Possible three-dimensional nematic odd-parity superconductivity in Sr 2 RuO 4
PPossible three-dimensional nematic odd-parity superconductivity in Sr RuO Wen Huang and Hong Yao
1, 2, 3, ∗ Institute for Advanced Study, Tsinghua University, Beijing, 100084 China State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China Collaborative Innovation Center of Quantum Matter, Beijing 100084, China (Dated: October 16, 2018)The superconducting pairing in Sr RuO is widely considered to be chiral p -wave with (cid:126)d k ∼ ( k x + ik y )ˆ z ,which belongs to the E u representation of the crystalline D h group. However, this superconducting orderappears hard to reconcile with a number of key experiments. In this paper, based on symmetry analysis wediscuss the possibility of odd-parity pairing with inherent three-dimensional (3D) character enforced by theinter-orbital interlayer coupling and the sizable spin-orbit coupling (SOC) in the material. We focus on a yetunexplored E u pairing, which contains finite ( k z ˆ x , k z ˆ y ) -component in the gap function. Under appropriatecircumstances a novel time-reversal invariant nematic pairing can be realized. This nematic superconductingstate could make contact with some puzzling observations on Sr RuO , such as the absence of spontaneousedge current and no evidences of split transitions under uniaxial strains. Introduction .—Superconductivity in Sr RuO was discov-ered [1] in 1994 and was immediately proposed to be of spin-triplet pairing in relation to the possible remnant ferromag-netic correlations in the material [2, 3]. Over the years, mul-tiple measurements show evidences of spin-triplet [4, 5], odd-parity pairing [6], with the additional feature of time-reversalsymmetry breaking (TRSB) [7, 8]. These point to a possiblechiral p -wave pairing [9–16] in the E u representation, repre-sented by the pairing function (cid:126)d k = ( k x ± ik y )ˆ z . Here “ ± ”indicate the time-reversed pair of degenerate chiral states, andthe direction of the (cid:126)d -vector, ˆ z in this case, denotes the struc-ture of Cooper pairing in spin space (see later). If confirmed,Sr RuO will be a solid state analog of the well-known liq-uid He A-phase [17]. This state is topologically nontrivial,wherein the Cooper pairs carry nonvanishing quantized or-bital angular momentum. It supports exotic excitations suchas chiral edge states and Majorana zero modes in supercon-ducting vortex cores. The latter is marked by non-abelianbraiding statistics crucial for topological quantum computa-tion [18, 19].However, the chiral p -wave pairing still currently standsin conflict with a number of experimental observations. Aprominent example is the absence of spontaneous edge current[20–22]. Existing measurements place an upper bound for theedge current over three orders of magnitude smaller than pre-dicted for an isotropic single-band chiral p -wave model [23].Other inconsistencies include but are not limited to: abundantresidual density of states going against the fully-gapped natureof a chiral p -wave [24, 25]; signatures reminiscent of the Paulilimiting behavior [26–29]; the absence of split transitions inthe presence of external perturbations expected to lift the de-generacy of the two E u components of the chiral order pa-rameter, such as an in-plane magnetic field [29] and in-planeuni-axial strains [30, 31], etc.Recent years have seen a broad spectrum of theoretical at-tempts to resolve various aspects of the puzzle [32–68]. How-ever, a consensus is still lacking regarding the exact pair-ing symmetry in Sr RuO . To this end, we take a differ-ent angle and study a possible alternative (cid:126)d -vector in the E u representation on account of the weak inter-orbital interlayertunneling and the sizable spin-orbit coupling (SOC) [69–71]between the Ru 4 d t g -orbitals – which introduce consider-able three-dimensional spin-orbit entanglement as reportedin photo-emission studies [70]. In addition to the pairingin the channel ( k x ˆ z, k y ˆ z ), the (cid:126)d -vector should contain finite ( k z ˆ x, k z ˆ y ) pairing, thereby constituting a full 3D supercon-ducting pairing. As we shall see, the