskite superconductor Sr3−xSnO.

I. INTRODUCTION

Search for topological materials is one of recent trendsin condensed matter physics.1–8 A promising direction forthis purpose is a multi-orbital material with strong spin-orbit coupling. Such a system can be topologically non-trivial by band inversion due to spin-orbit coupling. Forinstance, in the case of topological insulators (TIs), theZ2 topological indices are directly related to the numberof time-reversal invariant momenta at which band inver-sion takes place.9,10 Following this insight, a number ofTIs including Bi2Se3 and Bi2Te3 have been discoveredexperimentally.1,4

In the case of superconductors, on the other hand, evena single orbital system may possess non-trivial topolog-ical phase because the charge conjugate counterpart ofthe single band coexists in the Bogoliubov de GennesHamiltonian.11–17 Any spin-triplet superconductor canbe topological if the Fermi surface is properly chosen.17

However, search for spin-triplet superconductivity itselfis a challenge because of its strongly correlated origin.Although there are several promising candidates of spin-triplet superconductors, no spin-triplet superconductorhas been established yet.

Recently, it has been recognized that multi-orbital sys-tems may solve this difficulty as they allow another mech-anism of topological superconductivity:8 Using inter-orbital pairing interaction as well as spin-orbit coupling,a multi-orbital system may host odd-parity Cooper pairseven without strong correlation,18 which implies topo-logical superconductivity.17–19 In this new searching di-rection, various topological superconductivity in dopedtopological materials have been discussed recently.18,20–37

In this paper, we explore one more another class oftopological phases that can be realized only in multi-orbital systems. Spin-orbit coupled electrons may be-

have as higher order spin states, as a result of mix-ture of spin and orbital angular momentum. In crys-tals, the higher spin state can be identified as a J =3/2 state, since discrete crystalline rotation allows atmost fourfold degeneracy of the J = 3/2 spin. Super-conductivity of J = 3/2 electrons has been discussedfor single-orbital half-Heusler systems.38–45 Whereas thehalf-Heuslers may host interesting higher spin Cooperpairs, their actual realization is restricted because thesystems support only the spin degrees of freedom.

Here we consider alternatively J = 3/2 electrons inmulti-orbital systems. A class of relevant materials isantiperovskite A3BX, where A is (Ca, Sr, La), B is (Pb,Sn), X is (C, N, O). The antiperovskite materials sup-port two different J = 3/2 electrons near the Fermilevel, i.e. d-orbital electrons of A atom and p-orbitalones of B. The first principle calculations show thatband-inversion of these two orbitals may occur at theΓ-point, being accompanied with three-dimensional (3D)Dirac points with a tiny gap.46–48 As a result of the bandinversion, these materials become topological crystallineinsulators.48 It was also discovered very recently thatone of the antiperovskite topological materials, Sr3SnO,shows superconductivity with the hole doping.49–52 Al-though the viewpoint from J = 3/2 electrons was miss-ing, possible topological superconductivity in Sr3−xSnOwas suggested theoretically,49 using the analogy of thesuperconducting Dirac semimetal, Cd3As2.35,36

Below, we develop a theory of the higher spin topolog-ical materials. We assume cubic symmetry, which allowsthe four-dimensional representation corresponding to thefourfold degeneracy of J = 3/2. Based on the low energyeffective Hamiltonian, we find that the system hosts fourdifferent topological crystalline insulating phases, wherethe phase boundaries are characterized by two differentpatterns of gapless Dirac points: In addition to octahe-dral Dirac points discussed previously,46–48 there appear

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cubic Dirac points at one of the phase boundaries.

Bearing in mind the normal state properties in theabove, we then proceed to study superconducting statesof the doped higher spin topological materials. We firstclassify possible momentum-independent gap functions,based on the cubic symmetry. The gap functions maycontain spin J = 2 (spin-quintet) and J = 3 (spin-septet)components, as well as ordinary J = 0 (spin-singlet) andJ = 1 (spin-triplet) ones, since the Cooper pairs areformed by J = 3/2 electrons. Different from the singleorbital case like half-Heusler materials, additional orbitaldegrees of freedom make it possible to obtain any higherspin gap function in the framework of weakly correlatedconstant gap functions. Then, we evaluate the supercon-ducting critical temperature Tc for each gap function byusing simple density-density pairing interactions. It isfound that the system supports odd-parity superconduc-tivity if the inter-orbital pairing interaction is dominant.The J = 0 odd-parity pairing state with A1u representa-tion gives the highest Tc among odd-parity superconduct-ing states, but the T1u paring state consisting of J = 3and J = 1 Cooper pairs also has relatively high Tc.

We reveal that the simplest J = 0 odd parity pairingstate shows a new class of topological superconductivitydubbed as high winding topological superconductivity:As a three-dimensional time-reversal invariant supercon-ductor, the odd parity pairing state has a non-zero three-dimensional winding number like the 3He-B phase. How-ever, different from the 3He-B because of the higher spinnature of J = 3/2 electrons, it never support the mini-mal non-zero value but hosts a variety of higher windingnumber depending on the model parameters. We alsopresent the topological phase diagram of the supercon-ducting state and determine the topological numbers. Wefinally discuss experimental signals of the high windingsuperconductivity in the case of Sr3−xSnO. We find thatSr3−xSnO is in the vicinity of topological phase transi-tion, which predicts a characteristic nodal structure ofthe superconducting gap if it realizes the J = 0 odd par-ity pairing state.

This paper is organized as follows. In Sec. II, we ex-amine the normal state of higher spin topological ma-terials. Using the multi-orbital k · p Hamiltonian, weclarify the phase diagram of the J = 3/2 topologicalcrystalline insulators. We reproduce the results for theantiperovskite oxides,46–48 clarifying the topological ori-gin of Dirac points in antiperovskites. We also find thata novel cubic pattern of Dirac points is possible, in addi-tion to the octahedral Dirac points in antiperovskites. InSec. III, we examine superconductivity of doped higherspin topological insulators. We classify gap functions interms of irreducible representation of Oh point group,and evaluate the superconducting critical temperaturefor each gap function. We demonstrate that the A1u

(J = 0) representation is the most stable odd parity pair-ing state, and the T1u (J = 3, or 1) is the second one.Section IV is devoted to identify topological crystallinesuperconductivity of the A1u state, followed by the con-

clusion given in Sec. V.

II. NORMAL STATE

A. Effective Hamiltonian

First, we formulate higher spin topological insulators.Spin-orbit coupled electrons may behave as higher spinfermions with J = (2l + 1)/2 (l = 1, 2, . . . ) as a resultof mixture of spin and orbital angular momentum. Forelectrons in crystals, however, a more precise definitionof the higher spin fermion is necessary since rotation isdiscretized. Then, for a crystal, a spin J fermion canbe defined as a band fermion belonging to a (2J + 1)-dimensional representation of the point group of the crys-tal at the Γ point. From this definition, it is foundthat only a J = 3/2 fermion is possible as a higher spinfermion since point groups allow at most four dimensionalrepresentations at Γ. Furthermore, the crystal should becubic since the cubic symmetry is necessary to obtainthe four-dimensional representation. For this reason, weconsider a cubic crystal in the following. We also as-sume inversion symmetry for simplicity, which specifiesthe cubic symmetry as the Oh group.

For ordinary spin 1/2 electrons, a topological insu-lating phase is obtained by the band inversion betweenorbitals with different parity at the Γ point. We canconsider a similar band inversion mechanism even forJ = 3/2 electrons: The Oh group hosts two differentfour-dimensional, i.e. J = 3/2 representations, G3/2g

and G3/2u, which correspond to the even parity bandand the odd parity one, respectively. When these bandsare inverted, a gap between them closes at Γ, and thuswe can naturally expect a non-trivial topological phasetransition by the band inversion.

This argument specifies a system for higher spin topo-logical insulator: It is symmetric under the Oh group,and consists of two different J = 3/2 bands correspond-ing to the G3/2g and G3/2u representations of the Ohgroup. Moreover, time-reversal symmetry should be as-sumed when we consider an insulator.

It should be noted that a class of antiperovskite materi-als A3BX with A=(Ca, Sr, La), B=(Pb. Sn) and X=(C,N, O) realizes such a higher spin system. Near the Fermilevel, they host a d-orbital band of A atom and a p-orbitalone of B, which belong to the G3/2g and G3/2u represen-tations of the Oh group, respectively. Therefore, our the-ory is directly applicable to the antiperovskite materials.Taking the application to the antiveprovsikes into consid-eration, we below call the G3/2g (G3/2u) representationas d-orbital (p-orbital), but our theory is not restrictedto this specific class of materials.

