Inter-orbital topological superconductivity in spin-orbit coupled superconductors with inversion symmetry breaking
Yuri Fukaya, Shun Tamura, Keiji Yada, Yukio Tanaka, Paola Gentile, Mario Cuoco
aa r X i v : . [ c ond - m a t . s up r- c on ] M a y Inter-orbital topological superconductivity in spin-orbit coupled superconductors withinversion symmetry breaking
Yuri Fukaya, Shun Tamura, Keiji Yada, Yukio Tanaka, Paola Gentile, and Mario Cuoco Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan SPIN-CNR, I-84084 Fisciano (Salerno), Italy and Dipartimento di Fisica ”E. R. Caianiello”,Universitá di Salerno, I-84084 Fisciano (Salerno), Italy
We study the superconducting state of multi-orbital spin-orbit coupled systems in the presence ofan orbitally driven inversion asymmetry assuming that the inter-orbital attraction is the dominantpairing channel. Although the inversion symmetry is absent, we show that superconducting statesthat avoid mixing of spin-triplet and spin-singlet configurations are allowed, and remarkably, spin-triplet states that are topologically nontrivial can be stabilized in a large portion of the phasediagram. The orbital-dependent spin-triplet pairing generally leads to topological superconductivitywith point nodes that are protected by a nonvanishing winding number. We demonstrate that thedisclosed topological phase can exhibit Lifshitz-type transitions upon different driving mechanismsand interactions, e.g., by tuning the strength of the atomic spin-orbit and inversion asymmetrycouplings or by varying the doping and the amplitude of order parameter. Such distinctive signaturesof the nodal phase manifest through an extraordinary reconstruction of the low-energy excitationspectra both in the bulk and at the edge of the superconductor.
I. INTRODUCTION
Spin-triplet pairing is at the core of intense investiga-tion especially because of its foundational aspect in un-conventional superconductivity and owing to its tightconnection with the occurrence of topological phases withzero-energy surface Andreev bound states marked byMajorana edge modes . Some of the fundamentalessences of topological spin-triplet superconductivity arebasically captured by the Kitaev model and its gener-alized versions where non-Abelian states of matter andtheir employment for topological quantum computationcan be demonstrated . Another remarkable elementof odd-parity superconductivity is given by the poten-tial of having active spin degrees of freedom makingsuch states of matter also appealing for superconduct-ing spintronics applications based on spin control andcoherent spin manipulation of Cooper pairs . Theinterplay of magnetism and spin-triplet superconductiv-ity can manifest within different unconventional physicalscenarios, such as the case of the emergent spin-orbitalinteraction between the superconducting order parame-ter and interface magnetization , the breakdown ofthe bulk-boundary correspondence , and the anomalousmagnetic and spin-charge current effects occurringin the proximity between chiral or helical p -wave andspin-singlet superconductors. Achieving spin-triplet ma-terials platforms, thus, sets the stage for the developmentof emergent technologies both in nondissipative spintron-ics and in the expanding area of quantum devices.Although embracing strong promises, spin-triplet su-perconductivity is quite rare in nature and the mecha-nisms for electron pairs gluing are not completely set-tled. The search for spin-triplet superconductivity hasbeen performed along different routes. For instance, sci-entific exploration has been focused on the regions of thematerials phase diagram that are in proximity to ferro- magnetic quantum phase transitions , as in the caseof heavy fermions superconductivity, i.e., UGe , URhGe,and UIr , or in materials on the verge of a magnetic in-stability, e.g., ruthenates .Another remarkable route to achieve spin-triplet pair-ing relies on the presence of a source of inversion symme-try breaking, both at the surface/interface and in thebulk, or alternatively, in connection with noncollinearmagnetic ordering . Paradigmatic examples alongthese directions are provided by quasi-one-dimensionalheterostructures whose interplay of inversion and time-reversal symmetry breaking or noncollinear magnetismhave been shown to convert spin-singlet pairs into spin-triplet ones and in turn to topological phases .Similar mechanisms and physical scenarios are also en-countered at the interface between spin-singlet super-conductors and inhomogeneous ferromagnets with evenand odd-in time spin-triplet pairing that are generallygenerated . Semimetals have also been indicated as fun-damental building blocks to generate spin-triplet pairingas theoretically proposed and demonstrated in topologi-cal insulators interfaced with conventional superconduc-tors or by doping Dirac/Weyl phases , e.g., in the caseof Cu-doped Bi Se in anti-perovskites materials ,as well as Cd As .Generally, there are two fundamental interactions totake into account in inversion asymmetric microscopicenvironments: i) the Rashba spin-orbit coupling dueto inversion symmetry breaking at the surface or inter-face in heterostructures, and ii) the Dresselhaus couplingarising from the inversion asymmetry in the bulk of thehost material . For the present analysis, it is worthnoting that typically in multi-orbital materials, it is thecombination of the atomic spin-orbit interaction with theinversion symmetry-breaking sources that effectively gen-erates both Rashba and Dresselhaus emergent interac-tions within the electronic manifold close to the Fermilevel. Another general observation is that the lack of in-version symmetry is expected to lead to a parity mixingof spin-singlet and spin-triplet configurations withan ensuing series of unexpected features ranging fromanomalous magneto-electric effects to unconventionalsurface states , topological phases , and non-trivialspatial textures of the spin-triplet pairs . Such symme-try conditions in intrinsic materials are, however, funda-mentally linked to the momentum dependent structureof the superconducting order parameter. In contrast,when considering multi-orbital systems, more channelsare possible with emergent unconventional paths for elec-tron pairing that are expected to be strongly tied to theorbital character of the electron-electron attraction andof the electronic states close to the Fermi level.Orbital degrees of freedom are key players in quantummaterials when considering the degeneracy of d -bands ofthe transition elements not being completely removed bythe crystal distortions or due to the intrinsic spin-orbitalentanglement triggered by the atomic spin-orbit cou-pling. In this context, a competition of different andcomplex types of order is ubiquitous in realistic mate-rials, such as transition metal oxides, mainly owing tothe frustrated exchange arising from the active orbitaldegrees of freedom. Such scenarios are commonly en-countered in materials where the atomic physics plays asignificant role in setting the character of the electronicstructure close to the Fermi level. As the d -orbitals havean anisotropic spatial distribution, the nature of the elec-tronic states is also strongly dependent on the system’sdimensionality. Indeed, two-dimensional (2D) confinedelectron liquids originating at the interface or surface ofmaterials generally manifest a rich variety of spin-orbitalphenomena . Along this line, understanding how elec-tron pairing is settled in quantum systems exhibiting astrong interplay between orbital degrees of freedom andinversion symmetry breaking represents a fundamentalproblem in unconventional superconductivity, and it canbe of great relevance for a large class of materials.In this study, we investigate the nature of the super-conducting phase in spin-orbit coupled systems in theabsence of inversion symmetry assuming that the inter-orbital attractive channel is dominant and sets the elec-trons pairing. We demonstrate that the underlying in-version symmetry breaking leads to exotic spin-tripletsuperconductivity. Isotropic spin-triplet pairing config-urations, without any mixing with spin-singlet, gener-ally occur among the symmetry allowed solutions and areshown to be the ground-state in a large part of the pa-rameters space. We then realize an isotropic spin-tripletsuperconductor whose orbital character can make it topo-logically non trivial. Remarkably, the topological phaseexhibits an unconventional nodal structure with uniquetunable features. An exotic fingerprint of the topologi-cal phases is that the number and k -position of nodes canbe controlled by doping, orbital polarization, through thecompetition between spin-orbit coupling and lattice dis-tortions, and temperature (or equivalently, the amplitudeof the order parameter). The paper is organized as follows. In Sec. II, we in-troduce the model Hamiltonian and present the classi-fication of the inter-orbital pairing configurations withrespect to the point-group and time-reversal symmetries.Section III is devoted to an analysis of the stability of thevarious orbital entangled superconducting states and theenergetics of the isotropic superconducting states. Sec-tion IV focuses on the electronic spectra of the energet-ically most favorable phases and the ensuing topologicalconfigurations both in the bulk and at the boundary. Fi-nally, in Sec. V, we provide a discussion of the resultsand few concluding remarks. II. MODEL AND SYMMETRYCLASSIFICATION OF SUPERCONDUCTINGPHASES WITH INTER-ORBITAL PAIRING