interplay between thesetwo pairing channels brings about an interesting possibilityof a novel time-reversal invariant (TRI) nematic supercon-ducting phase. Such a state is doubly degenerate, possessessymmetry-imposed point-nodal quasiparticle excitations (butcould in principle support accidental nodal lines), and exhibitsa broken rotational symmetry with respect to the underlyingtetragonal crystal symmetry. In addition, in comparison withthe chiral p -wave order, the nematic pairing could better ex-plain the absence of split transitions under external perturba-tions that lift the degeneracy of the two E u components, as weshall explain later.In a similar vein, odd-parity nematic superconductivity hasbeen proposed in the doped topological insulator, Bi Se [72–74], which has strong SOC and whose resultant rotationalsymmetry breaking has been reported in a few measurements[75–79]. The Gingzburg-Landau theory .—The generic two-component odd-parity superconducting pairing function inthe E u representation reads, ˆ∆ k = i ( φ (cid:126)d , k · (cid:126)σ + φ (cid:126)d , k · (cid:126)σ ) σ y . (1)where φ , label the order parameters associated with the twocomponents, (cid:126)d i, k are real vectors denoting the spin structureof the two components of Cooper pairing. The components of (cid:126)d i, k contain appropriate form factors [e.g. (4)] which form atwo-dimensional E u representation of the underlying tetrag-onal crystalline space group D h [80, 81] . Throughout thework we assume only intraband Cooper pairing near the Fermilevel, as is appropriate for a weak-coupling superconductor.In the absence of SOC, the (cid:126)d -vectors can be written in aseparable form, (cid:126)d i, k = (cid:126)df i, k , due to full spin rotational in- a r X i v : . [ c ond - m a t . s up r- c on ] O c t TABLE I. Irreducible odd-parity representations of the D h groupand the corresponding basis functions.irrep. basis function ( (cid:126)d k ) A u k x ˆ x + k y ˆ y ; k z ˆ zA u k y ˆ x − k x ˆ yB u k x ˆ x − k y ˆ yB u k y ˆ x + k x ˆ yE u ( k x , k y )ˆ z ; ( k z ˆ x, k z ˆ y ) variance. Here, the form factors f i, k act as the basis func-tions of the corresponding symmetry group. In the presenceof SOC, Eq. (1) is more appropriately expressed in the pseu-dospin basis, whereby we would have duly accounted for theeffects of SOC. In particular, since the spin and orbital degreesof freedom are now entangled, the (cid:126)d -vector is locked with themomentum of the Bloch electron with pseudo-spin indices.In other words, the elements of the symmetry group operatesimultaneously on the spin and the spatial coordinates.We start our analyses with a phenomenological Gingzburg-Landau free energy. Up to the quartic order: f = r ( T − T c ) (cid:0) | φ | + | φ | (cid:1) + β (cid:0) | φ | + | φ | (cid:1) + β | φ | | φ | + β (cid:48) ( φ ∗ φ + φ φ ∗ ) . (2)Within mean-field, coefficients of the quartic terms determinethe stable superconducting state. In Ref. [82] we derive theexpressions for evaluating these coefficients, from where it isapparent that β and β are positive definite. By contrast, thesign of β (cid:48) depends on the structure of (cid:126)d i, k and can be roughlyapproximated by, β (cid:48) ≈ C (cid:68) ( (cid:126)d , k · (cid:126)d , k ) − | (cid:126)d , k × (cid:126)d , k | (cid:69) FS , (3)where C is a positive constant and (cid:104)· · · (cid:105) FS denotes an integralover the Fermi surface. Similar expression was also obtainedin Ref. [74]. By inspection, if β (cid:48) > , φ and φ preferentiallydevelop a π/ phase difference, i.e. φ = ( φ , φ ) = φ (1 , ± i ) ,thereby breaking time-reversal invariance, as for the chiral p -wave order; whilst if β (cid:48) < , the system favors a TRI orderparameter, φ = φ (1 , ± or φ (1 , . Note that the theoryapplies equally well to single- and multi-band models. Interlayer-coupling-enforced 3D (cid:126)d -vector .