# Inter-orbital topological superconductivity in spin-orbit coupled superconductors with inversion symmetry breaking

###### Abstract

We study the superconducting state of multi-orbital spin-orbit coupled systems in the presence of an orbitally-driven inversion asymmetry assuming that the inter-orbital attraction is the dominant pairing channel. Although the inversion symmetry is absent, we show that superconducting states that avoid mixing of spin-triplet and spin-singlet configurations are allowed, and, remarkably, spin-triplet states which are topologically non-trivial can be stabilized in a large portion of the phase diagram. The orbital dependent spin-triplet pairing generally leads to topological superconductivity with point nodes that are protected by a non-vanishing winding number. We demonstrate that the disclosed topological phase can exhibit Lifshitz-type transitions upon different driving mechanisms and interactions, e.g. by tuning the strength of the atomic spin-orbit and inversion asymmetry couplings or by varying the doping and the amplitude of order parameter. Such distinctive signatures of the nodal phase manifest through an extraordinary reconstruction of the low-energy excitation spectra both in the bulk and at the edge of the superconductor.

## I Introduction

Spin-triplet pairing is at the core of intense investigation especially for its foundational aspect in unconventional superconductivity Sigrist and Ueda (1991); Maeno et al. (1994); Tou et al. (1998); Kashiwaya et al. (2011) and due to its tight connection with the occurrence of topological phases with zero-energy surface Andreev bound states Buchholtz and Zwicknagl (1981); Hara and Nagai (1986); Hu (1994); Tanaka and Kashiwaya (1995); Kashiwaya and Tanaka (2000); Löfwander et al. (2001) marked by Majorana edge modes Kwon et al. (2004); Schnyder et al. (2008); Ryu et al. (2010); Qi and Zhang (2011); Tanaka et al. (2012); Leijnse and Flensberg (2012); Beenakker (2013); Sato and Fujimoto (2016); Sato and Ando (2017). Some of the fundamental essences of topological spin-triplet superconductivity are basically captured by the Kitaev model Kitaev (2001) and its generalized versions where non-Abelian states of matter and their employment for topological quantum computation can be demonstrated Kitaev (2001); Nayak et al. (2008); Wilczek (2009); Alicea (2012). Another remarkable element of odd-parity superconductivity is given by the potential of having active spin degrees of freedom making such state of matter also appealing for superconducting spintronics applications based on spin control and coherent spin manipulation of Cooper pairs Bergeret et al. (2005); Buzdin (2005); Khaire et al. (2010); Robinson et al. (2010); Eschrig et al. (2003); Eschrig and Löfwander (2008); Linder and Robinson (2015). Interplay of magnetism and spin-triplet superconductivity can manifest within different unconventional physical scenarios such as for the case of the emergent spin-orbital interaction between the superconducting order parameter and interface magnetization Gentile et al. (2013); Terrade et al. (2016), the breakdown of the bulk-boundary correspondence Mercaldo et al. (2016), and the anomalous magneticRomano et al. (2013, 2016) and spin-charge currentsRomano et al. (2017) effects occurring in the proximity between chiral or helical p-wave and spin-singlet superconductors. Achieving spin-triplet materials platforms, thus, set the stage for the development of emergent technologies both in non-dissipative spintronics and in the expanding area of quantum devices.

Although embracing strong promises, spin-triplet superconductivity is quite rare in nature and the mechanisms for electron pairs gluing are not completely settled. Search for spin-triplet superconductivity has been performed along different routes. For instance, scientific exploration has been focused on the regions of the materials phase diagram which are in proximity to ferromagnetic quantum phase transitions Fay and Appel (1980); Pfleiderer (2009), as in the case of heavy fermions superconductivity (i.e. , URhGe, , etc.), or in materials on the verge of a magnetic instability, e.g. ruthenates Maeno et al. (1994); Mackenzie and Maeno (2003).