One of the most common crystal structures of transi-tion metal oxides is the perovskite structure, with tran-sition metal (TM) elements surrounded by oxygen (O)in an octahedral environment. For cubic symmetry, ow-ing to the crystal field potential generated by the oxy-gen around the TM, the fivefold orbital degeneracy isremoved and d orbitals split into two sectors: t g , i.e., yz , zx , and xy , and e g , i.e., x − y and z − r . In thepresent study, the analysis is focused on two-dimensional(2D) electronic systems with broken out-of-plane inver-sion symmetry and having only the t g orbitals (Fig. 1)close to the Fermi level to set the low energy excitations.For highly symmetric TM-O bonds, the three t g bandsare directional and basically decoupled, e.g., an electronin the d xy orbital can only hop along the y or x directionthrough the intermediate p x or p y orbitals. Similarly, the d yz and d zx bands are quasi-one-dimensional when con-sidering a 2D TM-O bonding network. Furthermore, theatomic spin-orbit interaction (SO) mixes the t g orbitalsthus competing with the quenching of the orbital angu-lar momentum due to the crystal potential. Concerningthe inversion asymmetry, we consider microscopic cou-plings that arise from the out-of-plane oxygen displace-ments around the TM. Indeed, by breaking the reflectionsymmetry with respect to the plane placed in betweenthe TM-O bond , a mixing of orbitals that are even andodd under such a transformation is generated. Such crys-tal distortions are much more relevant and pronounced in2D electron gas forming at the interface of insulating po-lar and nonpolar oxide materials or on their surface andthey result in the activation of an effective hybridization,which is odd in space, of d xy and d yz or d zx orbitalsalong the y or x directions, respectively. Although thepolar environment tends to amplify the out-of-plane oxy-gen displacements with respect to the position of the TMion, such types of distortions can also occur at the inter-face of nonpolar oxides and in superlattices .Thus, the model Hamiltonian, including the t g hop-ping connectivity, the atomic spin-orbit coupling, and theinversion symmetry breaking term, reads as H = X k ˆ C ( k ) † H ( k ) ˆ C ( k ) , (1) H ( k ) = H ( k ) + H SO ( k ) + H is ( k ) , (2)where ˆ C † ( k ) = h c † yz ↑ k , c † zx ↑ k , c † xy ↑ k , c † yz ↓ k , c † zx ↓ k , c † xy ↓ k i isa vector whose components are associated with the elec-tron creation operators for a given spin σ [ σ = ( ↑ , ↓ ) ],orbital α [ α = ( xy, yz, zx ) ], and momentum k in theBrillouin zone. In Fig. 1(a), we report a schematic illus- FIG. 1. (a) d yz , d zx , and d xy -orbitals with L = 2 orbital angu-lar momentum. (b) Schematic image of the orbital dependenthopping amplitudes for ε yz , ε xy , and the orbital connectivityassociated with the inversion asymmetry term ∆ is . Here, wedo not explicitly indicate the intermediate p -orbitals of theoxygen ions surrounding the transition metal element that en-ter the effective d − d hopping processes. ε zx is obtained from ε yz by rotating π/ around z -axis. ∆ is corresponds to theodd-in-space hopping amplitude from d xy to d zx along the y -direction. Similarly, the odd-in-space hopping amplitude from d xy to d yz along the x -direction is obtained by π/ rotationaround the z -axis. (c) Sketch of the orbital mixing throughthe spin-orbit coupling term in the Hamiltonian. σ denotesthe spin state, and ¯ σ is the opposite spin of σ . ∆ t gives thelevel splitting between d xy -orbital and d yz /d zx -orbitals. (d)Schematic illustration of inter-orbital interaction. tration of the local orbital basis for the t g states. H ( k ) , H SO ( k ) , and H is ( k ) indicate the kinetic term, the spin-orbit interaction, and the inversion symmetry breakingterm, respectively. In the spin-orbital basis, H ( k ) is given by H ( k ) = − µ h ˆ l ⊗ ˆ σ i + ˆ ε k ⊗ ˆ σ , (3) ˆ ε k = ε yz ε zx
00 0 ε xy ,ε yz = 2 t (1 − cos k y ) + 2 t (1 − cos k x ) ,ε zx = 2 t (1 − cos k x ) + 2 t (1 − cos k y ) ,ε xy = 4 t − t cos k x − t cos k y + ∆ t , where ˆ l and ˆ σ are the unit matrices in orbital and spinspace, respectively. Here, µ is the chemical potential, and t , t , and t are the orbital dependent hopping ampli-tudes as schematically shown in Fig. 1(b). ∆ t denotesthe crystal field potential owing to the symmetry lower-ing from cubic to tetragonal symmetry. The symmetryreduction yields a level splitting between d xy orbital and d yz /d zx orbitals. H SO ( k ) denotes the atomic L · S spin-orbit coupling, H SO ( k ) = λ SO h ˆ l x ⊗ ˆ σ x + ˆ l y ⊗ ˆ σ y + ˆ l z ⊗ ˆ σ z i , (4)with ˆ σ i ( i = x, y, z ) being the Pauli matrix in spin space.In order to write down the L · S interaction, it is conve-nient to introduce the matrices ˆ l x , ˆ l y and ˆ l z , which arethe projections of the L = 2 angular momentum operatoronto the t g subspace, i.e., ˆ l x = i − i , (5) ˆ l y = − i i , (6) ˆ l z = i − i , (7)assuming { ( d yz , d zx , d xy ) } as orbital basis. Finally, asmentioned above, the breaking of the mirror plane in be-tween the TM-O bond, due to the oxygen displacements,leads to an inversion symmetry breaking term H is ( k ) ofthe type H is ( k ) = ∆ is h ˆ l y ⊗ ˆ σ sin k x − ˆ l x ⊗ ˆ σ sin k y i . (8)This contribution gives an inter-orbital process, due tothe broken inversion symmetry, that mixes d xy and d yz or d zx along x or y spatial directions [Fig. 1(b)]. H is re-sembles a Rashba-type Hamiltonian that, however, cou-ples the momentum to the orbital angular momentumrather than the spin. Its origin is due to distortions orother sources of inversion symmetry breaking that leadto local asymmetries deforming the orbital lobes and inturn antisymmetric hopping terms within the orbitals inthe t g sector. In this respect, it is worth pointing outthat it is the combination of the local spin-orbit cou-pling and the antisymmetric inversion symmetry inter-action that leads to a nontrivial momentum dependentspin-orbital splitting. While the original Rashba effect for the single-band system describes a linear spin split-ting and is typically very small, the multi-band char-acter of the model Hamiltonian yields a more complexspin-orbit coupled structure with significant splitting .Indeed, near the Γ point of the Brillouin zone, one canhave a linear spin splitting with respect to the momentumfor the lowest energy bands, but a cubiclike splitting inmomentum for the intermediate ones with enhancedanomalies when the filling is close to the transition fromtwo to four Fermi surfaces. The Rashba-like effects dueto the combined atomic spin-orbit coupling and the or-bitally driven inversion-symmetry term can be influencedby the application of an external electric field (e.g., viagating) in a dual way. On one hand, the gating directlymodifies the filling concentration and, on the other, itcan affect the deformation of the orbital lobes by chang-ing the amplitude of the polar distortion .In this paper, we set t = t ≡ t as a unit of energyfor convenience and clarity of presentation. The analysisis performed for a representative set of hopping param-eters, i.e., t /t = 0 . and ∆ t /t = − . . The primaryreason for the choice of the electronic parameters is thatwe aim to model superconductivity in transition-metalbased layered materials with low electron concentrationin the t g sector at the Fermi level both in the presence ofatomic spin-orbit and inversion symmetry breaking cou-plings. In this framework, the set of selected parametersis representative of a general physical regime where thehierarchy of the energy scales is such that ∆ t > ∆ is > λ SO and ∆ t ∼ t . The choice of this regime is also motivatedby the fact that this relation can be generally encoun-tered in d (or d ) layered oxides or superlattices in thepresence of tetragonal distortions with flat octahedra andinterface driven inversion-symmetry breaking potential.For instance, in the case of the two-dimensional elec-tron gas (2DEG) forming at the interface of two bandinsulators [e.g., the n -type 2DEG in LaAlO /SrTiO (LAO/STO)] or the 2DEG at the surface of a band in-sulator [e.g., in SrTiO (STO)], the energy scales forthe electronic parameters, as given by ab initio orspectroscopic studies , are such that the bare ∆ t ∼ - meV, ∆ is ∼ meV, and λ SO ∼ meV, while theeffective main hopping amplitudes (i.e., t ) can be in the - meV range. Similar electronic energies can bealso encountered in d layered oxides. Slight variationsof these parameters are expected; however, they do notalter the qualitative aspects of the achieved results. Wealso point out that our analysis is not intended for a spe-cific material case and that variations in the amplitudeof the electronic parameters that keep the indicated hi-erarchy do not alter the qualitative outcome and do notlead to significant changes in the results.The electronic structure of the examined model sys- −0.5−0.2500.250.5 ΓΜ X E / t Μ ππ − π Γ Μ XY k x k y