The single particle state of the J = 3/2 topologicalinsulator is represented by an eight component spinorck ≡ cjz,σz,k, where jz = ±3/2,±1/2 denotes the z-component of J corresponding to the four dimensionalrepresentation of the Oh group, σz specifies the parity

3

of the representation, i.e. σz = 1 (σz = −1) for the d-orbital (p-orbital), and k is the momentum. Generatorsof the Oh group consist of q-fold discrete crystal rotationsCq,n (q = 2, 3, 4) and spatial inversion P , which act onthe Hamiltonain H0(k) as

Cq,nH0(Cq,n[k])C−1q,n =H0(k) with Cq,n =e−i

2πq J·n,(1)

PH0(−k)P−1 = H0(k) with P = σz. (2)

Here n denotes the rotation axis, Ji=x,y,z is the 4×4 J =3/2 spin operator, and σi is the 2×2 Pauli matrix in theorbital space. Since the p (d) orbital changes (does notchange) the sign under inversion, P is given as P = σz.We also have time-reversal symmetry,

T H0(−k)T −1 = H0(k) with T = C2,yK. (3)

with complex conjugate operator K.In the following analysis, we use the low energy effec-

tive Hamiltonian for the J = 3/2 electrons. To constructthe effective Hamiltonian, we consider possible scalar ma-trices with respect to Oh symmetry. First, we note thatσz is a scalar, while σx and σy are pseudo scalars: Anyof σi (i = x, y, z) is invariant under rotation, but σx andσy change their signs under inversion. Second, becausethe spin operators Ji (i = x, y, z) and their third order

polynomials Ji ≡ 53

∑j 6=i JjJiJj −

76Ji behave as pseudo

vectors for the Oh point-group, the inner product k · Jand k · J are pseudo scalars.53 Finally, we can constructa scalar as a product of any two of pseudo scalars in theabove. Consequently, the effective Hamiltonian for lowerorder of momentum k is given as48

H0(k) = m(k)σz + k · (v1J + v2J)σx, (4)

where the σy term is absent due to the time-reversal sym-

metry. Note that J term reduces the continuous rota-tional symmetry to the discrete Oh one. The effectivemass m(k) in Eq. (4) is given as

m(k) = −m0 + αk2 +[k · (β1J+β2J)

] [k · (β3J+β4J)

],

up to the second order of k, where 2m0 is the energy gapbetween p- and d-orbitals at Γ point. In the main textbelow, we focus on the case where the band inversiontakes place at Γ point: We take m0 > 0, α > 0, andβi = 0 (i = 1, 2, 3, 4), which can reproduce the bandstructure of Sr3SnO qualitatively. The case without theband inversion will be considered in appendix E.

Note that H0(k) can not be an ordinary TI, althoughthe band inversion can occur at Γ point: The Z2 numberfor TI is always trivial due to the fourfold degeneracyof J = 3/2 spin at Γ. However, the system can be atopological crystalline insulator, as shown below.

B. Octahedral Dirac Points

By diagonalizing the Hamiltonian (4), we obtain theband spectrum. First, let us focus on the band spectrum

FL

(c)

(d)

-2

+2

03-1/3 1/2

-2

+2

03-1/3

(f)(e)

(a) (b)

FLInnerFS

OuterFS

FIG. 1: Energy dispersion near the Γ point for (a) v2 = 3v1

and (b) v2 = v1/2. For the hole doped antiperovskite mate-rial Sr3−xSnO, the Fermi level is located below the fourfolddegeneracy point E = −m0 as indicated by green line. (c)and (d) show octahedral and cubic Dirac points for v2 = 3v1

and for v2 = v1/2 respectively. Dots denote the Dirac points.Transparent panels indicate the mirror plane defining topo-logical phase transition with creation of the Dirac points. (e),(f) the Chern number for the mirror sector carrying 〈Mx〉 = iand 〈Mx+y〉 = i defined on the mirror plane.

along the primary axis, say the kz-axis. On the kz-axis,we have [C4,z, H0(0, 0, kz)] = 0 due to fourfold rotation

symmetry of the Oh group, C4,zH0(−ky, kx, kz)C−14,z =

H0(k), and thus the eigenvalue 〈C4,z〉 = e−ijz2 π is a good

quantum number. The Hamiltonian is subdivided intofour sectors with different jz,

H0,jz (kz) = mjz (kz)σz + bjzkzσx, (5)

where mjz (kz) = m−jz (kz) is an even function of kz andbjz is the following constant,

b±3/2 = ±(3v1 − v2)/2, (6)

b±1/2 = ±(v1 + 3v2)/2. (7)

From time-reversal and inversion symmetry, the sectorswith opposite jz form a Kramers pair with the same spec-tra. The first term of Eq.(5) gives a parabolic energy dis-persion for each orbital, and the second one mixes them.As seen in Eq.(6) (Eq.(7)), the orbital mixing in thejz = ±3/2 (jz = ±1/2) sector vanishes when v2 = 3v1

(v1 = −3v2). In these cases, there appear Dirac pointsin the band spectrum.

Figure 1(a) illustrates the band spectra for v2 = 3v1.The parabolic dispersions of p- and d-orbitals with jz =

4

±3/2 linearly cross each other at kz = ±k0 with k0 sat-isfying m±3/2(k0) = 0. As a result, there appear twoDirac points on kz axis. From the Oh group symme-try, their counterparts also exist on the kx and ky axis,respectively, forming vertices of an octahedron in the mo-mentum space. See Fig. 1(c). On the other hand, p- andd-orbitals in the jz = ±1/2 sectors hybridize with eachother and thus they do not form a Dirac point. Similarly,when v2 = −v1/3, octahedral Dirac points appear in thejz = ±1/2 sectors.

In contrast to ordinary Dirac semimetals like Cd3As2,these Dirac points are easily gapped out by small pertur-bation even if it preserves crystalline symmetry. In thissense, they are not stable, but their existence is robustnonetheless: They indeed appear as gap-closing pointsaccompanied by topological phase transition. A keytopological number is the mirror Chern number. Con-sider one of three equivalent horizontal mirror reflectionsof the Oh group, say the mirror reflection with respect tothe yz plane:

MxH0(−kx, ky, kz)M−1x = H0(k), (8)

with Mx = PC2,x. See Fig. 1 (c). On the mirror invari-ant kx = 0 plane, it holds that [Mx, H0(0, ky, kz)] = 0,so H0(0, ky, kz) is block diagonal in the diagonal basisof Mx. Each block-diagonal subsector H0,±(0, ky, kz)has a definite eigenvlaue of Mx = ±i and it may hostits own Chern number ν±Ch. The mirror Chern num-ber νσh for the horizontal mirror reflection is defined byνσh = (ν+

Ch − ν−Ch)/2. From time-reversal symmetry, the

total Chern number ν+Ch + ν−Ch vanishes, and thus νσh

takes an integer.

As summarized in Fig. 1(e), the mirror Chern numberis obtained as48

νσh = 2sign(b3/2b1/2), (9)

which jumps by four when Dirac points appear at v2 =3v1 or v2 = −v1/3. In other words, each of four Diracpoints on the kx = 0 plane change the mirror Chern num-ber by one. Therefore, these Dirac points are caused bythe topological phase transition. This means that theirpresence is robust against small perturbation as long asthe mirror reflection symmetry is retained: The smallperturbation may open a gap in the Dirac points, butit merely shifts the phase boundary of the topologicalphase transition, which hosts gapless Dirac points again.Except at the topological phase transition, the systemsupports a non-zero mirror Chern number, and thus thenormal state is a topological crystalline insulator if theFermi level is in the band gap.

The first principle calculations show that the antiper-ovskites support the octahedral Dirac points with a tinygap in the jz = 3/2 sector.46–48 Therefore, they are nearthe topological phase transition at v2 = 3v1.48

C. Cubic Dirac Points

The effective Hamiltonian (4) also reveals that anotherclass of Dirac points is possible in higher spin topologicalmaterials. To see this, consider the diagonal axis, saythe (111) axis. From the threefold rotation symmetry of

the Oh group, 〈C3,n111〉 = e−i23 j111π is a good quantum

number along the (111) axis, where n111 = (x + y +

z)/√

3 and j111 = ±1/2,±3/2. Note that j111 = ±3/2falls into the same subsector with 〈C3,n111〉 = −1. TheHamiltonian in the j111 subsector takes the same form asEq. (5), but kz, jz, and bjz in Eq. (5) are replaced with

k111 = (kx+ky+kz)/√

3, j111, and b′j111 . The interorbitalcoupling b′j111 is given by

b′±1/2 = ±v1 − 2v2

2, (10)

b′±3/2 =

[(3

2v1 +

1

3v2

)ηz +

5√

2

6v2ηx

]. (11)

Here ηi is the Pauli matrix in the j111 = ±3/2 space.Whereas b′±3/2 does not vanish for any vi, b

′±1/2 be-

comes zero when v2 = v1/2. In the latter case, the p-and d-orbitals do not hybridize with each other alongthe (111) axis, forming a Dirac point in Fig. 1(b). Bytaking into account Oh symmetry, we totally have eightDirac points at the vertices of a cube, as illustrated inFig. 1(d).