— Thus far, theassignment of the possible (cid:126)d -vector in Sr RuO has beenlargely dictated by the considerations of its layered structure.In particular, the quasi-2D character of the electronic struc-ture naturally leads one to conjecture the absence of interlayerCooper pairing that takes the form of, e.g. ∆ k ∼ k z in the p -wave channel. However, no particular symmetry constraintprohibits such a pairing. Indeed, interlayer pairings of oneform or another have been considered in a few microscopicmodels formulated in different contexts [83–85].According to the classification in Table I, when a k z -likepairing does develop, the superconducting state in the E u rep-resentation is more appropriately described by the following FIG. 1. Projection of a representative lowest order vertex into thepseudospin basis. The wavy lines denote the bare or projectedCoulomb interactions. The diagram on the left originates from thebare Coulomb repulsion between the spin-up xz and xy electrons.The first and second line on the right hand side are for spin-orbitentangled three-orbital models without and with direct inter-orbital xy - xz/yz hopping t z , respectively. The summation over the bandindices µ/ν are implicit. Note that only intraband pairing is consid-ered. In addition, due to the peculiar form of the SOC, when t z = 0 the spin-up (down) xy -orbital is projected solely to the pseudospin-down (up) states in the band basis. (cid:126)d -vector, (cid:126)d , k = k x ˆ z + (cid:15)k z ˆ x and (cid:126)d , k = k y ˆ z + (cid:15)k z ˆ y , (4)where (cid:15) is a non-universal real constant determined by therelative strength of the effective interactions responsible forthe respective ( k x , k y )ˆ z and ( k z ˆ x, k z ˆ y ) pairings, respectively.This alternative (cid:126)d -vector texture was alluded in [56].Most previous quasi-two-dimensional spin-orbit coupledmodels of Sr RuO do not support coexisting ( k x , k y )ˆ z and ( k z ˆ x, k z ˆ y ) pairings. This is due to the absence of direct inter-orbital hopping (or hybridization) between the xy and the xz/yz -orbitals in those models. To understand this, we studythe corresponding single-particle Hamiltonian, H = (cid:88) k ,s ψ † k ,s ˆ H s ( k ) ψ k ,s , (5)where the sub-spinor ψ k ,s = ( c xz, k ,s , c yz, k ,s , c xy, k , − s ) T with c a, k ,s annihilating a spin- s electron on the a -orbital ( a = xz, yz, xy ), s = ↑ and ↓ denote up and down spins, and, ˆ H s ( k ) = ξ xz, k λ k − isη iηλ k + isη ξ yz, k − sη − iη − sη ξ xy, k , (6)with ξ a, k given in Ref. [82]. Here λ k is the inter-orbital hy-bridization between the quasi-1D xz - and yz -orbitals; and η is the strength of SOC. The eigenstates of (5) constitute thepseudospin electrons in the band representation. It is conve-nient to define the states associated with ψ k , ↑ ( ↓ ) pseudospin-up (down) particles. In this case, each pseudospin electrondoes not carry weight of more than one spin species of anyorbital. Inversely, the decomposition of any individual spinspecies of any orbital belong with only one pseudospin species FIG. 2. Spin-resolved orbital weights across the γ -band Fermi sur-face at two different values of k z in the three-band tight-bindingmodel with t z = 0 . t . The corresponding tight-binding model isgiven in Ref. 82. Presented data are for the pseudospin-up states asdefined in the text. Only the weights of the yz and xy -orbitals areshown for clarity. In the horizontal axis θ k stands for the angle ofthe Fermi momentum w.r.t. the x -axis. Inset: Cross-section of the γ -band Fermi surface at the two k z ’s shown in (a) and (b). on the bands. Taking into account the on-site Coulomb inter-actions between the t g orbitals, a low-energy effective ac-tion in the Cooper channel can be constructed perturbativelyby projecting the Coulomb interactions and the associatedspin/charge-fluctuation-mediated interactions onto the Fermilevel [34, 47, 59].A close inspection of the projection reveals that the bareCoulomb interactions do not lead to scatterings from equal-pseudospin Cooper pairs to opposite-pseudospin pairs, norvice versa. More specifically, as explained in Ref. 82 andexemplified diagramatically in the first line of Fig 1, interac-tions such as the following are absent at this order: Γ k , q a † ν, k , ¯ ↑ a † ν, − k , ¯ ↑ a µ, − q , ¯ ↑ a µ, q , ¯ ↓ (7)where Γ k , q denotes the projected effective interaction and a † µ, ¯ σ ( a µ, ¯ σ ) creates (annihilates) a