ππ − π Γ Μ XY k x k y ππ − π Γ Μ XY k x k y (a) (b)(c) (d)(b)(c)(d) FIG. 2. (a) Band structure close to the Fermi energy in thenormal state at λ SO /t = 0 . and ∆ is /t = 0 . . (b)-(d) Fermisurfaces at (b) µ/t = − . , (c) µ/t = 0 . , and (d) µ/t = 0 . . tem can be accessed by direct diagonalization of the ma-trix Hamiltonian. Representative dispersions for λ SO /t =0 . and ∆ is /t = 0 . are shown in Fig. 2(a). We ob-serve six non degenerate bands due to the presence ofboth H SO ( k ) and H is ( k ) . Once the dispersions are de-termined, one can immediately notice that the numberof Fermi surfaces and the structure can be varied by tun-ing the chemical potential µ . Indeed, for µ/t = − . , µ/t = 0 . , and µ/t = 0 . one can single out all the mainpossible cases with two, four, and six Fermi sheets, asgiven in Figs. 2(b), (c), and (d), respectively. For theexplored regimes of low doping, all the Fermi surfacesare made of electron-like pockets centered around originof the Brillouin zone ( Γ ). The dispersion of the lowestoccupied band has weak anisotropy as it has a dominant d xy character (Fig. 2(a)); moreover, moving to higherelectron concentrations, the outer Fermi sheets exhibit ahighly anisotropic profile that becomes more pronouncedwhen the chemical potential crosses the bands mainlyarising from the d yz and d zx -orbitals.After having considered the normal state properties,we concentrate on the possible superconducting statesthat can be realized, their energetics and their topolog-ical behavior. The analysis is based on the assumptionthat the inter-orbital local attractive interaction is theonly relevant pairing channel that contributes to the for-mation of Cooper pairs. Then, the intra-orbital pair-ing coupling is negligible. Such a hypothesis can bephysically applicable in multi-orbital systems because theintra-band Coulomb interaction is typically larger thanthe inter-band one. Indeed, in the t g restricted sectorthe Coulomb interaction matrix elements of low-energylattice Hamiltonian can be evaluated by employing theHubbard-Kanamori parametrization in terms of U , U ′ and J H , after symmetrizing the Slater-integrals withinthe t g shell assuming a cubic splitting of the t g and e g orbitals. U corresponds to the intra-orbital Coulombrepulsion, whereas U ′ (with U ′ = U − J H in a cu-bic symmetry) is the inter-orbital interaction which isreduced by Hund exchange, J H . Hence, one has thatthe inter-orbital Coulomb repulsion is generally alwayssmaller than the intra-orbital one. Estimates for transi-tion metal oxide materials in d , d or d configurations,being relevant for the t g shell and thus for our work,indicate that U ∼ . eV and U ′ ∼ . eV . Thus, it isplausible to expect that the Coulomb repulsion tends tofurther suppress the electron pairing that occurs withinthe same band. In addition, in the case of having theelectron-phonon coupling as a source of electrons attrac-tion, it is shown that the effective inter- and intra-orbitalattractive interaction can be of the same magnitude (seeAppendix for more details).In this framework, we point out that topological super-conductivity is proposed to occur, owing to inter-orbitalpairing, in Cu-doped Bi Se for an inversion symmetriccrystal structure . Here, although similar inter-orbitalpairing conditions are considered, we pursue the super-conductivity in low-dimensional configurations, e.g., atthe interface of oxides, with the important constraint ofhaving a broken inversion symmetry. Concerning the or-bital structure of the pairing interaction, owing to thetetragonal crystalline symmetry, the coupling betweenthe d xy -orbital and d yz / d zx -orbital is equivalent, andthus one can assume that only two independent chan-nels of attraction are allowed, as shown in Fig. 1(d).Indeed, V xy denotes the interaction between the d xy and d yz / d zx -orbitals, while V z refers to the coupling betweenthe d yz and d zx -orbitals. Then, the pairing interaction isgiven by H I = V xy X i [ n xy,i n yz,i + n xy,i n zx,i ]+ V z X i n yz,i n zx,i , (9) n α,i = c † α ↑ i c α ↑ i + c † α ↓ i c α ↓ i , (10)where i denotes the lattice site. A. Irreducible representation and symmetryclassification
In this subsection, we classify the inter-orbital super-conducting states according to the point group symme-try. The system upon examination has a tetragonal sym-metry associated with the point group C v , marked byfour-fold rotational symmetry C and mirror symmetries M yz and M zx . Based on the rotational and reflectionsymmetry transformations, all the possible inter-orbitalisotropic pairings can be classified into five irreducible representations of the C v point group as summarizedin Table I. For our purposes, only solutions that do not TABLE I. Irreducible representation of the inter-orbitalisotropic superconducting states for the tetragonal group C v .In the columns, we report the sign of the order parameterupon a four-fold rotational symmetry transformation, C , andthe reflection mirror symmetry M yz , as well as the explicitspin and orbital structure of the gap function. In the E repre-sentation, + and − of the subscript mean the doubly degen-erate mirror-even ( + ) and mirror-odd ( − ) solutions, respec-tively. C v C M yz orbital basis function ( d xy , d yz ) d ( xy,yz ) y A + + ( d xy , d zx ) d ( xy,zx ) x = − d ( xy,yz ) y ( d yz , d zx ) d ( yz,zx ) z A + − ( d xy , d yz ) d ( xy,yz ) x ( d xy , d zx ) d ( xy,zx ) y = d ( xy,yz ) x B − + ( d xy , d yz ) d ( xy,yz ) y ( d xy , d zx ) d ( xy,zx ) x = d ( xy,yz ) y ( d xy , d yz ) d ( xy,yz ) x B − − ( d xy , d zx ) d ( xy,zx ) y = − d ( xy,yz ) x ( d yz , d zx ) ψ ( yz,zx ) E ± i ± ( d xy , d yz ) ψ ( xy,yz ) , d ( xy,yz ) z ( d xy , d zx ) ψ ( xy,zx )+ = ∓ id ( xy,yz ) z + d ( xy,zx ) z − = ∓ iψ ( xy,yz ) − ( d yz , d zx ) d ( yz,zx ) x , d ( yz,zx ) y break the time-reversal symmetry are considered and arereported in Table I. Then, the superconducting orderparameter associated to bands α and β can be classi-fied as an isotropic ( s -wave) spin-triplet/orbital-singlet d ( α,β ) -vector and s -wave spin-singlet/orbital-triplet withamplitude ψ ( α,β ) or as a mixing of both configurations.With these assumptions, one can generally describe theisotropic order parameter with spin-singlet and tripletcomponents as ˆ∆ α,β = i ˆ σ y h ψ ( α,β ) + ˆ σ · d ( α,β ) i , (11)with α and β standing for the orbital index, and havingfor each channel three possible orbital flavors. Further-more, owing to the selected tetragonal crystal symmetry,one can achieve three different types of inter-orbital pair-ings. The spin-singlet configurations are orbital tripletsand can be described by a symmetric superposition of op-posite spin states in different orbitals. On the other hand,spin-triplet components can be expressed by means of thefollowing d -vectors: d ( xy,yz ) = (cid:16) d ( xy,yz ) x , d ( xy,yz ) y , d ( xy,yz ) z (cid:17) , d ( xy,zx ) = (cid:16) d ( xy,zx ) x , d ( xy,zx ) y , d ( xy,zx ) z (cid:17) , d ( yz,zx ) = (cid:16) d ( yz,zx ) x , d ( yz,zx ) y , d ( yz,zx ) z (cid:17) , with d ( α,β ) indicating the spin-triplet configuration builtwith α and β -orbitals. In general, independently of theorbital mixing, spin-triplet pairing can be expressed in amatrix form as ∆ T = ∆ ↑↑ ∆ ↑↓ ∆ ↓↑ ∆ ↓↓ ! = − d x + id y d z d z d x + id y ! , (12)where the d -vector components are related to the pair-ing order parameter with zero spin projection along thecorresponding symmetry axis. The three components d x = ( − ∆ ↑↑ + ∆ ↓↓ ) , d y = i (∆ ↑↑ + ∆ ↓↓ ) and d z = ∆ ↑↓ are expressed in terms of the equal spin ∆ ↑↑ and ∆ ↓↓ , andthe anti-aligned spin ∆ ↑↓ gap functions. As the compo-nents of the d -vector are associated with the zero spinprojection of spin-triplet configuration, if the d -vectorpoints along a given direction, the parallel spin config-urations lie in the plane perpendicular to the d -vectororientation. In the presence of time-reversal symmetry,the superconducting order parameter should satisfy thefollowing relations: ∆ ↓↓ α,β = h ∆ ↑↑ α,β i ∗ , (13) ∆ ↑↓ α,β = − h ∆ ↓↑ α,β i ∗ , (14)with the appropriate choice of the U(1) gauge. In addi-tion, the pairing order parameter has four-fold rotationalsymmetry and mirror reflection symmetry with respectto the yz and zx planes as dictated by the point group C v . Thus, it has to be transformed according to thefollowing relations: C ˆ∆ C t = e i nπ ˆ∆ ,M yz ˆ∆ M tyz = ± ˆ∆ , where n equals to for A representation, for B repre-sentation, , and for E representation. Such propertiesare very important to distinguish the symmetry of the so-lutions obtained by the Bogoliubov-de Gennes equation.The energy gap functions are then explicitly constructedby taking into account the corresponding irreducible rep-resentations. For the one-dimensional representations,the A state is given by d ( xy,zx ) x = − d ( xy,yz ) y , ∆ ↑↑ xy,yz = ∆ ↓↓ xy,yz = id ( xy,yz ) y , ∆ ↑↑ xy,zx = − ∆ ↓↓ xy,zx = − d ( xy,zx ) x , ∆ ↑↓ yz,zx = ∆ ↓↑ yz,zx = d ( yz,zx ) z , while for the A representation, d ( xy,zx ) y = d ( xy,yz ) x , ∆ ↑↑ xy,yz = − ∆ ↓↓ xy,yz = − d ( xy,yz ) x , ∆ ↑↑ xy,zx = ∆ ↓↓ xy,zx = id ( xy,zx ) y , the B representation, d ( xy,zx ) x = d ( xy,yz ) y , ∆ ↑↑ xy,yz = ∆ ↓↓ xy,yz = id ( xy,yz ) y , ∆ ↑↑ xy,zx = − ∆ ↓↓ xy,zx = − d ( xy,zx ) x , and the B representation, d ( xy,zx ) y = − d ( xy,yz ) x , ∆ ↑↑ xy,yz = − ∆ ↓↓ xy,yz = − d ( xy,yz ) x , ∆ ↑↑ xy,zx = ∆ ↓↓ xy,zx = id ( xy,zx ) y , ∆ ↑↓ yz,zx = − ∆ ↓↑ yz,zx = ψ ( yz,zx ) . Finally, for the E representation, there are doubly degen-erate mirror-even (+) and mirror-odd ( − ) solutions: ψ ( xy,zx )+ = ∓ id ( xy,yz ) z + ,d ( xy,zx ) z − = ∓ iψ ( xy,yz ) − , ∆ ↑↓ xy,yz = α − ψ ( xy,yz ) − + α + d ( xy,yz ) z + , ∆ ↓↑ xy,yz = − α − ψ ( xy,yz ) − + α + d ( xy,yz ) z + , ∆ ↑↓ xy,zx = α + ψ ( xy,zx )+ + α − d ( xy,zx ) z − , ∆ ↓↑ xy,zx = − α + ψ ( xy,zx )+ + α − d ( xy,zx ) z − , ∆ ↑↑ yz,zx = − α − d ( yz,zx ) x − + iα + d ( yz,zx ) y + , ∆ ↓↓ yz,zx = α − d ( yz,zx ) x − + iα + d ( yz,zx ) y + , where α + and α − denote arbitrary constants for the lin-ear superposition. As a consequence of the symmetryconstraint and of the inter-orbital structure of the or-der parameter, different types of isotropic spin-tripletand singlet-triplet mixed configurations can be obtained.Equal spin-triplet and opposite spin-triplet pairings aremixed in the A representation. On the other hand, inthe B representation, equal spin-triplet and spin-singletpairings are mixed. For the A and B representations,only equal spin-triplet pairings are allowed, and all typesof pairings can be realized in the E representation. Itis worth noting that A , B , and E representations havepairings between all the orbitals in the yz - zx and xy - yz/zx channels, while A and B can make electron pair-ings only in the xy - yz/zx channel, that is, by mixingthe d xy and d yz / d zx -orbitals as shown in Table I. Thissymmetry constraint is important when searching for theground-state configuration. III. ENERGY GAP EQUATION AND PHASEDIAGRAM
In order to investigate which of the possible symmetry-allowed solutions is more stable energetically, we solve theEliashberg equation within the mean field approximationby taking into account the multi-orbital effects near thetransition temperature. The linearized Eliashberg equa-tion within the weak coupling approximation is given by Λ∆ στα,γ = − k B TN V α,γ X k ′ ,iε m F ασ,γτ ( k ′ , iε m ) , (15) V xy,yz = V xy,zx = V yz,xy = V zx,xy ≡ V xy ,V yz,zx = V zx,yz ≡ V z ,F ασ,γτ ( k ′ , iε m ) (16) = X β,δ X σ ′ ,τ ′ ∆ σ ′ τ ′ β,δ G σσ ′ α,β ( k ′ , iε m ) G ττ ′ γ,δ ( − k ′ , − iε m ) , where Λ is the eigenvalue of the linearized Eliashbergequation. Here, σ , τ , σ ′ , and τ ′ denote the spinstates and α , β , γ , and δ stand for the orbital in-dices. F ασ,γτ ( k ′ , iε m ) is the anomalous Green’s func-tion. As we assume an isotropic Cooper pairing, whichis k -independent, the summation over momentum andMatsubara frequency in Eq. (16) gets simplified. Fi-nally, the problem is reduced to the diagonalization ofthe × matrix. We then study the relative stabil-ity of the irreducible representations as listed in TableI. An analysis of the energetically most favorable super-conducting states is performed as a function of V z /V xy ,assuming that V xy /t = − . and for a given temperature T /t = 5 . × − . When we keep the ratio V z /V xy , theeigenvalue Λ is proportional to V xy within the mean-fieldapproximation. The choice of the representative coupling V xy /t = − . is guided by the fact that one aims to ac-cess a physical regime for the superconducting phase thatin principle can be compared to realistic superconduct-ing materials in the weak coupling limit. For instance, ifone chooses t ∼ − meV, which is common in ox-ides, and considering that the superconducting transitiontemperature T c is obtained when the magnitude of thegreatest eigenvalue gets close to 1, then one would find T c to be of the order of − mK, which is reasonablefor the 2DEG superconductivity at the oxide interface.Figure 3 shows the superconducting phase diagramfor representative amplitudes of the spin-orbit coupling, λ SO /t = 0 . , and inversion asymmetry interaction, ∆ is /t = 0 . , while varying both the chemical potentialand the ratio of the pairing couplings V z /V xy . Owing tothe inequivalent mixing of the orbitals in the paired con-figurations, it is plausible to expect a significant compe-tition between the various symmetry allowed states andthat such an interplay is sensitive not only to the pairingorbital anisotropy, but also to the structure and the num-ber of Fermi surfaces. A direct observation is that for V z larger than V xy , the A phase is stabilized with respectto the B phase because it contains a d ( yz,zx ) z channel ofa spin-triplet pairing in the yz - zx sector that is absentin B phase. However, such a simple deduction does notdirectly explain why the A phase wins the competitionwith other superconducting phases, e.g., the B and Ephases, which also can gain condensation energy by pair-ing electrons in the yz - zx sector. As a different type µ / t V z / V xy B B A A A B FIG. 3. Phase diagram as a function of V z /V xy at λ SO /t =0 . , ∆ is /t = 0 . , T /t = 5 . × − , and V xy /t = − . .The brown solid line is the border between A and B states.The black solid line indicates the value of the chemical po-tential for which the number of Fermi surfaces changes. Theblack dotted lines correspond to the values of the chemicalpotentials used in Fig. 2 for the normal state Fermi surfaces. of d -vector orientation enters into the A and E config-urations, yet in the B state, the yz - zx channel has aspin-singlet pairing, one can deduce that the interplaybetween the spin of the Copper pairs and that of thesingle-electron states close to the Fermi level is relevantto single out the most favorable superconducting phase.The boundary between the A and B phases exhibitsa sudden variation when one tunes the chemical potentialacross the value for which the number of Fermi surfaceschanges. Such an abrupt transition is, however, plau-sible when passing through a Lifshitz point in the elec-tronic structure of the normal state because other pairingchannels get activated at the Fermi level. The relationbetween the modification of the superconducting stateand electronic topological or Lifshitz transition that theFermi surface can undergo is a subject of general interest.Indeed, there are many theoretical studies and experi-mental signatures pointing to a subtle interplay of Lif-shitz transitions and superconductivity in cuprates ,heavy-fermion superconductors and more recently iniron-based superconductors . In those cases, majorchanges of the superconducting state seem to occur whengoing through a Lifshitz transition because Fermi pocketscan appear or disappear at the Fermi level and in turnlead to different physical effects.Here, along this line of investigation, the role of theelectron filling is also quite important and sets the com-petition between the energetically most stable phases.Indeed, one can notice that the A ( B ) phase is sta-bilized for higher (lower) V z /V xy and lower (higher) µ .Furthermore, we find that, in the case of two Fermi sur-faces, the A state is further stabilized by decreasing thechemical potential and moving to a regime of extremelylow concentration. On the other hand, a transition tothe B phase is achieved by electron doping. In the dop-ing regime of four bands at the Fermi level, the A - B boundary evolves approximately as a linear function of V z /V xy . This implies that the A configuration tendsto be less stable and a higher ratio V z /V xy is needed toachieve such a configuration at a given chemical poten-tial. Finally, approaching the doping regime of six Fermisurfaces, the A - B boundary becomes independent ofthe amplitude of µ . It is remarkable that the doping cansubstantially alter the competition between the A and B phases, thus manifesting the intricate consequencesof the spin-orbital character of the electronic structureclose to the Fermi level.To explicitly and quantitatively demonstrate the en-ergy competition among all the symmetry allowedphases, one can follow the behavior of the eigenvaluesof the linearized Eliashberg equations as a function ofthe ratio V z /V xy (Fig. 4). Figures. 4(a)-(c) show the V z / V xy T h e e i g e nv a l u e A representationB representation (a) (b) (c) T h e e i g e nv a l u e B representationA representationE representation T h e e i g e nv a l u e V z / V xy V z / V xy FIG. 4. Evolution of the eigenvalue of the Eliashberg matrixequation as a function of V z /V xy at (a) µ/t = − . , (b) µ/t = 0 . , and (c) µ/t = 0 . at λ SO /t = 0 . , ∆ is /t = 0 . , T /t = 5 . × − , and V xy /t = − . . (a) A representation isdominant. (b)and(c) B is dominant for small amplitude ofthe ratio V z /V xy . eigenvalues of the Eliashberg matrix equation for all theirreducible representations as a function of V z /V xy whenthe number of Fermi surfaces is (a) two ( µ/t = − . ),(b) four ( µ/t = 0 . ), and (c) six ( µ/t = 0 . ) as indicatedby the dotted lines in Fig. 3. With the increase in V z , themagnitude of the eigenvalues of the irreducible represen-tations including the yz - zx channel, i.e., A , B , and Erepresentations, increases in all the cases with two, four,and six Fermi surfaces. On the other hand, the eigen-values of the A and B representations are independentof V z , as V z is irrelevant for this pairing channel. Whenthe number of Fermi surfaces is two, the A representa-tion is the most dominant pairing for all V z . Althoughthe magnitude of the eigenvalues for the B and E rep-resentations also increases with V z , these solutions neverbecome dominant as compared with the A state. Whenthe number of Fermi surfaces is four or six, the eigenvalue of the B phase is larger than that of the A representa-tion for lower V z .Finally, we have investigated the phase diagram byscanning a larger range of temperatures for few repre-sentative cases of pairing interaction and filling concen-tration (see Appendix). The results are not significantlychanged except in a region of extremely high tempera-ture, corresponding to an unphysically large amplitudeof the pairing interaction. There, although B keeps be-ing the most stable state, the largest eigenvalues indicatea competition between the B and B rather than the B and A configurations. IV. TOPOLOGICAL PROPERTIES ANDENERGY EXCITATION SPECTRUM IN THEBULK AND AT THE EDGE