The cubic Dirac points are also understood as gapclosing points for topological phase transition. The rel-evant topological index in the present case is the mirrorChern number for the diagonal mirror reflection of theOh group. For instance, consider the diagonal mirror re-flection with respect to the kx+ky = 0 plane in Fig. 1(d).In a manner similar to the octahedral Dirac case, we canevaluate the mirror Chern number νσd with respect to thediagonal mirror plane. The result is shown in Fig. 1(f).At v2 = v1/2, where there are four Dirac points on thisplane, the mirror Chern number jumps by four.

It should be noted here that two of octahedral Diracpoints appear on the diagonal mirror plane when v2 =3v1 or v2 = −v1/3. As a result, the diagonal mirrorChern number in Fig. 1(f) jumps by two at v2 = 3v1 andv2 = −v1/3.

III. SUPERCONDUCTIVITY

A. Oh classification of the gap function

With carrier doping, topological insulators can be su-perconductors at low temperature. Here we consider su-perconducting states of doped higher spin topological in-sulators.

To describe the superconducting states, we use theNambu space spanned by the eight component spinor ckand its time-reversal hole partner ck ≡ T c−k = C2,yc

†−k.

5

In this basis (ck, ck), the Bogoliubov-de Gennes Hamil-tonian is written as

H(k)=

(H0(k)− µ ∆(k)

∆†(k) −H0(k) + µ

), (12)

where we have used T H0(−k)T −1 = H0(k). Here thegap function ∆(k) satisfies

(∆(k)C2,y)T = −∆(−k)C2,y, (13)

from the Fermi statistics of ck. Since T commutes withany point group operation, ck and ck transform in thesame manner under the point group. Therefore, in thisbasis, the point group acts on the gap function ∆(k) inthe same manner as the normal Hamiltonian H0(k).

We classify here the multi-component gap function∆(k). For simplicity, we assume that ∆(k) is indepen-dent of the momentum k like the conventional BCS the-ory. The higher spin and multi-orbital natures of the sys-tem enable us to host a variety of unconventional Cooperpairs even in the weakly correlated case.

The gap function is expanded as

∆ =∑i,ν

∆i,νΦi,ν , Φi,ν =ϕi(J)⊗ σν

N, (14)

where ϕi(J) is a set of 4× 4 matrices spanning the spinspace, σν (ν = 0, x, y, z) is the Pauli matrix in the or-

bital space, and N =√

tr[(ϕi ⊗ σν)2]/8 is the normal-ization constant. A convenient basis of ϕi(J) is thespherical harmonic Yl,m(J), which is defined by Yl,m±1 =∓[J±, Yl,m] with J± = Jx ± iJy. In the present case, thespin of a Cooper pair is given by 3/2⊗3/2 = 0⊕1⊕2⊕3,so the azimuthal and magnetic quantum numbers l andmin Yl,m take l = 0, 1, 2, 3 and m = −l, . . . , l. Reconstruct-ing the basis Yl,m for each l in terms of representationsof the O group, we obtain ϕi(J) in Table I. Here we usethe O group rather than Oh since spatial inversion actson the spin space trivially as the identity operator.

It should be noted that a possible combination of ϕi(J)and σν in the right hand side of Eq.(14) is restrictedby the constraint (13). Since ϕi(J) for even (odd) lsatisfies [ϕi(J)C2,y]T = −ϕi(J)C2,y ([ϕi(J)C2,y]T =+ϕi(J)C2,y), it is combined only with the symmetricPauli matrices σ0, σx, and σz (the anti-symmetric Paulimatrix σy). As a result, we have 28 different Φi,ν inEq.(14). In the group theory, they are classified into eightirreducible representations of the Oh group, as shown inTable. II.

Since the spatial inversion operator is P = σz, an in-terorbital pairing state, which is written as ∆ with σx orσy, becomes an odd parity pairing state even if the gapfunction is independent of k. This type of odd paritypairing states is possible only in multi-orbital systems.These odd parity pairing states are expected to be topo-logically non-trivial.17,19

TABLE I: Classification of spin basis for the 4 × 4 matri-ces by O point group symmetry. The columns correspond toirreducible representation of O point group symmetry withindices for the amplitude l of coupled angular momentum fortwo 3/2-spins, and the basis matrix.

O(l) Basis ϕi(J)

A(0)1 1

T(1)1 {Jx, Jy, Jz}

E(2) {2J2z−J2

x−J2y , J

2x−J2

y}

T(2)2 {JxJy+JyJx, JyJz+JzJy, JzJx+JxJz}

A(3)2 JxJyJz+JyJzJx+JzJxJy+JzJyJx+JxJzJy+JyJxJz

T(3)1 {Jx, Jy, Jz}

T(3)2 {JyJxJy−JzJxJz, JzJyJz−JxJyJx, JxJzJx−JyJzJy}

B. Critical temperatures

Now we evaluate the superconducting transition tem-perature Tc for each representation in Table II. Below, weassume the following density-density pairing interactions,

Hint =−∫d3r

{U [n2

d(r) + n2p(r)]+2V nd(r)np(r)

},(15)

where the nσz =∑jzψ†jz,σz (r)ψjz,σz (r) is the density of

the electron with orbital σz defined by the field operatorψjz,σz (r) =

∫d3ke−ik·rcjz,σz,k.

The transition temperature Tc is obtained by solvingthe linearlized gap equation. For each representation inTable II, we have18,36 (see also appendix),

detX(Tc) = 0, (16)

TABLE II: Eight classes of whole basis of the gap functionby Oh point-group symmetry. The columns denote intra orinterorbital pairing, irreducible representation of Oh point-group symmetry and the basis matrix given as the directproducts of spin basis in O representation and Pauli matricesfor the orbital basis.

Orbital Oh Basis ϕi(J)⊗ σνA1g A

(0)1 ⊗ σ0, A

(0)1 ⊗ σz

Intra Eg E(2) ⊗ σ0, E(2) ⊗ σzT2g T

(2)2 ⊗ σ0, T

(2)2 ⊗ σz

A1u A(0)1 ⊗ σx

A2u A(3)2 ⊗ σy

Inter Eg E(2) ⊗ σxT1u T

(1)1 ⊗ σy, T

(3)1 ⊗ σy

T2u T(2)2 ⊗ σx, T

(3)2 ⊗ σy

6

0.05

0.3

2 4 6 8 100

0

(a)

(b)

FIG. 2: Transition temperature for even (a) and odd (b) par-ity superconducting states. The shaded region of v2/v1 cor-responds to the doped antiperovskite Sr3−xSnO. Here µ =−2m0, α = 6.25v2

1/m0, and βi = 0.

where X(T ) is given by

Xα′,β′(T )=δα′,β′− Vα′D0

64kBT

×∑λ1,λ2

∫kF

d2k〈uλ2,k|Φβ′ |uλ1,k〉〈uλ1,k|Φα′ |uλ2,k〉.

(17)

Here α′ and β′ specifies a different basis ϕi(J)⊗σν/N ≡Φα′ (if exist) in the representation. For instance, the A1g

gap have two different bases, A(0)1 ⊗σ0 and A

(0)1 ⊗σz, so α′

runs α′ = 1, 2. Vα′ = U (V ) for intra (inter) orbital pair-ing, D0 is the dimensionless density of states on the Fermisurface, kB is the Boltzmann constant, and |uλ,k〉 is thenormalized single particle state, [H0(k)− µ] |uλk〉 = 0,where λ = ±1 is the pseudo spin PT |uλ,k〉 = λ|u−λ,k〉.We show the numerical solutions of Eq. (16) in Fig. 2.The results for even parity states and those for odd onesare shown separately since Tc’s of even (odd) parity statesare independent of the inter-orbital (intra-orbital) pair-ing interaction V (U). It is found that the spin-singletA1g (A1u) representation exhibits the highest Tc amongeven (odd) parity states, but a higher spin pairing stateof the T1u representation also shows a relatively high Tc.