Another remarkable route to achieve spin-triplet pairing relies on the presence of a source of inversion symmetry breaking, both at the surface/interface and in the bulk, or, alternatively, in connection with non-collinear magnetic ordering Martin and Morpurgo (2012); Pöyhönen et al. (2014); Pientka et al. (2013a); Braunecker and Simon (2013); Klinovaja et al. (2013); Kim et al. (2014); Pientka et al. (2014); Nadj-Perge et al. (2013); Nakosai et al. (2013); Pientka et al. (2013b); Heimes et al. (2014); Mendler et al. (2015). Paradigmatic examples along these directions are provided by quasi one-dimensional heterostructures whose interplay of inversion and time-reversal symmetry breaking or non-collinear magnetism have been shown to convert spin-singlet pairs into spin-triplet ones and in turn to topological phasesSau et al. (2010); Alicea et al. (2010); Lutchyn et al. (2010); Oreg et al. (2010); Nadj-Perge et al. (2013). Similar mechanisms and physical scenarios are also encountered at the interface between spin-singlet superconductors and inhomogeneous ferromagnets with even and odd-in time spin-triplet pairing which are generally generated Bergeret et al. (2005). Semimetals have also been indicated as fundamental building blocks to generate spin-triplet pairing as theoretically proposed and demonstrated in topological insulators interfaced with conventional superconductors or by doping Dirac/Weyl phases Volovik (2003), e.g. in the case of Cu-doped Hor et al. (2010); Wray et al. (2010); Kriener et al. (2011); Fu and Berg (2010a); Sasaki et al. (2011); Hao and Lee (2011); Yamakage et al. (2012) in anti-perovskites materials Oudah et al. (2016), as well as CdAs Aggarwal et al. (2015); Wang et al. (2015), etc.

On a general ground there are two fundamental interactions to take into account in inversion asymmetric microscopic environments: i) the Rashba spin-orbit coupling Rashba (1960) due to inversion symmetry breaking at the surface or interface in heterostructures, and ii) the Dresselhaus coupling arising from the inversion asymmetry in the bulk of the host material Dresselhaus (1955). For the present analysis, it is worth noting that typically in multi-orbital materials it is the combination of the atomic spin-orbit interaction with the inversion symmetry-breaking sources that effectively generates both Rashba and Dresselhaus emergent interactions within the electronic manifold close to the Fermi level. Another general observation is that the lack of inversion symmetry is expected to lead to a parity mixing of spin-singlet and spin-triplet configurations Gorkov and Rashba (2001); Frigeri et al. (2004) with an ensuing series of unexpected features ranging from anomalous magneto-electric Lu and Yip (2008) effects, to unconventional surface states Vorontsov et al. (2008), topological phases Tanaka et al. (2009); Sato and Fujimoto (2009); Schnyder et al. (2010), as well as non-trivial spatial textures of the spin-triplet pairs Ying et al. (2017). Such symmetry conditions in intrinsic materials are however fundamentally linked to the momentum dependence structure of the superconducting order parameter. In contrast, when considering multi-orbital systems, more channels are possible with emergent unconventional paths for electron pairing that are expected to be strongly tight to the orbital character of the electron-electron attraction and of the electronic states close to the Fermi level.

Orbital degrees of freedom are key players in quantum materials when considering the degeneracy of -bands of the transition elements not being completely removed by the crystal distortions or due to the intrinsic spin-orbital entanglement Oleś (2012) triggered by the atomic spin-orbit coupling. In this context, a competition of different and complex types of order is ubiquitous in realistic materials (e.g. transition metal oxides) mainly due to the frustrated exchange arising from the active orbital degrees of freedom. Such scenarios are commonly encountered in materials where the atomic physics plays a significant role in setting the character of the electronic structure close to the Fermi level. Since the -orbitals have an anisotropic spatial distribution, the nature of the electronic states is also strongly dependent on the system dimensionality. Indeed, two-dimensional (2D) confined electron liquids originating at the interface or surface of materials generally manifest a rich variety of spin-orbital phenomena Hwang et al. (2012). Along this line, understanding how electron pairing is settled in quantum systems exhibiting a strong interplay between orbital degrees of freedom and inversion symmetry breaking represents a fundamental problem in unconventional superconductivity and it can be of great relevance for a large class of materials.