In the previous section, we confirmed that both the A and B pairings can be energetically stabilized in alarge region of the parameter space. Thus, it is relevantto further consider the nature of the electronic structureof these superconducting phases in order to provide keyelements and indications that can be employed for thedetection of the most favorable inter-orbital supercon-ductivity. The analysis is based on the solution of theBogoliubov-de Gennes (BdG) equation for the evalua-tion of the low-energy spectral excitations both in thebulk and at the edge of the superconductor for both the A and B phases. The matrix Hamiltonian in momen-tum space is given by H BdG ( k ) = H ( k ) ˆ∆ˆ∆ † − H ∗ ( − k ) ! . (17)with H ( k ) being the normal state Hamiltonian. A. Bulk energy spectrum and topologicalsuperconductivity
In order to determine the excitation spectrum, we solvethe BdG equations for both the A and B configurations.For convenience, we introduce the gap amplitude | ∆ | ,and we set the components of the d -vectors to be d ( xy,yz ) y = − d ( xy,zx ) x = d ( yz,zx ) z = | ∆ | , (18)for A and d ( xy,yz ) y = d ( xy,zx ) x = | ∆ | , (19)for B state. Here, the parameter | ∆ | /t = 1 . × − isset as a scale of energy.We start focusing on the doping regime of four bandsat the Fermi level. In this case, the A state has a fullygapped electronic structure for all the bands at the Fermilevel as demonstrated by the inspection of the in-plane −1 −1 −1 −1 k x k y Γ π /2 −π /2 −π /2 π /2 Fermi surface00 π −π θ π −π θ g a p / | ∆ | g a p / | ∆ | π −π θ g a p / | ∆ | π −π θ g a p / | ∆ | (a)(b) (c)(d) (e) θ (b)(c)(d)(e) FIG. 5. (a) Fermi surfaces at µ/t = 0 . in the normal state.[(b)-(e)] A quasi-particle energy gap along the Fermi surfaceas a function of the polar angle θ as shown in (a) for λ SO /t =0 . , ∆ is /t = 0 . , and | ∆ | /t = 1 . × − corresponding tothe Fermi surfaces in (a). angular dependence of the gap magnitude [Figs. 5(b)-(e)]. In particular, we notice that the gap amplitude isnot isotropic and orbital dependent when moving fromthe outer to the inner Fermi surface [Figs. 5(b)-(e)]. Thenodal state (Fig. 6), on the other hand, exhibits a moreregular behavior of the gap amplitude which is basicallyorbital independent and point nodes occurring only alongthe diagonal of the Brillouin zone on the various Fermisurfaces.It is interesting to further investigate the nature of thenodal B phase by determining whether the existence ofthe nodes is related to a non-vanishing topological invari-ant. As the model Hamiltonian owes particle-hole andtime-reversal symmetry, one can define a chiral operator ˆΓ as a product of the particle-hole ˆ C and time-reversal ˆΘ operators. As the chiral symmetry operator anticom-mutes with H BdG ( k ) , by employing a unitary transforma-tion rotating the basis in the eigenbasis of ˆΓ , the Hamilto-nian can be put in an off-diagonal form with antidiagonalblocks. Hence, the determinant of each block can be putin a complex polar form and, as long as the eigenvaluesare non-zero, it can be used to obtain a winding numberby evaluating its trajectory in the complex plane. On a −10 −5 −10 −5 −10 −5 −10 −5 k x k y Fermi surface1D winding number = +1 π /2 −π /2 −π /2 π /2 1D winding number = −100 0 −π π g a p / | ∆ | θ −π π g a p / | ∆ | θ −π π g a p / | ∆ | θ −π π g a p / | ∆ | θ (a)(b) (c)(d) (e) θ Γ (b)(c)(d)(e) FIG. 6. (a) Fermi surfaces and position of the nodes at µ/t =0 . . We indicate the winding numbers defined at each node.(b)-(e) indicate the quasi-particle energy spectra for the B state with λ SO /t = 0.10, ∆ is /t = 0 . , and | ∆ | /t = 1 . × − at the corresponding Fermi surfaces shown in (a). general ground, we point out that the number of singu-larities in the phase of the determinant is a topologicalinvariant because it cannot change without the am-plitude going to zero, thus implying a gap closing anda topological phase transition. For this symmetry class,then, one can associate and determine the winding num-ber around each node by following, for instance, the ap-proach already applied successfully in Refs. [102–104].The chiral, particle-hole, and time-reversal operators areexpressed as ˆΓ = − i ˆ C ˆΘ , (20) ˆ C = I × ˆ I × ! = ˆ l ⊗ ˆ σ x ⊗ ˆ τ , (21) ˆΘ = ˆ l ⊗ i ˆ σ y ⊗ ˆ τ . (22)Here, ˆ I × and ˆ τ denote the × unit matrix and theidentity matrix in the particle-hole space, respectively.As we consider time-reversal symmetric pairings, the chi-ral operator anticommutes with the Hamiltonian: { H BdG ( k ) , ˆΓ } = 0 . (23)One can then introduce a unitary matrix ˆ U Γ that diago-0 π / 2 π / 2 k x k y k x k y C (a) (b)+1 : −1 : FIG. 7. (a) Fermi surfaces at λ SO /t = 0 . , ∆ is /t = 0 . , µ/t = 0 . , and point nodes position (winding number) at ∆ /t = 1 . × − . (b) Zoomed view of the plot in (a) and acontour of the integral C . nalizes the chiral operator ˆΓ : ˆ U † Γ ˆΓ ˆ U Γ = ˆ I × − ˆ I × ! , (24) ˆ U † Γ = ˆ U Γ = 1 √ ˆ I × ˆ l ⊗ ˆ σ y ˆ l ⊗ ˆ σ y − ˆ I × ! . (25)In this basis the BdG Hamiltonian is block antidiagonal-ized by ˆ U Γ , ˆ U † Γ H BdG ( k ) ˆ U Γ = q ( k )ˆ q † ( k ) 0 ! , (26) ˆ q = H ( k ) h ˆ l ⊗ ˆ σ y i − ˆ∆ . (27)Then, the determinant of the ˆ q ( k ) matrix block can beput in a complex polar form, and as long as the eigen-values are nonzero, it can be used to obtain the windingnumber W by evaluating its trajectory in the complexplane as W = 12 π I C dθ ( k ) , (28) θ ( k ) ≡ arg[det ˆ q ( k )] .C in Eq. (28) is a closed line contour that encloses agiven node as schematically shown in Fig. 7(b). Fromthe explicit calculation, we find that the amplitude of W is ± [see Figs. 6(a) and 7(a)]. If the nodes have anonzero winding number, edge states appear due to thebulk-edge correspondence. It is known that the follow-ing index theorem is satisfied: for any one-dimensionalcut in the Brillouin zone that is indicated by a givenmomentum k k that is parallel to the edge, one has that w ( k k ) = n + − n − , with n + and n − being the number ofthe eigenstates associated to the eigenvalues +1 and − of the chiral operator ˆΓ , respectively. The number of edgestates is equal to | w ( k k ) | when considering a boundaryconfiguration with a conserved k k . We can easily showthat W which is given in Eq. (28) and w ( k k ) are deeply −3 −2 −1 −0.3−0.2−0.100.10.20.30.410 −3 −2 −1 −0.3−0.2−0.100.10.20.30.410 −3 −2 −1 −0.3−0.2−0.100.10.20.30.4 10 −3 −2 −1 −3 −2 −1 µ / t | ∆ | / t FS = 6FS = 4FS = 2FS = 2FS = 4FS = 6n = 6 (a)(b)(c)
FS = Γ −M) (d)(e) n = 6 n = 4FS = 2FS = 4FS = 6 FS = 6FS = 4FS = 2FS = 6FS = 4FS = 2n = 2n = 6n = 4 µ / t µ / t n = 4 n = 2n = 2 n = 0n = 4 n = 2n = 2 n = 0n = 4n = 4 n = 6 n = 4n = 4 n = 2n = 2n = 2 n = 0n = 0n = 2 n = 2n = 0n = 0n = 0n = 0n = 2n = 2 n = 2n = 2n = 4n = 6 n = 4 n = 2n = 0n = 0 λ SO / t ∆ i s / t (a)(b)(c)(d) (e) (f) | ∆ | / t | ∆ | / t | ∆ | / t | ∆ | / t FIG. 8. Phase diagram of the Lifshitz transitions for thenodal B phase at (a) λ SO /t = 0 . and ∆ is /t = 0 . ,(b) λ SO /t = 0 . and ∆ is /t = 0 . , (c) λ SO /t = 0 . and ∆ is /t = 0 . , (d) λ SO /t = 0 . and ∆ is /t = 0 . , and (c) λ SO /t = 0 . and ∆ is /t = 0 . . (f) Schematic plot of the cor-respondence between the spin-orbit and the inversion asym-metry couplings and the panels (a)-(e). Black and red labelsdenote the number of Fermi surfaces in the normal state andthe point nodes in the superconducting phase that are locatedalong the diagonal of the Brillouin zone from Γ to M, respec-tively. The black dotted line and the orange solid line indi-cate the two-to-four Fermi surface separation and the four-to-six Fermi surface boundary in the normal state. Circles andsquares set the transition lines for the nodal superconductorbetween configurations having different number of nodes inthe excitation spectrum. linked: w ( k ) − w ( k ) = − W sgn( k − k ) where W is thetotal winding number around the nodes between k k = k and k k = k . Thus, nonzero W on the nodes meansnonzero w ( k k ) and the existence of the zero energy edgestate with appropriate choice of the crystal plane. Suchrelation sets the main physical connection between thewinding number and the properties of the topological su-perconductor.We generally find that two to six point nodes can oc-1cur along the Γ -M direction, and their number is relatedto that of the Fermi surfaces. Interestingly, the positionof the point nodes is not fixed and pinned to the linesof the Fermi surface in the normal state. In general,their position along the diagonal of the Brillouin zonedepends on the amplitude | ∆ | and indirectly on the val-ues of the spin-orbit and inversion asymmetry couplings.Thus, two adjacent point nodes with opposite windingnumbers can, in principle, be moved until they mergeand then disappear by opening a gap in the excitationspectrum. This behavior is generally demonstrated inFig. 