These results can be understood by spin and orbitaltextures of the Fermi surfaces in the normal state. InFig. 3, we show the expectation values of 〈J〉 and 〈σ〉on the Fermi surfaces with respect to the single particlestate |uλk〉. The hole doped antiperovskites with µ <−m0 support two Fermi surfaces around the Γ point, andeach Fermi surface has twofold degeneracy with differentspin and orbital textures, due to PT symmetry. Theorbital texture 〈σ〉 indicates the mixing between d- and p-orbitals: When 〈σ〉 ‖ z, the corresponding single particlestate satisfies σz = ±1, so it should be either d- or p-orbital and there is no orbital mixing. On the other hand,

Spin Orbital

Fermi surface

FIG. 3: (center) Fermi surfaces of antiverovskites. For µ <−m0, there are two Fermi surfaces. (top) The spin and orbitaltextures of the inner Fermi surface. (bottom) The spin andorbital textures of the outer Fermi surface. The green (blue)arrow represents the expectation value of the spin (orbit).The dashed arrows indicate T , P and PT partners. Theparameters are µ = −2m0, v2 = 2.9v1, α = 6.25v2

1/m0, βi = 0and m0 > 0.

when 〈σ〉 ‖ x, d- and p-orbitals are fully mixed in equalweight.

First, consider the A1g state. Since it is spin-singlet,the Cooper pair has an anti-parallel spin configuration.Thus, it is formed by an electron and its time-reversalpartner, which have opposite spins. In this case, there isno obstruction for them to form an intra orbital pairing.Indeed, as shown in Fig. 3, because time-reversal does notchange the orbital, the electron and the partner have thesame 〈σ〉. As a result, the A1g state is naturally realizedin the presence of the intra-orbital pairing interaction.

In a similar manner, the A1u state is naturally realizedin the presence of the inter-orbital pairing interaction.

7

The A1u state is also spin-singlet, so its Cooper pair isformed again between an electron and its time-reversalpartner. However, for the inter-orbital pairing state, theorbital mixing is necessary, which gives interesting nodalstructure of the gap depending on the model parameters.For instance, when v2 ∼ 3v1, the outer Fermi surface doesnot show the orbital mixing at the intersections with theprimary axis (i.e. the ki=x,y,z axis). See Fig. 3. Cor-respondingly, there are gap-nodes at the intersections inFig. 4 (b). Later, we will show that this nodal structureis not accident but it has a topological origin. In Fig. 5,we show the phase diagram obtained by our calculation.It is found that the A1u phase is realized when V is muchlarger than U .

Up to this time, a time-reversal partner has been con-sidered for a Cooper pair. However, it is not the only pos-sible partner. An electron may also form a Cooper pairwith its spatial inversion partner. Such a Cooper pairinevitably has a parallel spin configuration with J = 1, 3since inversion does not flip the spin. As shown in Ta-ble II, the A2u, T1u and T2u states contain this class ofCooper pairs, but only T1u hosts a simple J = 1 compo-nent. Furthermore, it also hosts a J = 3 component, andthus by adjusting the ratio of these two components, T1u

can optimize the gap function so as to be consistent withthe spin and orbital textures on the Fermi surfaces, asmuch as possible. For these reasons, T1u shows a higherTc. We illustrate the optimized superconducting gap ofthe T1u state in Fig. 4.

It is worth noting that Tc of T1u is comparable to thatof A1u, which suggests that T1u could win by improvingthe spin and orbital textures slightly. Indeed, a similarexchange of the pairing symmetry has been reported inthe case of doped topological insulators: By adding thewarping term, the spin-triplet E1u symmetry beats thespin-singlet A1u one.21 Therefore, the present calculationdoes not exclude the possibility of T1u.

C. Higher spin Cooper pair

As mentioned in the above, the T1u state supports bothof the J = 1 and J = 3 components. In Fig. 6, we showthe mixing ratio of these two components at Tc in ournumerical solution of the gap equation. It is found thatthe higher spin J = 3 component cannot be neglected inthe whole region of v2/v1, and it can even dominate thegap function in some regions. Therefore, the T1u stateactually realizes a higher spin pairing state. The higherspin nature of the pairing state could be tested by spinsensitive experiments.

It should be noted that T1u state realizes a nematic su-perconducting state which spontaneously breaks the Ohgroup symmetry: As illustrated in Fig. 4, T1u supports ananisotropic superconducting gap breaking the Oh group.The nematic feature also can be used to identify the T1u

state by thermal transport measurements.23

IV. TOPOLOGICAL SUPERCONDUCTIVITYIN THE A1u STATE

In this section, we study topological properties of theA1u (J = 0) superconducting state, in terms of topolog-ical indices and numerically obtained surface states.

A. 3D winding number

We first examine the topological properties indepen-dent of crystal symmetry. The BdG Hamiltonian (12)for A1u is rewritten as

H(k) = [H0(k)− µ]τz + ∆0σxτx, (18)

where τi are the Pauli matrices in the Nambu space. Be-low we take ∆0 > 0 by using gauge transformation. Be-cause A1u keeps time-reversal symmetry, the BdG Hamil-tonian also exhibits time-reversal symmetry

T H(k)T −1 = H(−k) with T = C2,yK, (19)

1

0

(b)

(a)

(e)

(d)

1

0

1

1

1

1

0

0

0

0

(c) (f)

FIG. 4: Energy gap E of A1u representation at the inner (a)and outer (b) Fermi surface for v2 = 3v1. Those for v2 = 3v1

(dashed) and 4v1 (solid) on kzkx cut are displayed in (c). (d-f)is the same plot as (a-c) for T1u representation.

8

0

0.5

1

0 2 4 6 8 10

FIG. 5: Phase diagram of odd and even parity states spannedby U/V and v2/v1. Curves are phase boundary for typicalvalues of chemical potential. The other parameters are takenthe same as Fig. 3.

0

1

0 2 4 6 8 10

FIG. 6: Mixing ratio of angular momenta l = 1 and l = 3pairs for T1u gap function ∆ = ∆0

2√5(R1Jz + R3Jz) ⊗ σy.

Normalization condition is given by R21 +R2

3 = 1. Parametersare the same as Fig. 2.

as well as particle-hole symmetry inherent to supercon-ductors,

CH(k)C−1 =−H(−k) with C =

(0 C†2,y

C2,y 0

)K. (20)

By combining these symmetries, we also have chiral sym-metry

ΓH(k)Γ = −H(k) with Γ = iT C = −τy. (21)

These symmetries do not depend on any particularcrystal structure, and thus they specify the most gen-eral symmetry-protected topological number.14 In thepresent case, the general topological number is the three-

dimensional winding number,

w3D =

∫d3k

48π2εαβγtr

[ΓH−1∂αHH

−1∂βHH−1∂γH

].

(22)

Below, we consider the weak pairing limit, ∆0/µ � 1,like ordinary superconductors.

First we numerically evaluate w3D: It is found thatwhen the absolute value of the chemical potential |µ| isless than the critical value µc,

µc =√m2

0 + ∆20 (23)

w3D is zero

w3D(|µ| < µc) = 0. (24)

At |µ| = µc, the BdG Hamiltonian in Eq.(18) becomesgapless at k = 0, then for |µ| > µc, we have

w3D(|µ| > µc) =

−2 sign(v1) for v2/v1 < −1/3

4 sign(v1) for −1/3 < v2/v1 < 1/2−4 sign(v1) for 1/2 < v2/v1 < 3

2 sign(v1) for v2/v1 > 3

.

(25)

For |µ| > µc, the system supports two Fermi surfaces(inner and outer Fermi surfaces) around the Γ point,as was illustrated in Fig. 3. When w3D changes atv2/v1 = −1/3, 3 (v2/v1 = 1/2), there appear six (eight)point nodes on the outer Fermi surface, each of whichcontributes to the change of w3D by 1 (−1). These re-sults imply that the A1u state hosts nontrivial topologicalsuperconductivity when |µ| > µc. In particular, it sup-ports a characteristic higher winding number |w3D| ≥ 1.As is discussed immediately below, the higher windingnumber of the topological superconductivity is a directconsequence of the higher spin of the present system.