In this paper, we investigate the nature of the superconducting phase in spin-orbit coupled systems in the absence of inversion symmetry assuming that the inter-orbital attractive channel is dominant and sets the electrons pairing. We demonstrate that the underlying inversion symmetry breaking leads to exotic spin-triplet superconductivity. Isotropic spin-triplet pairing configurations, without any mixing with spin-singlet, generally occur among the symmetry allowed solutions and are shown to be the ground-state in a large part of the parameters space. We then realize an isotropic spin-triplet superconductor whose orbital character can make it topologically non trivial. Remarkably, the topological phase exhibits an unconventional nodal structure with unique tunable features. An exotic fingerprint of the topological phases is that the number and -position of nodes can be controlled by doping, orbital polarization, through the competition between spin-orbit coupling and lattice distortions, and temperature (or equivalently the amplitude of the order parameter).

The paper is organized as follows. In Sect. II we introduce the model Hamiltonian and present the classification of the inter-orbital pairing configurations with respect to the point-group and time-reversal symmetries. Sect. III is devoted to the analysis of the stability of the various orbital entangled superconducting states and the energetics of the isotropic superconducting states. Sect. IV focuses on the electronic spectra of the energetically most favorable phases and the ensuing topological configurations both in the bulk and at the boundary. Finally, in Sect. V we provide a discussion of the results and few concluding remarks.

## Ii Model and symmetry classification of superconducting phases with inter-orbital pairing

One of the most common crystal structure of transition metal oxides is the perovskite one, with transition metal (TM) elements surrounded by oxygens (O) in an octahedral environment. For cubic symmetry, due to the crystal field potential generated by the oxygens around the TM, the fivefold orbital degeneracy is removed and -orbitals split into two sectors, i.e. (i.e. , , ) and (i.e. , ). In the present paper, the analysis is focused on two-dimensional (2D) electronic systems with broken out-of-plane inversion symmetry and having only the -orbitals (see Fig. 1) close to the Fermi level to set the low energy excitations. For highly symmetric TM-O bonds, the three -bands are directional and basically decoupled, e.g. an electron in the -orbital can only hop along the or direction through the intermediate or -orbitals. Similarly, the and -bands are quasi one-dimensional when considering a 2D TM-O bonding network. Furthermore, the atomic spin-orbit interaction (SO) mixes the -orbitals thus competing with the quenching of the orbital angular momentum due to the crystal potential. Concerning the inversion asymmetry, we consider microscopic couplings that arise from the out-of-plane oxygen displacements around the TM. Indeed, by breaking the reflection symmetry with respect to the plane placed in between the TM-O bond Khalsa et al. (2013), a mixing of orbitals that are even and odd under such transformation is generated. Such crystal distortions are much relevant and more pronounced in 2D electron gas forming at the interface of insulating polar and non-polar oxide materials or on their surface and they result into the activation of an effective hybridization, which is odd in space, of and or -orbitals along the or directions, respectively. Although the polar environment tends to amplify the out-of-plane oxygen displacements with respect to the position of the TM ion, such type of distortions can be also occurring at the interface of non-polar oxides and in superlattices Autieri et al. (2014).

Then, the model Hamiltonian, including the hopping connectivity, the atomic spin-orbit coupling and the inversion symmetry breaking term, reads as

(1) | ||||

(2) |

where is a vector whose components are associated to the electron creation operators for a given spin (), orbital () and momentum in the Brillouin zone.

In Fig. 1(a) we report a schematic illustration of the local orbital basis for the states. , and indicate the kinetic term, the spin-orbit interaction and inversion symmetry breaking term, respectively. In the spin-orbital basis is given by

(3) | |||

where is the unit matrix in spin space. Here, , and are the orbital dependent hopping amplitudes as schematically shown in Fig. 1(b). denotes the crystal field potential owing to the symmetry lowing from cubic to tetragonal. The symmetry reduction yields a level splitting between -orbital and -orbitals. denotes the atomic spin-orbit coupling,

(4) |

with being the Pauli matrix in spin space. In order to write down the interaction, it is convenient to introduce the matrices , and which are the projections of the angular momentum operator to the subspace, i.e.