8. A phase diagram can be determined in terms ofthe amplitude | ∆ | and the chemical potential µ . Thenodal superconductor can undergo different types of Lif-shitz transitions, and in general, those occurring in thenormal state are not linked to the nodal merging in thesuperconducting phase. Indeed, one of the characteristicfeatures of the nodal superconductor is that, by changingthe filling, through µ , one can drive a transition from twoto four and six point nodes independently of the numberof bands crossing the Fermi level in the normal state. Itis rather the strength of | ∆ | that plays an importantrole in tuning the nodal superconductor. An increase in | ∆ | tends to reduce the number of nodes until a fullygapped phase appears. As the critical lines are sensitiveto the spin-orbit λ SO and inversion asymmetry ∆ is cou-plings, one can get line crossings that allow for multiplemerging of nodes such that the superconductor can un-dergo a direct transition from six to two at µ/t ∼ . [Figs. 8(e) and (d)] or from four to zero point nodes,as for instance nearby the crossing between the blue andorange lines at µ/t ∼ . in the Fig. 8. As the positionsof the point nodes are fixed, each Fermi surface in thelimit of small | ∆ | and its distance in the Brillouin zoneincreases with level splitting by ∆ is and λ SO , a larger | ∆ | is required to annihilate the point nodes when both ∆ is and λ SO grow in amplitude as demonstrated by theshift of the green and blue critical lines in Figs. 8(a)-8(c)for different values of ∆ is , and Figs. 8(d), 8(b), and 8(e)in terms of λ SO . When considering these results in thecontext of two-dimensional superconductors that emergeat the surface or interface of band insulators we observethat the achieved topological transitions can be drivenby gate voltage and temperature, as µ and ∆ is are tun-able by electric fields, and the amplitude of | ∆ | can becontrolled by the temperature and the electric field aswell. B. Local density of states at the edge of thesuperconductor
Having established that the nodes in the B config-uration are protected by a nonvanishing winding num-ber, one can expect that flat zero-energy surface Andreevbound states (SABS) occur at the boundary of the su-perconductor.In this section, we investigate the SABS and the local density of states (LDOS) for two different terminations ofthe two-dimensional superconductor, i.e., the (100) and(110) oriented edges. We start by discussing the LDOSfor the (100) and (110) edges at representative values of λ SO /t = 0 . , ∆ is /t = 0 . , and | ∆ | /t = 1 . × − , andby varying the chemical potential in order to comparethe cases with a different number of point nodes in thebulk energy spectrum at µ/t = − . , µ/t = 0 . , and µ/t = 0 . as shown in Figs. 9(a)-9(c), respectively.As expected, the momentum-resolved LDOS indicatesthat zero-energy SABS can be observed but only for spe-cific orientations of the edge. Indeed, as reported in Figs.9(d)-9(i), one has zero-energy SABS (ZESABS) for the(110) boundary while they are absent for the (100) edge.The reason for having inequivalent SABS edge modes isdirectly related to the presence of a nontrivial windingnumber that is protecting the point nodes. For the (110)edge, isolated point nodes exist in the surface Brillouinzone, and they have winding numbers with opposite sign.Thus, the ZESABS, which connects the nodes with a pos-itive and negative winding number, emerge in the gap.On the other hand, when considering the (100) orientedtermination, the winding numbers for positive k x andnegative k x are completely opposite in sign, and theycancel each other when projected on the (100) surfaceBrillouin zone. Thus, flat zero-energy states cannot oc-cur for the (100) edge. Nevertheless, helical edge modesare observed inside the energy gap as demonstrated inFig. 9(d). This is because the Majorana edge modeswith positive and negative chirality can couple, get split,and acquire a dispersion. The differences in the edgeABS also manifest in the momentum integrated LDOS.For the (110) edge, owing to the presence of the ZESABS,the LDOS normalized by its normal state value at E = 0 shows pronounced zero-energy peaks (see dash-dottedline in Figs. 9(j), (k), and (l)). On the other hand, forthe (100) boundary, they lead to a broad peak or exhibitmany narrow spectral structures reflecting the complexdispersion of the edge states.Finally, we discuss the | ∆ | dependence of LDOS atzero energy, i.e., E = 0 . For the (110) edge, the zero-energy peak mainly originates from the zero-energy flatband. The height of the zero-energy peak can then becharacterized by (i) the strength of the localization of theedge state and (ii) the total length of the ZESABS withinthe surface Brillouin zone. The strength of the localiza-tion is defined by the inverse of the localization length /ξ and /ξ ∝ | ∆ | . In other words, the peak height gener-ally increases with | ∆ | . On the other hand, as shown inFig. 8, the extension in the momentum space of the zero-energy flat states becomes shorter with increasing | ∆ | .For simplicity, one can focus on the two Fermi surfaceconfiguration. In this case, the total length of the zero-energy flat band is roughly estimated as δk (1 −| ∆ | / | ∆ c | ) for | ∆ | < | ∆ c | and zero for | ∆ | > | ∆ c | , where δk is theFermi surface splitting along the Γ -M direction and | ∆ c | is a critical value above which the point-nodes disappear.Then, the height of the zero-energy peak is proportional2 FIG. 9. Momentum-resolved and angular averaged LDOS for B representation. The Fermi surfaces and the position of thepoint nodes are shown for (a) µ/t = − . , (b) µ/t = 0 . , and (c) µ/t = 0 . . The momentum ( k k ) resolved LDOS at the(100) oriented surface for (d) µ/t = − . , (e) µ/t = 0 . , and (f) µ/t = 0 . . The momentum ( k k ) resolved LDOS at the (110)oriented surface for (g) µ/t = − . , (h) µ/t = 0 . , and (i) µ/t = 0 . . LDOS normalized by its normal state value at E = 0 [(DOS N ( E = 0) ] at the (100) and (110) oriented surfaces, and in the bulk for (j) µ/t = − . , (k) µ/t = 0 . , and (l) µ/t = 0 . .The red solid line, blue dash-dotted line, and black dashed line denote the LDOS at the (100) oriented surface, (110) orientedsurface, and in the bulk, respectively. Other parameters are λ SO /t = 0 . , ∆ is /t = 0 . , and | ∆ | /t = 1 . × − . -3 -2 -1 (110) surface µ / t DO S S C ( E = ) / DO S N ( E = ) | ∆ |/ t FIG. 10. The LDOS at E = 0 for the (110) oriented surface forthe B representation as a function of | ∆ | /t at λ SO /t = 0 . and ∆ is /t = 0 . . The red solid line, blue dotted line, andgreen dashdotted line correspond to µ/t = − . , µ/t = 0 . ,and µ/t = 0 . . to | ∆ | (1 − | ∆ | / | ∆ c | ) for | ∆ | < | ∆ c | and vanishes for | ∆ | > | ∆ c | . This is a nonmonotonic dome-shaped be-havior of the ZELDOS as a function of | ∆ | . The ex-plicit profile can be seen in Fig. 10 at µ/t = − . and µ/t = 0 . . For µ/t = 0 . , the point nodes still existin this parameter regime, and the height of the zero en-ergy peak develops with | ∆ | . Thus, we have that theZESABS get strongly renormalized and are tunable by avariation in the electron filling ( µ ) and amplitude of theorder parameter | ∆ | as shown in Fig. 8. V. DISCUSSION AND SUMMARY
We investigated and determined the possible super-conducting phases arising from inter-orbital pairing inan electronic environment marked by spin-orbit couplingand inversion symmetry breaking while focusing on mo-mentum independent paired configurations. One remark-able aspect is that, although the inversion symmetry isabsent, one can have symmetry-allowed solutions thatavoid mixing of spin-triplet and spin-singlet configura-tions. Importantly, states with only spin-triplet pairingscan be stabilized in a large portion of the phase diagram.Within those spin-triplet superconducting states, weunveiled an unconventional type of topological phase intwo-dimensional superconductors that arises from theinterplay of spin-orbit coupling and orbitally driveninversion-symmetry breaking. For this kind of a modelsystem, atomic physics plays a relevant role and in-evitably tends to yield orbital entanglement close to theFermi level. Thus we assumed that local inter-orbitalpairing is the dominant attractive interaction. As alreadymentioned, this type of pairing in the presence of inver-sion symmetry breaking allows for solutions that do notmix spin-singlet and triplet configurations. The orbital-singlet/spin-triplet superconducting phase can have a topological nature with distinctive spin-orbital finger-prints in the low-energy excitations spectra that makeit fundamentally different from the topological configu-ration that is