To clarify the origin of the higher winding number, weevaluate w3D in terms of the Fermi surface properties: Ingeneral, under the weak pairing limit, a bulk topologicalnumber of superconductors can be attributed to that ofthe Fermi surfaces. In the present case, we have54

w3D =1

2

∑n

sign [vnF] sign [∆n] νnCh, (26)

where n is summed for all disconnected Fermi surfaces,vnF is the Fermi velocity of the n-th Fermi surface, ∆n isthe expectation value of the gap function

∆n = 〈un,k|∆|un,k〉 (27)

with the one-particle state |un,k〉 on the n-th Fermi sur-face, and νCh is the first Chern number of the Fermisurface,

νnCh =

∫k=knF

d2k′εαβ∂k′αanβ(k) (28)

9

with

anα = −i〈un,k|∂k′αun,k〉, (29)

where k′α is the two dimensional momenta on the Fermisurface. Here we take into account Kramers partnersseparately in the summation of n. See Appendix C formore details. From Eq. (26), w3D is evaluated as thetopological number defined on the Fermi surfaces.

The relation between the J = 3/2 spin and the higherwinding number becomes obvious in the spherical sym-metric case at v2 = 0. In this case, the eigen functions ofthe normal state Hamiltonian H0(k) are given by

|u±jz,k〉 = R(k,J) |jz〉 ⊗ |Ψ±(ρk)〉σ . (30)

Here R(k, j) = e−iJzφke−iJyθk with the polar and az-imuthal angles (θk, φk) of k is the rotation matrix thatdiagonalizes the spin-dependent part k ·J of H0(k), and|jz〉 ∈ {|±3/2〉 , |±1/2〉} is the eigenstate of Jz. Theorbit-dependent part of the wave functions is given by

|Ψ±(ρk)〉σ = e−iσyρk |±〉, (31)

with ρk = 12 arctan[v1kjz/m(k)] and the eigenstate |±〉

of σz. It is found that the states |u+jz,k〉 (|u−jz,k〉) give

the electron (hole) branches of the normal spectra, sothey define the Fermi surfaces for µ > µc (µ < −µc).Therefore, Eq. (26) leads to

w3D =∑jz

wjz3D, (32)

with

wjz3D = ±1

2sign(∆±jz,k)νjzCh, (33)

where the double sign ± corresponds to the case withµ ≷ ±µc, ∆±jz,k = 〈u±jz,k|∆0σx|u±jz,k〉, and νjzCh is the first

Chern number of |u±jz,k〉 on each Fermi surface. Using thepolar coordinates of the Fermi surface, we can evaluateνjzCh analytically,

νjzCh =1

2π

∫dθkdφk

(−i

⟨∂u±jz,k∂θk

∣∣∣∣∣∂u±jz,k

∂φk

⟩+ h.c.

)= 2jz, (34)

and Eq.(30) also yields

sign(∆±jz,k) = ±sign(jzv1). (35)

Thus, we have

wjz3D = sign(v1)|jz|, (36)

both for µ > µc and for µ < µc.The last expression clearly indicates that a higher spin

provides a higher winding number. Using this expression,we can also evaluate the winding number for each Fermi

surface: For v2 = 0, the outer (inner) Fermi surface con-sists of the jz = ±1/2 (jz = ±3/2) components, so weobtain

wout = w1/23D + w

−1/23D = sign(v1),

win = w3/23D + w

−3/23D = 3sign(v1), (37)

where wout (win) denotes the winding number for theouter (inner) Fermi surface. These winding numbers keepthe same values for −1/3 < v2/v1 < 1/2 since either agap closing or a touching of Fermi surfaces does not oc-cur until v2/v1 reaches at the boundary of the region atv2/v1 = 1/2 or −1/3. Here we note that the total wind-ing number wout +win reproduces the numerical result ofEq.(25) in the same region of −1/3 < v2/v1 < 1/2.

Using the result in the above, we can also identify thewinding numbers (wout, win) for other regions of v2/v1.First, to evaluate them in the region 1/2 < v2/v1 < 3, weuse a “duality” relation. As is explained in Appendix D,there is a unitary transformation which maps the param-eter (v1, v2) to (−3v1/5 − 4v2/5,−4v1/5 + 3v2/5). Thistransformation exchanges the region of −1/3 < v2/v1 <1/2 with that of 1/2 < v2/v1 < 3, keeping the energyspectra and reversing the winding numbers. Thus, com-bining the duality relation with Eq.(37), we obtain

wout = −sign(v1), win = −3sign(v1), (38)

for 1/2 < v2/v1 < 3. Then, to determine the wind-ing numbers for the remaining two regions v2/v1 < −1/3and v2/v1 > 3, we use properties of the topological phasetransitions at v2/v1 = −1/3 and v2/v1 = 3. As was men-tioned in the above, at the topological phase transitions,there appear six point nodes on the outer Fermi surface,which change |wout| by 6.64 Requiring also that the totalwinding number wout + win reproduces Eq.(25), we canuniquely determine the winding numbers as

wout = −5sign(v1), win = 3sign(v1), (39)

for v2/v1 < −1/3, and

wout = 5sign(v1), win = −3sign(v1), (40)

for v2/v1 > 3. In both cases, the Fermi surfaces havecharactersitic higher winding numbers.

B. Mirror Chern number

The present system has the Oh group crystalline sym-metry, which provides more detailed topological informa-tion.55–59 To be specific, we first consider the horizontal(or equivalently vertical) mirror reflection Mx with re-spect to the kx-axis.

The A1u gap function is odd under the mirror reflec-tion. In this case, the mirror reflection operator for theBdG Hamiltonian is given by Mxτz, where Mx is the mir-ror operator for the normal Hamiltonian H0(k).60 On the

10

FL

FL(a) (b)

0 0

Electron Hole

::

::

::

::

FIG. 7: Level structures and their angular momentum in〈Mxτz〉 = i sector at Γ point. (a) and (b) are for electronhole sectors respectively.

mirror invariant plane kx = 0, the BdG Hamiltonian canbe block diagonal in the diagonal basis of Mxτz. Then,the mirror Chern number νσh for the BdG Hamiltonianis defined in the same manner as νσh in Sec.II.

First, let us evaluate νσh in the limit µ = 0 and ∆0 = 0.In this limit, the BdG Hamiltonian reduces to H0(k)τz,so νσh can be evaluated as νσh = 2νσh . Therefore, fromEq.(9), we have

νσh(µ=0) = 4sign(b3/2b1/2). (41)

For small ∆0 and µ, νσh keeps the same value unless thegap of the system closes.

When |µ| = µc, the gap of the system closes, and νσhchanges. Using fourfold rotation symmetry around the kxaxis, we obtain the following useful formula to calculateνσh .35

νσh =∑jx

Njx,+(Γ)jx mod 4, (42)

where Njx,+(Γ) is the number of negative energy stateswith spin jx at the Γ point in the Mxτz = i sector of theBdG Hamiltonian. In the weak coupling limit ∆0 → 0,a negative energy state in the Mxτz = i sector reduceto either an electron state in the Mx = i sector or ahole state in the Mx = −i sector below the Fermi level.Therefore, Njx,+(Γ) can be evaluated as the number ofthose electron and hole states. Then, as summarized inFig. 7, when |µ| exceeds m0, the jx = 3/2,−1/2 electron(jx = 1/2,−3/2 hole) bands in the Mx = i (Mx = −i)sector go above (below) the Fermi level at Γ. Therefore,the mirror Chern number jumps by

∆νσh(|µ| > µc) = −(

3

2− 1

2

)+

(1

2− 3

2

)mod 4

= −2 mod 4 (43)

We can also evaluate νσh numerically as

νσh(|µ| > µc) = 2sign(b3/2b1/2), (44)

which is consistent with Eq. (43).According to the bulk-edge correspondence, the non-

zero mirror Chern number implies the existence of sur-face states. Here we calculate the energy spectrum in the

slab geometry in Fig. 8 with finite size L along z axis.This system has the horizontal mirror reflection symme-try with respect to the x direction even in the presenceof the surface. Along the high symmetric line kx = 0,we numerically solve the BdG equation for each mirrorsector by replacing the momentum kz → −i∂z, and usingthe Gauss-Lobatto expansion method.61

As shown in left column of Fig. 9, when |µ| < µc thespectrum along the kx = 0 line exhibits four branchesof surface states. The number of the branches coincideswith |νσh |. Furthermore, the surface states change thechirality (or the connectivity with the bulk states) whenνσh changes the sign: For −1/3 < v1/v2 < 3, each surfacestate connects the bulk bands upward when going fromleft to right, but for v1/v2 < −1/3 or for v1/v2 > 3 itconnect them downward. On the other hand for |µ| > µc,as shown in left column of Fig. 10, the surface states hastwisted spectrum. This originates from the multiple Ma-jorana fermions protected by higher 3D winding number|win/out| > 1 discussed in Sec. IV A. In this case, whenthe surface state cuts the zero energy upwards (down-wards) going from left to right, it accumulate the mirrorChern number by 1 (−1). The total mirror Chern num-ber counted from Fig. 10 is consistent with Eq. (44).