µ -band pseudospin- σ elec-tron. Here the bar atop the spin symbol denotes pseudospinbasis. It can be further verified that effective interactions like(7) remain absent even when higher order scattering processesare considered. As a consequence, the x/y and z -componentsof the (cid:126)d -vector of the concerned pseudospin-triplet channelare decoupled. Hence the ( k x , k y )ˆ z and ( k z ˆ x, k z ˆ y ) pairingsin general need not coexist, or shall condense at different tem-peratures if they do at low- T .However, in real material there always exists finite, albeitrelatively weak, xy and xz/yz hybridization once interlayercoupling is considered. As is evident in Fig 2 and as ex-plained in more detail in Ref. 82, even a relatively weakinterlayper hopping t z between the xy and xz/yz orbitalscould yield strong modifications to the details of the electronicstructure. The resultant pseudospins, especially those withmomenta around the Brillouin zone diagonal, possess sizableweights of both spin species of all orbitals. Remarkably, thisis achieved without introducing noticeable corrugation to thethree-dimensional Fermi surface (inset of Fig. 2). We stressthat the significant k z -dependence of the spin-orbit entangle-ment was also observed in spin-resolved photo-emission stud-ies [70]. The corresponding action now permits scattering FIG. 3. The phase diagram as a function of (cid:15) . The chiral and nematicphases are shaded in light and dark grey, respectively. We assumecylindrical Fermi surface with radius 1 and replace k z by sin k z (tak-ing the range of k z to be [ − π, π ] in the integral). In this calculation β (cid:48) changes sign at (cid:15) c ≈ . . processes like (7), as illustrated in the second line of Fig. 1(see also Ref. 82). In this case, the gap functions ( k x , k y )ˆ z and ( k z ˆ x, k z ˆ y ) are inherently coupled and should emerge si-multaneously at a single T c .Note that although the realistic pairing functions are likelymore anisotropic than shown in (4), however, assuming thegeneral relevance of the ( k z ˆ x, k z ˆ y ) pairing, here we are onlyinterested in the nontrivial consequences of the resultant 3Dodd-parity pairing. Odd-parity nematic pairing .—We proceed to discuss thestable superconducting states associated with (4) in a single-band model for illustration. These (cid:126)d -vectors lead to a simpleexpression for (3), β (cid:48) ≈ C (cid:10) k x k y − (cid:15) k z ( k x + k y + (cid:15) k z ) (cid:11) FS . (8)Depending on the value of the anisotropic parameter (cid:15) , β (cid:48) cantake either signs. In Fig. 3, we sketched the sign of β (cid:48) as afunction of (cid:15) , assuming a cylindrical Fermi surface with radius k F (cid:107) = 1 and taking k z → sin k z . Note that the critical value (cid:15) c at which β (cid:48) = 0 will in general be different in a more real-istic model. As discussed, (cid:15) = (cid:15) c separates the TRSB and TRIphases. In the latter case, either a diagonal nematic state with φ = φ (1 , ± or a horizontal one with φ = φ (1 , / (0 , could be more stable, depending on the details of the realisticband and gap structures [82].For small | (cid:15) | , β (cid:48) > and the system favors a TRSB chiral-like pairing ( (cid:15) = 0 returns the ordinary chiral p -wave) witha nodeless isotropic superconducting gap. This state is non-unitary, and it generates finite edge current. We do not furtherexplore this possibility, but emphasize its intrinsic 3D natureif indeed realized in Sr RuO .Below we focus on the TRI states, e.g. the horizon-tal nematic state φ = φ (1 , . Its gap function | ∆ k | =∆ (cid:112) (cid:15) k z + k x reflects a breaking of the lattice C symme-try down to C , with gap minima at k x = 0 and maxima at k x = ± k F at each k z . Hence it may be termed a “TRI ne-matic pairing”. This state possesses nodal points at TRI k z ,i.e. k z = 0 and k z = π upon replacing k z by appropriatelattice harmonics in the gap function. Note that in other ne-matic states, the nodal-point directions are properly rotated.The representative gap structure at various k z is shown in Fig.4 for the two different types of nematic states.One appealing