usually obtained in single band noncen-trosymmetric superconductors. Here, a remarkable find-ing is that, contrary to the common view that an isotropicpairing structure leads to a fully gapped spectrum, anodal superconductivity can be achieved when consid-ering an isotropic spin-triplet pairing. Although in a dif-ferent context, we noticed that akin paths for the genera-tion of an anomalous nodal-line superconductor can alsobe encountered when local spin-singlet pairing occur inantiferromagnetic semimetals .In the present study, for a given symmetry, the super-conducting phase can exhibit point nodes that are pro-tected by a nonvanishing winding number. The moststriking feature of the disclosed topological superconduc-tivity is expressed by its being prone to both topologicaland Lifshitz-type transitions upon different driving mech-anisms and interactions, e.g., when tuning the strengthof intrinsic spin-orbit and orbital-momentum couplingsor by varying doping and the amplitude of order pa-rameter by, for example, varying the temperature. Theessence of such a topologically and electronically tunablesuperconductivity phase is encoded in the fundamentalobservation of having control of the nodes position in theBrillouin zone. Indeed, the location of the point nodesis not determined by the symmetry of the order parame-ter in the momentum space, as occurs in the single bandnoncentrosymmetric system, but rather it is a nontrivialconsequence of the interplay between spin-triplet pair-ing and the spin-orbital character of the electronic struc-ture. In particular, their position and existence in theBrillouin zone can be manipulated through various typesof Lifshitz transitions, if one varies the chemical poten-tial, the amplitude of the spin-triplet order parameter,the inversion symmetry breaking term, and the atomicspin-orbit coupling. While electron doping can induce achange in the number of Fermi surfaces, such electronictransition is not always accompanied by a variation inthe number of nodes within the superconducting state.This behavior allows one to explore different physical sce-narios that single out notable experimental paths for thedetection of the targeted topological phase. Owing to thestrong sensitivity of the topological and Lifshitz transi-tions with respect to the strength of the superconductingorder parameter, one can foresee the possibility of observ-ing an extraordinary reconstruction of the superconduct-ing state both in the bulk and at the edge by employingthe temperature to drive the pairing order parameter to avanishing value, i.e., at the critical temperature, startingfrom a given strength at zero temperature. Then, a sub-stantial thermal reorganization of the superconductingphase can be obtained. While a variation in the numberof nodes in the low energy excitations spectra cannot beeasily extracted by thermodynamic bulk measurements,we find that the electronic structure at the edge of the su-perconductor generally undergoes a dramatic reconstruc-4tion that manifests into a non-monotonous behavior ofthe zero bias conductance or in an unconventional ther-mal dependence of the in-gap states. Another impor-tant detection scheme of the examined spin-triplet su-perconductivity emerges when considering its sensitivityto the doping or to the strength of the inversion symme-try breaking coupling, which can be accessed by applyingan electrostatic gating or pressure. Such gate/distortivecontrol can find interesting applications, especially whenconsidering two-dimensional electron gas systems.Another interesting feature of the multiple-nodes topo-logical superconducting phase is given by the strong sen-sitivity of the edge states to the geometric termination, asdemonstrated in Fig. 9. This is indeed a consequence ofthe presence of nodes with an opposite sign winding num-ber within the Brillouin zone. Hence, when consideringthe electronic transport along a profile that is averagingdifferent terminations, it is natural to expect multiplein-gap features.Owing to the multi-orbital character of the supercon-ducting state, we expect that non-trivial odd-in-timepair amplitudes are also generated . In particu-lar, we predict that both local odd-in-time spin-singletand triplet states can be obtained in the bulk and at theedge. The local spin-singlet odd-in-time pair correlationsare an exquisite consequence of the multi-orbital super-conducting phase. Accessing the nature of their competi-tion/cooperation and its connection to the nodal super-conducting phase is a general and relevant problem inrelation to the generation, manipulation, and control ofodd-in-time pair amplitudes.It is also relevant to comment on the impact of anintra-orbital pairing on the achieved results. Here, thereare few fundamental observations to make. Firstly, onemay ask whether the topological B phase is robustto the adding of an extra pairing component which inthe intra-orbital channel is most likely to have a spin-singlet symmetry. For this circumstance, one can startby pointing out that for any intra-orbital pairing com-ponent that does not break the chiral symmetry protect-ing the nodal structure of the superconducting state, theB configuration can only undergo a Lifshitz-type tran-sition associated with the merging of nodes having oppo-site sign in the winding number. Moreover, specificallyfor the B irreducible representation, the intra-orbitalspin-singlet component would have a d x − y -wave sym-metry ( ∼ cos k x − cos k y ) and thus its amplitude would bevanishing along the Γ -M direction of the Brillouin zonewhere the nodes of the B phase are placed. Hence, theintra-orbital component cannot affect at all the nodalstructure of the B phase. From this perspective, theB phase is remarkably robust to the inclusion of spin-singlet intra-orbital pairing components. In Appendix,the intra-orbital spin-singlet pairings other than B rep-resentation ( d x − y -wave) are discussed.Concerning the experimental consequences of the topo-logical superconducting phase, one can observe that,apart from the direct spectroscopic access to the tem- perature dependence of the edge states, the use ofa superconductor-normal metal-superconductor (S-N-S)junction can also contribute to design of experiments todirectly probe the peculiar behavior of the B phase. Inparticular, by scanning its temperature dependent prop-erties, since the B state can undergo a series of Lifshitztransitions within the superconducting phase by gappingout part of the nodes, a dramatic modification of theAndreev spectrum at the S-N boundary is expected tooccur. Hence, upon the application of a phase differencebetween the superconductors in the S-N-S junction, theJosephson current is expected to exhibit an anomaloustemperature behavior. In particular, the abrupt changesin the Andreev bound states will drive a rapid variation inthe Josephson current through the S-N-S junction whenthe superconductor undergoes transitions in the numberof nodes.Finally, we point out that the examined model Hamil-tonian is generally applicable to two-dimensional lay-ered materials, in the low/intermediate doping regime,having t g d -bands at the Fermi level and subjected toboth atomic spin-orbit coupling and inversion symme-try breaking, for instance owing to lattice distortionsand bond bending. Many candidate material cases canbe encountered in the family of transition metal ox-ides. There, unconventional low-dimensional quantumliquids with low electron density can be obtained byengineering a 2DEG at polar/nonpolar interfaces be-tween two band insulators, on the surface of band insu-lators (i.e., STO) or by designing single monolayer het-erostructures, ultrathin films or superlattices. A paradig-matic case of superconducting 2DEG is provided by theLAO/STO heterostructure . Recent experimentalobservations by tunneling spectroscopy have pointed outthat the superconducting state can be unconventionalowing to the occurrence of in-gap states with peaks atzero and finite energies . Although these peaks maybe associated with a variety of concomitant physicalmechanisms, e.g., surface Andreev bound states , theanomalous proximity effect by odd-frequency spin-tripletpairing , and bound states owing to the presenceof magnetic impurities , their nature can provide keyinformation about the pairing symmetry of the super-conductor. Furthermore, the observation of Josephsoncurrents across a constriction in the 2DEG confirms afundamental unconventional nature of the superconduct-ing state . A common aspect emerging from thetwo different spectroscopic probes is that the supercon-ducting state seems to have a multi-component character.Although it is not easy to disentangle the various contri-butions that may affect the superconducting phase in the2DEG, we speculate that the proposed topological phasecan be also included within the possible candidates foraddressing the puzzling properties of the superconduc-tivity of the oxide interface.5 ACKNOWLEDGMENTS
This work was supported by a JSPS KAKENHI(Grants No. JP15H05853, No. JPH06136, and No.JP15H03686), and the JSPS Core-to-Core program "Ox-ide Superspin", and the project Quantox of QuantERAERA-NET Cofund in Quantum Technologies, imple-mented within the EU H2020 Programme.