Similarly, the diagonal mirror Chern number νσd withrespect to the kx+ky direction can be defined for the BdGHamiltonian. For µ = 0, it is evaluated as the doublevalue of that in Fig. 1(f), which persists as long as µreaches the critical value µc,

νσd(|µ| < µc) =

+4 for − 1

3 <v2v1< 1

2 ,

−4 for 12 <

v2v1< 3,

0 otherwise .

(45)

In addition, for |µ| > µc, we numerically find that

νσd(|µ|>µc) =

+2 for − 1

3 <v2v1< 1

2 ,

−2 for 12 <

v2v1< 3,

0 otherwise .

(46)

The quasiparticle spectra along the mirror invariant linekx + ky = 0 for the slab geometry are shown in rightcolumns of Figs. 9 and 10, all of which are consistent withthe above values of the diagonal mirror Chern number.

FIG. 8: System geometry for slab system.

11

0.8

-0.8

0

0.8

-0.8

0

0.8

-0.8

0

-1 10 1 10

(b)

(c)

(d)

0.8

-0.8

0

(a)

-0.5 0.50 -0.5 0.50

FIG. 9: Energy spectrum of slab in different topologicalphases of small |µ| regime along kx (left) and k110 = (kx −ky)/√

2 (right) direction. For the visibility, we display thesurface states localized only at z = −L/2 highlighted by redcurves. The momentum unit is k0 = m0/v1. Insets of (b)are the zoom for center of Brilloiuin zone and close to E = 0.Here, ∆0 = 0.025m0, µ = −0.75m0, v2/v1 = −0.5 (a), −0.2(b), 2.5 (c), and 4 (d), α = 0.64v2

1/m0 (a,b) and 6.25v21/m0

(c,d).

In particular, for the µ = −0.75m0, surface dispersionsof Figs. 9(a) and (d) cut the zero energy upward as manytimes as downwards going from right to left, so there isno topological protection.

In Fig 10, all the surface states for |µ| > m0 passthrough the high symmetry point ky = 0 (k110 = 0)at E = 0. The origin of this property will be explainedin the next section.

It is worth noting that even in the parameter regionm0α < 0, where the band inversion of two orbital at theΓ point does not occur in normal state, the mirror Chernnumbers for |µ| > µc take the same value as Eq. (44)and (46) although those for |µ| < µc are νσh = νσd =

0.8

-0.8

0

0.8

-0.8

0

0.8

-0.8

0

-1 10 1 10

(b)

(c)

(d)

0.8

-0.8

0

(a)

-0.5 0.50 -0.5 0.50

FIG. 10: Energy spectrum of slab in different topologicalphases for large |µ| regime. The round and square symbolsdenote helical Majorana fermion corresponding to those ininsets of Fig. 11. The open symbols indicate pairs of helicalMajorana fermions with opposite helicity, which can annihi-late by continuous deformation. The other conditions are thesame as Fig. 9.

0, as also discussed in the appendix E. This is becausethe topological phase transition from |µ| < µc to |µ| >µc does not originate from the band inversion, but fromshrink of Fermi surface at fourfold degenerated point.

C. 1D winding number

The system also has fourfold rotation symmetry, fromwhich one can define another topological invariant. TheA1u gap function is invariant under the fourfold rotationalong the primary axis, so the BdG Hamiltonian triviallyrealizes the fourfold symmetry as,

C4,zH(ky,−kx, kz)C−14,z = H(k), C4,z = C4,zτ0. (47)

12

++

+ ++

+- ---

-

-+ +++

+

+- ----+ +

++

--

- --

=[-2, -2, 0, -1, 1] [4, 2, 2, 1, 1] [-4, 2, -2, 1, 1] [2, -2, 0, 1, -1]

[0, -4, 0, 0, 0] [0, 4, 4, 0, 0] [0, 4, -4, 0, 0] [0, -4, 0, 0, 0]

0 1/2 3-1/30

Octahedral Dirac Cubic Dirac Octahedral Dirac

(a) (b) (c) (d)

(e) (f) (g) 1

0

FIG. 11: (a) Topological phase diagram of the A1u pairing state for v1 > 0. The A1u state may host the 3D winding numberw3D as a time-reversal invariant superconductor. From the Oh group symmetry, it also support the two different mirror Chernnumbers, νσh and νσd , and the rotation 1D winding number wjz1D. If we take v1 < 0, the winding numbers w3D and wjz1D

change the sign. The insets indicate location of helical Majorana fermion in surface Brillouin zone, which carries by windingnumber ±1. Top and bottom rows in |µ| > µc phase show contributions from outer and inner Fermi surfaces respectively. Gapstructures on the outer Fermi surface (e) v2 = 3v1, (f) v2 = v1/2, and (g) v2 = v1/2. Red dots indicate the point node of thesuperconducting gap. Here, µ = −2m0, βi = 0, m0 > 0, and α = 6.25v2

1/m0 for (e,f) and α = 0.64v21/m0 for (g).

On the primary axis kx = ky = 0, the BdG Hamiltonian

is block diagonal in the eigen basis C4,z = e−ijz2 π,

Hjz (kz) = H0,jz (kz)τz + ∆0σxτx. (48)

We also find that each sectorHjz (kz) keeps chiral symme-

try Γ = iT C in Eq.(21), since it holds that [Γ, C4,z] = 0.Thus on the basis where Γ is diagonal, Hjz (kz) takes theform

Hjz =

[0 hjz (kz)

h†jz (kz) 0

]. (49)

Then, one can introduce the 1D winding number

wjz1D =1

2πIm

{∫dkz∂kz ln[det(hjz )]

}, (50)

with

det(hjz )=µ2−m2jz (kz)+(bjzkz)

2−∆20+2ibjzkz∆0, (51)

where bjz is given by Eqs. (6) and (7). It can be calcu-lated analytically as

wjz1D =

{0 for |µ| < µcsign(bjz ) for |µ| > µc

. (52)

13

It is also found that the total winding number vanishes,∑jzwjz1D = 0, because of b−jz = −bjz , and the 1D wind-

ing number is related to mirror Chern number as

2w3/21D w

1/21D = νσh mod 4 (53)

When |µ| > µc, |wjz1D| = 1 for each jz. This means thatthere must be four zero modes at kx = ky = 0 in eachmirror subsector, which explains the reason why the sur-face states in one mirror sector in Figs. 10 pass the highsymmetric points ky = 0 or k110 = 0 when µ = −1.25m0.

D. Topological phase diagram

We summarize the obtained topological phase diagramin Fig. 11. On the bottom of Fig. 11 with |µ| = 0, thetopological indices and the corresponding surface statesoriginate from those in the normal state. As alreadydiscussed in Sec. II, there appear octahedral or cubicDirac points at topological phase transition points in thisregime. On the other hand, for |µ| > µc, as can be seenin Fig. 11(f) [(e) and (g)], there appear eight (six) pointnodes of superconducting gap on the outer Fermi sur-face at v2 = v1/2 (3v1 and −v1/3). These point nodesare also caused by topological phase transitions wherethe bulk topological number w3D changes by eight (six).This kind of nodal structures has been overlooked in thegroup theoretical classification of gap nodes62, but ourtopological analysis here clarifies the robustness of thepoint nodes at the same level as the bulk Dirac points inthe normal states. The presence of point dips near thetopological phase transitions gives a chance to confirmthe A1u state by low temperature measurements of thespecific heat, the NMR spectrum and so on.

In Fig.11, we also illustrate patterns of helical Majo-rana fermions composed by electrons of the outer andinner Fermi surfaces, respectively. The plus and the mi-nus signs in the Fig.11 denote the helicity of the Majornafermions, where the total helicities for the outer (inner)Fermi surface should be the same as the winding num-ber wout (win). As explained below, these configurationsare determined by comparing the surface state spectra inFig.10 with the winding numbers and taking into accountrequirement from symmetry.