feature of the nematic phase is the absenceof spontaneous current at the edges due to TRI. In addition,at (100) or (010) surfaces, dispersionless edge modes with en- k z = k z = π k z = π ( a ) k x k y k z = k z = π k z = π ( b ) k x k y FIG. 4. Representative gap structure on the approximately cylindricalFermi surface (black) for the (a) horizonal nematic and (b) diagonalnematic states with (cid:15) = 0 . . As indicated, the three curves shown ineach plot are for three values of k z . In these calculations, we havereplaced k z in the gap function by sin k z . ergy ∝ (cid:15)k z emerge for each k z (Fig 5a) for the case of diag-onal nematic pairing. Note that the horizontal nematic statewith φ ∼ (1 , supports surface states on the (100)-surface,but not on the (010)-surface. These surface states could beassociated with the conductance peaks in the tunneling spec-tra [86, 87]. Taking the (100) surface in the diagonal nematicstate as an example, there should be two horizontal Majoranaarcs within k y ∈ [ − k F (cid:107) , k F (cid:107) ] / √ at k z = 0 and π (Fig.5b). By contrast, in the non-unitary chiral state, two singly-degenerate chiral edge modes appear (Fig.5c) and finite edgecurrent follows naturally. The zero modes in this case formtwo vertically connected and elongated Majorana loops, as de-picted in Fig. 5c. The peculiar surface spectra of the nematicand chiral states can be distinguished in photoemission andquasiparticle interference studies. Furthermore, the nematicstates form the bases of a U (1) × Z field theory. The Z symmetry permits the formation of domains characterized bythe two degenerate pairing functions, much like what has beenproposed for chiral p -wave. This may be consistent with thesignatures of domain formation observed in some experiments[11, 88–91]. Discussions and summary .—Guided by symmetry con-siderations, we argued that the odd-parity E u pairing inSr RuO acquires an inherent 3D form ( k x ˆ z + (cid:15)k z ˆ x, k y ˆ z + (cid:15)k z ˆ y ) in the presence of finite SOC and inter-orbital interlayercoupling. This leads to an appealing possibility of a novel TRInematic odd-parity pairing, thereby providing an alternativeperspective to understand the perplexing superconductivity inthis material.Besides resolving the notorious edge current problem, thenematic pairing may also explain some other outstanding puz-zles. For example, compared with the chiral pairing, the ne-matic state could stand a better chance to explain the absenceof split transitions under perturbations that break the degener-acy of the two E u components [29–31]. With applied uniaxialstrain along any generic direction, the splitting is expected forthe scenario with the chiral ground state where a sequence oftwo transitions spontaneously break distinct symmetries: the U (1) symmetry at the upper transition, and the time-reversal FIG. 5. (a) low-energy spectra at fixed k z = 0 for the diagonal ne-matic odd-parity pairing at (cid:15) = 0 . in a geometry with two (100)surfaces. In the lattice BdG calculations, the k i ’s in the pairing func-tions are replaced by the simple lowest order harmonic sin k i . Notethat the flat edge modes acquire some dispersion due to finite sizeeffects. (b) sketched arcs spanned by the surface zero modes in thenematic phase. (c) sketched Majorana loops formed by the Majoranazero modes in the chiral phase. symmetry at the lower one. For the case of diagonal nematicpairing, a genuine second transition occurs only when a uni-axial strain is applied exactly parallel to the (100)- or (010)-direction. In this scenario, the lower transition breaks a reflec-tion symmetry about the vertical plane parallel to the (100)-or (010)-direction. However, for any amount of misalignmentin applied strain, as could be the case in a real experiment, nosharp lower transition should occur [92], although a smearedcrossover may appear to a degree that depends on the level ofmisalignment. Note also that, in the case of the horizontal ne-matic state, applying external strain along the (100)-directioncannot give rise to split transitions.Nevertheless, this nematic pairing needs to withstand thetest of various