VI. APPENDIX
In this section we address three different issues relatedto the presented results. Firstly, we investigate how amodification of the pairing interaction affects the phasediagram and the relative competition between the var-ious configurations by scanning a larger range of tem-peratures at representative cases of filling concentration.Then, we consider the classification of the irreducible rep-resentations of the superconducting phases in the pres-ence of an intra-orbital attractive interaction. Moreover,we demonstrate that the intra-orbital and inter-orbitalpairing interactions mediated by phonons have the sameamplitude.Starting from the impact of the pairing interactionon the phase diagram, in Fig. 11 we show that at agiven temperature the maximal eigenvalue in the variousirreducible representations scales with the values of V z and V xy at λ SO /t = 0 . , ∆ is /t = 0 . and µ/t = 0 . .When we keep the ratio V z /V xy , the eigenvalue Λ is pro-portional to V xy within the mean field approximation.Hence, the phase diagram is basically determined by theratio V z /V xy . In addition, since the transition tempera-ture T c is achieved when the magnitude of the greatesteigenvalue gets close to 1, then, according to this relation,one can identify the regime of temperatures which is closeto the superconducting transition by suitably scaling thepairing interactions. In this way, the corresponding irre-ducible representation with the largest eigenvalue is themost stable according to the solution of the gap equation.In order to understand how a change in the criti-cal temperature can affect the relative stability, in Fig.12 we report the eigenvalues for the various irreduciblerepresentations at λ SO /t = 0 . , ∆ is /t = 0 . and µ/t = 0 . as a function of T /t at two different ratio (a) V z /V xy = 1 . and (b) V z /V xy = 0 . . We notice thatthe most stable configuration is not affected by a changein temperature or the strength of the pairing coupling.However, the eigenvalues of the B and E representationbecome larger than that of A above T /t ∼ . × − (see Fig. 12(b)), thus affecting the competition betweenthe A and B configurations. Otherwise, the analysis atdifferent temperatures demonstrate that even for largervalues of the pairing interaction the phase diagram is notmuch affected.Concerning the role of the intra-orbital spin-singletpairing, in Ref. [1], we can classify the possible ir-reducible representations for the tetragonal group C v V z / V xy V z / V xy (a) (c) t h e e i g e nv a l u e V xy / t = −0.10 V xy / t = −1.0 A representationB representation A representationB representationE representation V z / V xy (b) V xy / t = −0.5 FIG. 11. The eigenvalues for various irreducible represen-tations as a function of V z /V xy at (a) V xy /t = − . , (b) V xy /t = − . and (c) V xy /t = − . , assuming that T /t =1 . × − , λ SO /t = 0 . , ∆ is /t = 0 . , and µ/t = 0 . . −5 −4 −3 −2 −1 −5 −4 −3 −2 −1 T / t t h e e i g e nv a l u e A representationB representation A representationB representationE representation (a) (b) V z / V xy = 1.00 V z / V xy = 0.70 T / t
FIG. 12. The eigenvalues for various irreducible representa-tions as a function of
T /t at (a) V z /V xy = 1 . and (b) V z /V xy =0 . , assuming that V xy /t = − . , λ SO /t = 0 . , ∆ is /t =0 . , and µ/t = 0 . . assuming both isotropic inter-orbital pairing and intra-orbital ones with isotropic and anisotropic structurescompatible with the symmetry configuration as shownin Table II.Finally, we consider the relative strength of the attrac-tive interaction in the inter- and intra-orbital channel asdue to electron-phonon coupling in a t g multi-orbitalsystem.Consider the electron phonon coupling in t g system, H ep = 1 √ N X k , q ,m,l,l ′ σ α mll ′ ( q ) c † k + q ,l,σ c k ,l ′ ,σ , (29)where m denotes the phonon mode, α mll ′ ( q ) is theelectron-phonon coupling constant, and l and l ′ standsfor orbital indices in the basis of yz , zx , and xy . Here,6 TABLE II. Irreducible representation of isotropic inter-orbitalsuperconducting states and intra-orbital spin-singlet oneswith isotropic and anisotropic structures for the tetragonalgroup C v . In the columns, we report the sign of the orderparameter upon a four-fold rotational symmetry transforma-tion, C , and the reflection mirror symmetry M yz , as well asthe explicit spin and orbital structure of the gap function. Inthe E representation, + and − of the subscript mean the dou-bly degenerate mirror-even ( + ) and mirror-odd ( − ) solutions,respectively. C v C M yz orbital basis function A + + ( d yz , d yz ) ψ ( yz,yz ) = const. ( d zx , d zx ) ψ ( zx,zx ) = ψ ( yz,yz ) ( d xy , d xy ) ψ ( xy,xy ) = const. ( d xy , d yz ) d ( xy,yz ) y ( d xy , d zx ) d ( xy,zx ) x = − d ( xy,yz ) y ( d yz , d zx ) d ( yz,zx ) z A + − ( d yz , d yz ) ψ ( yz,yz ) = sin k x sin k y (cos k x − cos k y )( d zx , d zx ) ψ ( zx,zx ) ( k x , k y ) = ψ ( yz,yz ) ( k y , − k x )( d xy , d xy ) ψ ( xy,xy ) = sin k x sin k y (cos k x − cos k y )( d xy , d yz ) d ( xy,yz ) x ( d xy , d zx ) d ( xy,zx ) y = d ( xy,yz ) x B − + ( d yz , d yz ) ψ ( yz,yz ) ∝ cos k x − cos k y ( d zx , d zx ) ψ ( zx,zx ) ( k x , k y ) = − ψ ( yz,yz ) ( k y , − k x )( d xy , d xy ) ψ ( xy,xy ) ∝ cos k x − cos k y ( d xy , d yz ) d ( xy,yz ) y ( d xy , d zx ) d ( xy,zx ) x = d ( xy,yz ) y B − − ( d yz , d yz ) ψ ( yz,yz ) ∝ sin k x sin k y ( d zx , d zx ) ψ ( zx,zx ) ( k x , k y ) = − ψ ( yz,yz ) ( k y , − k x )( d xy , d xy ) ψ ( xy,xy ) ∝ sin k x sin k y ( d xy , d yz ) d ( xy,yz ) x ( d xy , d zx ) d ( xy,zx ) y = − d ( xy,yz ) x ( d yz , d zx ) ψ ( yz,zx ) E ± i ± ( d xy , d yz ) ψ ( xy,yz ) , d ( xy,yz ) z ( d xy , d zx ) ψ ( xy,zx )+ = ∓ id ( xy,yz ) z + d ( xy,zx ) z − = ∓ iψ ( xy,yz ) − ( d yz , d zx ) d ( yz,zx ) x , d ( yz,zx ) y we consider only the diagonal elements, which are rele-vant to the attractive interaction. Off-diagonal ones arerelevant to the pair hopping, which enhance the transi-tion temperature. H ep = 1 √ N X k , q ,m,l,σ α mll ( q ) c † k + q ,l,σ c k ,l,σ . (30)The effective interaction in the Eliashberg equation dueto this electron-phonon coupling is given by V mll ′ ( q , ω n ) = − α mll ( q ) α ml ′ l ′ ( q ) D m ( q , ω n ) , (31)where D m ( q , ω n ) is the Green’s function of phonon D m ( q , ω n ) = 2 ω m ( q ) ω m ( q ) + ω n , (32)with phonon’s frequency ω m ( q ) and bosonic Matsubarafrequency ω n = 2 nπk B T . Here, we suppose the A modesare the most relevant to the interaction. In A modes, α myz,yz ( q ) = α mzx,zx ( q ) (33)in the tetragonal symmetry. Then, we have the relation V myz,yz ( q , ω n ) = V mzx,zx ( q , ω n ) = V myz,zx ( q , ω n ) = V mzx,yz ( q , ω n ) . 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