Let us first consider the region −1/3 < v2/v1 < 1/2,which includes the spherical symmetric point at v2 = 0.See Fig. 10(b) and Fig.11(b). In this region, the outerFermi surface with wout = 1 provides a single helical Ma-jorana fermion at the center of the surface Brillouin zone,which is indicated as the square symbol at k = 0 (k = kxor k111) in Figs. 10(b) and 11(b). On the other hand,the inner Fermi surface carries the unusual higher wind-ing number win = 3. In the spherical symmetric limitv2 = 0, we have a single surface state at the center of thesurface Brillouin zone again, but the surface spectrumshows a higher order k-dependence near k = 0. More-over, when v2 6= 0, the surface state breaks into several

helical Majorana fermions because the spherical symme-try reduces to the point group Oh. To be consistent withC4v symmetry on the surface, there should be minimallya single helical Majorana fermion at the center of thesurface Brillouin zone and four others at other points, asillustrated in Fig.11 (b). In terms of the winding num-ber, this splitting corresponds to win = −1 + 4. Fromthe actual surface spectrum in Fig.10(b), it is found thatthe four Majorana fermions are located on the diagonalmirror lines in the surface Brillouine zone. We also no-tice that the two center helical Majorana fermions on theinner and the outer Fermi surfaces do not mix althoughthey are located at the same position with opposite helic-ities. This is because they also have different topological

numbers, w1/21D = w

3/21D = 1. In order to generate the cen-

ter Majorana fermion and the satellite ones at the sametime, the surface spectrum is twisted around k = 0, ascan be seen in Fig. 10(b). The twisted surface state en-hances the density of states around E = 0, providing acharacteristic zero bias conductance peak in the tunnelconductance spectroscopy63.

We can also determine the configurations of surfaceMajorana fermions in other regions. In a manner sim-ilar to the argument for the bulk winding numbers inSec. IV A, we use the duality relation to determine theconfiguration in the region 1/2 < v2/v1 < 3. Sincethe duality keeps the spectrum but reverses the windingnumbers, the configuration in Fig.11 (c) is obtained fromFig. 11(b). Finally, the configurations in Figs.11(a) and(d) are determined by considering the topological phasetransitions at v2/v1 = −1/3 and 3. At these transitionpoints, there appear six point nodes on the outer Fermisurface. Being projected on the surface Brillouin zone,these point nodes change the helicity of the center Majo-rana fermions by two, and create four additional satelliteMajorana fermions in the outer Fermi surface excitations,as shown in Figs.11 (a) and (d). The obtained Majoranaconfigurations in Figs.11 (a), (c), and (d) are consistentwith the spectra in Figs.10 (a), (c) and (d), respectively.

E. Application to Sr3−xSnO

Finally, we discuss possible experimental signals of thehigh winding topological superconductivity in the caseof Sr3−xSnO. As we mentioned before, Sr3−xSnO sup-ports octahedral Dirac points in the normal state, so itcorresponds to our model with v2/v1 ∼ 3. Moreover,the superconductivity in antiperovskite Sr3−xSnO.49,50,52

is achieved in highly hole doped regime with µ < −µc.From our topological phase diagram in Fig. 11, if the sys-tem realizes the J = 0 odd parity superconductivity, thisimplies that Sr3−xSnO is in the vicinity of the topolog-ical phase transition between the wout = −1 phase andthe wout = 5 one. Therefore, as shown in Figs 4 (a)-(c), there should be point dips on the outer Fermi sur-face. Note that there is no gap node on the inner Fermisurface since the topological phase transition keeps the

14

value of win. The point node structure can be detectedby low temperature experiments such as the heat capac-ity measurement. We also would like to point out thatdifference in behaviors of the superconducting gap be-tween the outer Fermi surface and inner one might ex-plain the double superconducting transition reported inSr3−xSnO.49

V. CONCLUSIONS

We have investigated topological superconductivity indoped topological insulators of J = 3/2 electrons. An-tiperovskites realize this class of topological materials,the normal state of which hosts Dirac points at topo-logical quantum phase transition. In contrast to singleorbital half-Heuslers, the antiverovskites support multi-ple J = 3/2 bands, which allows to realize a variety ofunconventional superconductivity within the BCS typeof constant gap functions. Using the k · p Hamiltonian,we have demonstrated that the J = 0 odd-parity pairingstate with the A1u representation of Oh group exhibitsthe highest transition temperature Tc if the inter-orbitalpairing interaction is dominant. The A1u state showsa new class of topological superconductivity with higherwinding number w3D = ±2 or ±4. Moreover, it also dis-plays topological crystalline superconductivity with re-spect to the mirror reflections and fourfold rotation ofOh group.

We also have revealed that the nematic T1u state withJ = 3 Cooper pairs has a high Tc, which is comparableto that of A1u. This result suggests interesting possibil-ity of spin-septet nematic superconductivity in antiper-ovskites materials. In a manner similar to the case ofdoped Bi2Se3, the T1u phase might be stabilized againstthe A1u phase by taking into account higher order termsof the k · p Hamiltonian.

Acknowledgments

This work was supported by JSPS KAKENHIGrant numbers JP16K17755, JP16H06861, JP17H02922,JP17J08855 and JSPS Core-to-Core program. S.K. isalso supported by the Building of Consortia for the De-velopment of Human Resources in Science and Technol-ogy.

Appendix A: T1u matrices in J = 3/2 space

Here, we give the explicit forms of matrices useful todescribe J = 3/2 systems. First, the spin matrices in

J = 3/2 space is given by

Jx =1

2

0√

3 0 0√3 0 2 0

0 2 0√

3

0 0√

3 0

, (A1)

Jx =i

2

0 −

√3 0 0√

3 0 −2 0

0 2 0 −√

3

0 0√

3 0

, (A2)

Jz =1

2

3 0 0 00 1 0 00 0 −1 00 0 0 −3

. (A3)

In general, the 4×4 Hermitian matrix in J = 3/2 spaceis expanded by basis given in Table I. Among them, T1u

representation including spin matrices itself {Jx, Jy, Jz}behaves as vector under the spin rotation Cqn defined inEq (1). Here the other set of T1u basis is given by thethird order polynomials of spin matrix, which has explicitform as follows.

Jx ≡5

3(JyJxJy + JzJxJz)−

7

6Jx

=1

4

0√

3 0 −5√3 0 −3 0

0 −3 0√

3

−5 0√

3 0

, (A4)

Jy ≡5

3(JzJyJz + JxJyJx)− 7

6Jy

=i

4

0 −

√3 0 −5√

3 0 3 0

0 −3 0 −√

3

5 0√

3 0

, (A5)

Jz ≡5

3(JxJzJx + JyJzJy)− 7

6Jz

=1

2

−1 0 0 00 3 0 00 0 −3 00 0 0 1

. (A6)

By this definition, the two sets of T1u basis satisfies or-thogonal condition

J · J = J · J =15

4, J · J = 0. (A7)

Appendix B: Formulations for estimating transitiontemperature

In the system discussed in main text, the mean fieldHamiltonian is given as

H(k) =(c†k c†k

)H(k)

(ckck

), (B1)

15

where the spinor of annihilation operator is given byck = cjz,σz,k with the indices jz and σz for spin and or-bital respectively, its time-reversal hole partner is ck ≡T c−k = C2,yc

†−k, and H(k) is the BdG Hamiltonian

given in Eq. (12). Here, we can define the thermal andanomalous Green’s functions

Gk(τ, τ ′) = −⟨Tτ

[ck(τ)c†k(τ ′)

]⟩, (B2)

Fk(τ, τ ′) = −⟨Tτ

[ck(τ)c†k(τ ′)

]⟩, (B3)

where ck(τ) = eHτ/~cke−Hτ/~ is the Heisenberg opera-

tor for imaginary time τ . The imaginary time derivativeof these two Green’s functions gives the Gor’kov equation

[i~ωN−{H0(k)−µ}]Gk(ωN )−∆Fk(ωN )=~,[i~ωN+T {H0(−k)−µ}T −1

]Fk(ωN )−∆†Gk(ωN )=0,

(B4)

where Gk(ωN ) and Fk(ωN ) are given as the Fourier com-ponent such as

Gk(τ, τ ′) =1

β~∑N

eiωN (τ−τ ′)Gk(ωN ), (B5)

with Matsubara frequency ωN = (2N + 1)πkBT/~. Forthe U -V model (15) and the gap function (14) in themain text, the gap equation is described as

∆∗αΦ†α = − limη→0

kBT

~∑ωN

∫d3kVαe

iωNηFk(ωN ), (B6)

where α ≡ (i, ν) specifies basis of gap function given inEq. (14).