other existing measurements, which remainsto be carefully examined. Note that the absence of C anisotropy in thermodynamic measurements under in-planemagnetic fields [93, 94] could be consistent with nematicpairing because the externally applied field may drive a ro-tation of the nematic orientation. By contrast, in Bi Se su-perconductors the nematic orientation may be pinned by aweak structural distortion as reported in Ref. 95. The ne-maticity in Sr RuO could be revealed in measurements likethe angle-dependent in-plane Josephson tunneling [96] andthe visualization of single vortex structure in scanning tun-neling microscopy, as has been demonstrated for Cu x Bi Se [97, 98]. Regarding the contradiction between the point-nodalgap structure and the experimental indications of line-nodalpairing [24, 25], we note that the realistic gap function couldbe more anisotropic. In particular, it is in principle possible tohave, e.g. (cid:126)d k = g k ( k x ˆ z + (cid:15)k z ˆ x ) and similarly for (cid:126)d k , wherethe form factor g k carries horizontal or vertical line nodes.Such line nodes are not imposed by symmetry but could verylikely arise in reality, given the highly anisotropic electronicstructure in Sr RuO . Acknowledgements . We are grateful to Fan Yang and Li-DaZhang for stimulating discussions, and to Daniel Agterbergand Shingo Yonezawa for helpful communications. This workis supported in part by the MOST of China under Grant Nos.2016YFA0301001 and 2018YFA0305604 and by the NSFCunder Grant No. 11474175 (W.H. and H.Y.). W.H. also ac-knowledges the support by the C. N. 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SUPPLEMENTAL MATERIALS FOR “POSSIBLE THREE-DIMENSIONAL NEMATIC ODD-PARITYSUPERCONDUCTIVITY IN SR RUO ”I. EFFECTIVE TWO-DIMENSIONAL THREE-BAND MODELS We first present the effective multi-orbital model, constructed from the three Ru t g orbitals, which is commonly employedin previous studies of Sr RuO , and is a purely two-dimensional model without interlayer coupling. By symmetry direct inter-orbital xy - xz/yz hopping (hybridization) is absent, but the sizable spin-orbit coupling can be accounted for from the outset.Repeating the main text for completeness, the single particle Hamiltonian reads, H = (cid:88) k ,s ψ † k ,s ˆ H s ( k ) ψ k ,s , (S1)where the sub-spinor ψ k ,s = ( c xz, k ,s , c yz, k ,s , c xy, k , − s ) T with c a, k ,s annihilating a spin- s electron on the a -orbital ( a = xz, yz, xy ), s = ↑ and ↓ denote up and down spins, and, ˆ H s ( k ) = ξ xz, k λ k − isη iηλ k + isη ξ yz, k − sη − iη − sη ξ xy, k , (S2)with ξ xz, k = − t cos k x − t cos k y − µ , ξ yz, k = − t cos k x − t cos k y − µ , λ k = − t (cid:48)(cid:48) sin k x sin k y , ξ xy, k = − t (cid:48) (cos k x +cos k y ) − t (cid:48)(cid:48)(cid:48) cos k x cos k y − µ . Here λ k is the inter-orbital hybridization between the two quasi-1D xz - and yz -orbitals,and η is the strength of spin-orbit coupling (SOC). In the main text we use the set of parameters (˜ t, t (cid:48) , t (cid:48)(cid:48) , t (cid:48)(cid:48)(cid:48) , η, µ , µ ) =(0 . , . , . , . , . , , . t , which well reproduces the band structure and the Fermi surface geometry of Sr RuO . Note thatbecause SOC mixes different spins on the xy - and the other two orbitals, the spins are not good quantum numbers. However,thanks to the inversion symmetry the Kramers degeneracy on each band is preserved, it is therefore convenient to adopt apseudospin notation and denote the degenerate electrons pseudospin-up and down. In this case, the two pseudospin species arefully characterized by the respective sub-spinors ψ k ,s , e.g. the pseudospin-up electron is linearly composed only of xz ↑ , xz ↑ and xy ↓ electrons but carries not weight of xz ↓ , xz ↓ and xy ↑ . Similar statements can be made in reverse. For example, the xz ↑ , yz ↑ and xy ↓ electrons can only be decomposed into pseudo-spin up electrons in the band basis.To understand the projection of the pairing vertex onto the Fermi level, we again look at the example given in the main text,i.e. with the bare inter-orbital Coulomb interaction between xz ↑ and xy ↑ electrons: V n xz, ↑ n xy, ↑ → V c † xz, k , ↑ c † xy, − k , ↑ c xy, − q , ↑ c xz, q , ↑ = (cid:88) µ,ν Γ ν ¯ ↑ ,ν ¯ ↓ ; µ ¯ ↓ ,µ ¯ ↑ xz ↑ ,xy ↑ ; xy ↑ ,xz ↑ ( k , q ) a † ν, k , ¯ ↑ a † ν, − k , ¯ ↓ a µ, − q , ¯ ↓ a µ, q , ¯ ↑ , (S3)with the effective pairing vertex given by, Γ ν ¯ ↑ ,ν ¯ ↓ ; µ ¯ ↓ ,µ ¯ ↑ xz ↑ ,xy ↑ ; xy ↑ ,xz ↑ ( k , q ) = V (cid:104) ξ ν ¯ ↑ xz ↑ ( k ) ξ ν ¯ ↓ xy ↑ ( − k ) (cid:105) ∗ ξ µ ¯ ↓ xy ↑ ( − p ) ξ µ ¯ ↑ xz ↑ ( p ) . (S4)Here the coefficients ξ µ ¯ σ (cid:48) aσ ( k ) are the corresponding elements of the unitary transformation relating the orbital and band repre-sentations. Note in the above expressions we have used the fact that ξ µ ( ν )¯ ↓ xz ↑ = ξ µ ( ν )¯ ↑ xy ↑ ≡ when the xy - xz/yz hybridization isabsent. It then follows that effective interactions such as (7) in the main text cannot appear in the low-energy theory. This can beshown to be true for all other bare Coulomb interactions and higher-order scatterings. As a side remark, only intraband pairingis considered under the weak-coupling assumption in the present study. II. ROLE OF xy - xz/yz HOPPING (HYBRIDIZATION)
A more complete description of the electronic structure necessarily involves nonvanishing xy - xz/yz hopping (between samespin species). To make explicit its influence to the electronic structure, we consider the lowest order contribution which involvesinterlayer coupling. We expand the full single-particle Hamiltonian (S2) in the matrix form as follows, H = (cid:88) k Ψ † k ˆ H ( k ) ψ k , (S5)where the full spinor Ψ k = ( ψ T k , ↑ , ψ T k , ↓ ) = ( c xz k ↑ , c yz k ↑ , c xy k ↓ , c xz k ↓ , c yz k ↓ , c xy k ↑ ) T and, ˆ H ( k ) = ξ xz, k λ k − iη iη T z ( k ) λ k + iη ξ yz, k − η T z ( k ) − iη − η ξ xy, k T z ( k ) T z ( k ) 00 0 T ∗ z ( k ) ξ xz, k λ k + iη iη T ∗ z ( k ) λ k − iη ξ yz, k ηT ∗ z ( k ) T ∗ z ( k ) 0 − iη η ξ xy, k . (S6)Here T iz ( k ) represents the xy - xz/yz hopping terms. To leading order, they may be approximated by T z ( k ) =8 t z cos k x sin k y sin k z and T z ( k ) = 8 t z sin k x cos k y sin k z where t z is the corresponding interlayer hopping amplitude be-tween the 1D and 2D orbitals. Notably, since the interlayer hopping preserves the inversion symmetry, the Kramers degeneracyremains. Hence the notion of pseudospins remains valid. When these interlayer hoppings are absent, the Hamiltonian returns to(S2) and is block-diagonalized.This leads to a finite corrugation of the 3D Fermi surface along the k z -axis. As can be inferred from the inset of Fig 2,the corrugation is minuscule for a relatively weak t z , as is consistent with the experiments. In spite of this, the influence onthe pseudospin structure is noticeable. This is evident from the sizable mixture of spin and orbital species belonging to thetwo spinor subspace, which is not accessible in models with vanishing xy - xz/yz hopping. Due to the level-crossing of theunhybridized orbital dispersions around the BZ diagonals, the mixing is strongest in these regions. The k z -dependence of thepseudospin structure is also evident in Fig 2. Note that since T iz ( k ) vanishes at k z = 0 , the mixture of the two sub-spinors ψ k ,s is suppressed at this k z .The projection of the interactions follows similar procedure as in (S3). However, due to the mixing of all orbital and spinspecies at k z (cid:54) = 0 , the example given in the previous section now permits effective vertices Γ with generically finite ξ µ ( ν )¯ ↓ xz ↑ and ξ µ ( ν )¯ ↑ xy ↑ . As a consequence, effective interactions such as (7) in the main text are generally allowed in the effective action.As a side remark, besides making the E u -pairing inherently three-dimensional, the same mechanism also makes the A u statethree-dimensional, namely, (cid:126)d k = k x ˆ x + k y ˆ y + (cid:15)k z ˆ z according to the classification in Table I, which is similar to the B-phase of He. In addition to being fully-gapped, this state is singly-degenerate, hence no domain formation is expected.