Let us consider the linearized gap equation in the weakcoupling limit relevant for the temperature close to Tc.In this limit, the anomalous Green’s function Fk, namelythe correlation between particle and hole, is non-zero onlyat the Fermi surface and the basis α runs only over thesame irreducible representations of crystal symmetry. Inaddition, by solving (B4) for Gk and Fk respectively andsubstituting them successively, we obtain the approxi-mated anomalous Green’s function,

Fk(ωN ) ∼ −~G0k(−ωN )

∑α

∆∗αΦ†αG0k(ωN ), (B7)

where we use the single particle Green’s functionG0

k(ωN ) = [i~ωN −H0(k) + µ]−1

and the time-reversalsymmetry T H0(k)T −1 = H0(−k). By this approxima-tion we can linearize the gap equation (B6) as∑

β

Xαβ(T )∆∗β = 0, (B8)

with coefficient

Xαβ(T ) = δαβ +∑ωN

kBTD0Vα8

×∫kF

d2ktr[G0

k(ωN )Φ†βG0k(ωN )Φ†α

].

(B9)

The transition temperature satisfies conditions for a triv-ial solution of Eq. (B8),

detX(Tc) = 0. (B10)

Next, we derive explicit form of coefficient matrix interms of the solution of the Schrodinger equation for thenormal state

(H0(k)− µ)|un,k〉 = ξn(k)|un,k〉. (B11)

By using the orthonormality and completeness of thewave function,

〈un,k|un′,k〉 = δnn′ ,∑n

|un,k〉〈un,k| = 1, (B12)

it is easy to check that

G0k(ωN ) =

∑n

|un,k〉〈un,k|i~ωN − ξn(k)

. (B13)

This allows us to rewrite the coefficient (B9) to

Xαβ(T ) = δαβ−∑ωN

VαD0

8

∫kF

d2k∑n,n′

tanh (ξn(k)/2kBT )

4ξn(k)

×〈un′,k|Φ†α|un,k〉〈un,k|Φ†β |un′,k〉, (B14)

where we use expansion formula z−1 tanh(z/2) =4∑∞n=0[(2n + 1)2π2 + z2]−1. By taking ξλ(k) = 0 and

n to be labels for Kramers degenerated Fermi surfacesλ = ±, we can obtain the coefficient of Eq. (17) in themain text.

Appendix C: Formulations for 3D winding number

Here, we describe the details on the derivation of the3D winding number (33) in the main text, applying pro-cedure in Ref.54. The Hamiltonian and the chiral opera-tor take the form

H(k) =

(0 h(k)

h†(k) 0

)and Γ =

(1 00 −1

). (C1)

The off-diagonal blocks are given by

h(k) = H0(k)− µ+ i∆. (C2)

The single particle Hamiltonian can be written as thediagonal form in terms of their eigen functions (B11),

H0(k)− µ =∑n

ξn(k)|un,k〉〈un,k|. (C3)

Here, ξn(k) is eigen energy for n-th band. PT symmetryacting on the wave function as

PT |un,k〉 = eiζn |un,k〉 (C4)

16

gives double degeneracy ξn = ξn. In addition, the gapfunction is expanded as

∆ =∑n,n′

∆n,n′ |un,k〉〈un′,k| (C5)

with matrix element ∆n,n′ = 〈un,k|∆|un′,k〉. The time-reversal symmetry restrict ∆n,n′ to real. In the weakcoupling limit, matrix element is ∆n,n′ = 0 except forn′ = n and n′ = n. In addition, for the pairing statesformed by time-reversal partner including A1u, the ma-trix element of PT partner is also zero, ∆n,n = 0. In thiscase, the gap function (C5) also has diagonal form with

∆n,n′ = ∆nδn,n′ . (C6)

Note that the sign of ∆n(k) is invariant for any directionof k. By using (C3) and (C5) , Eq. (C2) is rewritten as

h(k) =∑n

(ξn + i∆n)Pn(k) (C7)

with projection operator

Pn(k) = |un,k〉〈un,k|. (C8)

The eigen values of (C7) are given by absolute valuesof their matrix elements E = ±|ξn + i∆n|. Here, thereplacement

h(k)→ Q(k) =∑n

ξn + i∆n

|ξn + i∆n|Pn(k) (C9)

=∑n

eiϑn(k)Pn(k), (C10)

which is called spectral flattening, does not change thewinding number. Here, the phase factor is given by

ϑn(k) = arccotξn(k)

∆n(k). (C11)

In terms of Q(k) in (C9), we can reformulate the 3Dwinding number (22) in the main text,

w3D =1

24π2

∫d3kεαβγTr

[(Q∂αQ

†)(Q∂βQ†)(Q∂γQ

†)].

(C12)

π

0

FIG. 12: The phase factor κn(k) and its path with increasingk.

In the weak coupling limit, the derivative of phase fac-tor (C11) with respect to the momentum k perpendicularto Fermi surface is

∂kϑn = −sign[vFn∆n

]πδ(k − kF), (C13)

with Fermi velocity of n-th Fermi surface vnF (see Fig. 12).In this case, Eq. (C12) is written as

w3D =∑n

1

8π2

∫d2k′dkεαβTr

[(Q∂k′αQ

†)(Q∂k′βQ†){

− i(∂kϑn)Pn +∑n′

ei(ϑn−ϑn′ )Pn∂kPn′

}](C14)

with the 2D momenta k′α parallel to Fermi surface. Thenon-trivial contribution of Eq. (C14) comes from wherethe phase factors ϑn from all the Q(k) are canceled witheach other. As a result, we obtain

w3D =1

2

∑n

sign[vF∆n,kF

]νjzCh (C15)

with first Chern number

νjzCh = − i

2π

∫kF

d2k′εαβ〈∂k′αun,k|∂k′βun,k〉. (C16)

Appendix D: Duality relation for k · p Hamiltonian

As can be easily checked, the identical vectors J andJ in spin space hold that

DJJiD†J = −3

5Ji −

4

5Ji, (D1)

DJ JiD†J = −4

5Ji +

3

5Ji, (D2)

with the unitary matrix acting on the spin J = 3/2 space

DJ =

0 0 1 00 0 0 −1−1 0 0 00 1 0 0

. (D3)

-2 -3/4 -1/3 1/2 3

FIG. 13: Relations between the systems with parameters v1

and v2/v1 mapped by Eq. (D4) and (D6). The color codeindicates the winding number w3D = −2 (light blue), 4 (ma-genta), −4 (orange), and 2 (light green), the same as Fig. 11

17

[-4,2,-2,1,1][4,2,2,1,1][-2,-2,0,-1,1] [2,-2,0,1,-1]

= [0, 0, 0, 0, 0]

1/2 3-1/30

FIG. 14: Topological phase diagram of the A1u pairing statefor m0α < 0. The other conditions are the same as Fig. 11.

Since the A1u gap function is invariant under DJ , thisunitary transformation maps the system with a set ofparameters v1 and v2 to that with another set,(

v1

v2

)DJ−−→

(v′1v′2

)=

(−[3v1 + 4v2]/5−[4v1 − 3v2]/5

), (D4)

which gives the “duality” of the BdG Hamiltonian. Inaddition, the operator DJ commutes with chiral operator

[Γ, DJ ] = 0. (D5)

Therefore, the systems at two sets of parameters in (D4)give the same winding number w3D.

In the similar manner, the inversion operator for thesuperconducting state maps the parameters as(

v1

v2

)P−→(v′1v′2

)=

(−v1

−v2

)with P = σzτz. (D6)

Note that this mapping does not change the v2/v1. Since

the operator P anti-commutes with chiral operator Γ,

{Γ, P} = 0, (D7)

then the systems at two sets of parameters in (D6) givethe opposite signed winding number w3D.

Finally, in Fig. 13, we summarize these relations inparameter space spanned by v1 and v2/v1 used in themain text. For v2/v1 > −3/4, the unitary transformationDJ maps a v2/v1 to opposite side of v2/v1 = 1/2 denotedby dashed line in Fig. 13. At the same time, it changesthe sign of v1. Furthermore, we can map a system with v1

to another with −v1 by P . Therefore, at the opposite sidewith respect to v2/v1 = 1/2 and with the same sign of thev1 the winding number w3D exhibits the opposite sign. Incontrast, for v2/v1 < −3/4, the unitary transformationDJ connects the parameters with the same sign of v1 (seeFig. 13), which have the same winding number w3D.

Appendix E: Topological phase diagram for m0α < 0

In the main text, we have assumed that m0α > 0.Here we provide the topological phase diagram of A1u

for m0α < 0. When m0α < 0, the normal Hamilto-nian (4) does not exhibit the band inversion at Γ point,so it is topologically trivial. Correspondingly, the A1u

superconducting state is also topologically trivial when|µ| < µc. However, a topological phase transition takesplace at |µ| = µc, and the system shows topological su-perconductivity for |µ| > µc. We summarize the obtainedtopological phase diagram in Fig. 14.

∗ Electronic address: takuto.kawakami@yukawa.kyoto-u.ac.jp

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