III. GINZBURG-LANDAU β -COEFFICIENTS For a generic two-component odd-parity pairing function ˆ∆ k = i ( φ (cid:126)d , k · (cid:126)σ + φ (cid:126)d , k · (cid:126)σ ) σ y , the free energy functional isobtained via a standard perturbative expansion in powers of the order parameter fields φ and φ . The Gorkov Greens functionin Nambu spinor space reads, ˆ G − ( iw n , k ) = (cid:18) ( iw n − ξ k ) σ ˆ∆ k ˆ∆ † k ( iw n + ξ k ) σ (cid:19) , (S7)where σ is the rank-2 identity matrix, w n = (2 n +1) πT is the Matsubara frequency with T near T c , around which the expansionis valid. Defining g + ( iw n , k ) = iw n − ξ k , g − ( iw n , k ) = iw n + ξ − k , ˆ G − = (cid:18) g − σ g − − σ (cid:19) and ˆ∆ = (cid:18) k ˆ∆ † k (cid:19) , the part ofthe expansion essential to our discussion reads, − T · tr ln ˆ G − = − T · tr ln ( ˆ G − + ˆ∆) = − T · tr ln ˆ G − (1 + ˆ G ˆ∆)= const. − T · tr ln (1 + ˆ G ˆ∆)= const. + T l ∞ (cid:88) l =1 tr [ ˆ G ˆ∆] l . (S8)Note only even-order contributions survive in the last line. The l = 1 term yields, T tr (cid:104) ˆ G ˆ∆ ˆ G ˆ∆ (cid:105) = T L d (cid:88) w n , k g + g − tr (cid:104) ˆ∆ k ˆ∆ † k (cid:105) = T L d (cid:88) w n , k g + g − tr (cid:104) ( φ (cid:126)d k · (cid:126)σ + φ (cid:126)d k · (cid:126)σ )( φ ∗ (cid:126)d k · (cid:126)σ + φ ∗ (cid:126)d k · (cid:126)σ ) (cid:105) = T L d (cid:88) w n , k g + g − tr (cid:104) | (cid:126)d k | | φ | σ + | (cid:126)d k | | φ | σ + ( (cid:126)d k · (cid:126)σ )( (cid:126)d k · (cid:126)σ ) φ ∗ φ + ( (cid:126)d k · (cid:126)σ )( (cid:126)d k · (cid:126)σ ) φ ∗ φ (cid:105) . (S9)Noting that ( (cid:126)d k · (cid:126)σ )( (cid:126)d k · (cid:126)σ ) = ( (cid:126)d k · (cid:126)d k ) σ + i(cid:126)σ · ( (cid:126)d k × (cid:126)d k ) , and that the last two terms in the last line can be shown tovanish upon k -summation. Therefore φ and φ do not couple at this quadratic order. Proceeding to the quartic order, l = 2 ,using (S9), T tr (cid:104) ˆ G ˆ∆ ˆ G ˆ∆ (cid:105) = T L d (cid:88) w n , k ( g + g − ) tr (cid:104) ˆ∆ k ˆ∆ † k (cid:105) = T L d (cid:88) w n , k ( g + g − ) tr (cid:110) | (cid:126)d k | | φ | σ + | (cid:126)d k | | φ | σ + ( (cid:126)d k · (cid:126)d k ) σ ( φ ∗ φ + φ ∗ φ ) + i(cid:126)σ · ( (cid:126)d k × (cid:126)d k )( φ ∗ φ − φ ∗ φ ) (cid:111) . (S10)Expanding the curly bracket we obtain the quartic terms. By comparing with (2) in the main text, it is readily seen that, β = T L d (cid:88) w n , k ( g + g − ) | (cid:126)d k | tr ( σ ) = T L d (cid:88) w n , k w n + ξ k ) | (cid:126)d , k | ∝ (cid:68) | (cid:126)d , k | (cid:69) FS , (S11) β = T L d (cid:88) w n , k w n + ξ k ) (cid:104) | (cid:126)d , k | | (cid:126)d , k | + 2 | (cid:126)d , k × (cid:126)d , k | (cid:105) × (S12) β (cid:48) = T L d (cid:88) w n , k w n + ξ k ) (cid:104) ( (cid:126)d , k · (cid:126)d , k ) − | (cid:126)d , k × (cid:126)d , k | (cid:105) . (S13)The (cid:104) ... (cid:105) FS in the second line of (S11) denotes a Fermi surface integral. Similar, although not exactly equivalent (due to adifferent form of free energy functional we adopt), expressions were obtained in Ref. [74]. Note that after completing theMatsubara frequency summation, the k -summation can be approximated as an average across the Fermi surface, as examplifiedin (S11). In addition, terms like | φ | ( φ ∗ φ + φ ∗ φ ) vanish, as their coefficients do not survive the k -summation. IV. HORIZONTAL VS DIAGONAL NEMATIC PAIRING
We consider the scenario where the nematic pairing is favored, i.e. β (cid:48) < . Rewriting the free energy functional (2) in themain text as follows, f = r ( T − T c )( | φ | + | φ | ) + β ( | φ | + | φ | ) + ( β − β + 4 β (cid:48) ) | φ | | φ | + β (cid:48) ( φ ∗ φ − φ ∗ φ ) , (S14)it is clear that, up to the quartic order, if β − β + 4 β (cid:48) = 0 , the states with ( φ , φ ) = φ (cos θ, sin θ ) are degenerate forall nematic θ , exhibiting a continuous degeneracy. This continuous degeneracy is present if the system respects the in-planerotational symmetry. However, real materials possess only discrete lattice rotational symmetry. As a result, β − β + 4 β (cid:48) (cid:54) = 0 and the continuous degeneracy is generically lifted. Through straightforward saddle point approximation, it can be shown thatthe diagonal nematic pairing with φ (1 , ± / √ is preferred when β − β + 4 β (cid:48) < , otherwise the horizontal nematic phasewith φ (1 , / (0 ,1)