(5) | ||||

(6) | ||||

(7) |

assuming as orbital basis. Finally, as mentioned above, the breaking of the mirror plane in between the TM-O bond, due to the oxygen displacements, leads to an inversion symmetry breaking term of the type

(8) |

This contribution gives an inter-orbital process, due to the broken inversion symmetry, that mixes and or along or spatial directions (Fig. 1(b)). For convenience and clarity of presentation, we set as a unit of energy. The analysis is performed for a representative set of hopping parameters, i.e. and . Slight variations are expected to do not alter the qualitative aspects of the achieved results.

The electronic structure of the examined model system can be accessed by direct diagonalization of the matrix Hamiltonian. Representative dispersions for and are shown in Fig. 2(a). We observe that there are six non degenerate bands due to the presence of both and . Once the dispersions are determined, it is immediate to notice that the number of Fermi surfaces and the structure can be varied by tuning the chemical potential. Indeed, for , and one can single out all the main possible cases with two, four and six Fermi sheets, as given in Figs. 2(b), (c) and (d), respectively. For the explored regimes of low doping, all the Fermi surfaces are made of electron-like pockets centered around origin of the Brillouin zone (). The lowest occupied band has a symmetric shape since it has a dominant character (Fig. 2(a)), while moving to higher electron concentrations the outer Fermi sheets exhibit an anisotropic profile which become more pronounced when the chemical potential crosses the bands mainly arising from the and -orbitals.

After having considered the normal state properties, let us concentrate on the possible superconducting states which can be realized, their energetics and topological behavior. The analysis is based on the assumption that the inter-orbital local attractive interaction is the only relevant pairing channel to contribute for the Cooper pairs formation. Then, the intra-orbital pairing coupling is negligible. Such hypothesis can be physically applicable in multi-orbital systems because the intra-band Coulomb interaction is typically larger than the inter-band one. Thus, it is plausible to expect that the Coulomb repulsion tends to suppress more the electron pairing that occurs within the same band. Other sources of enhancement of the inter-band pair scattering can arise from the coupling to specific phononic modes as related to the symmetry of electronic states in the presence of an inversion symmetry breaking coupling. For instance, we point out that topological superconductivity is proposed to occur, as due to inter-orbital pairing, in Cu-doped for an inversion symmetric crystal structure Fu and Berg (2010b). Here, although similar inter-orbital pairing conditions are considered, we pursue the superconductivity in low-dimensional configurations, e.g. at the interface of oxides, with the important constraint of having a broken inversion symmetry. Concerning the orbital structure of the pairing interaction, due to the tetragonal crystalline symmetry, the coupling between -orbital and /-orbital are equivalent, and thus one can assume that only two independent channels of attraction are allowed, as shown in Fig. 1(d). Indeed, denotes the interaction between and /-orbitals, while refers to the coupling between and -orbitals.

Then, the pairing interaction is given by

(9) | ||||

(10) |

where denotes the lattice site.

### ii.1 Irreducible representation and symmetry classification

In this subsection we classify the inter-orbital superconducting states according to the point group symmetry and by taking into account the breaking of the inversion symmetry. The system upon examination has tetragonal symmetry associated with the point group , marked by four-fold rotational symmetry and mirror symmetries and . Based on the rotational and reflection symmetry transformations, all the possible inter-orbital isotropic pairing can be classified into five irreducible representations of the point group as summarized in Table 1.

orbital | spin (-vector) | |||
---|---|---|---|---|

, | ||||

E | , | |||

, |

For our purposes, only solutions that do not break the time-reversal symmetry are considered and reported in Table I. Then, the superconducting order parameter associated to bands and can be classified as an isotropic (-wave) spin-triplet/orbital-singlet -vector and -wave spin-singlet/orbital-triplet with amplitude or as a mixing of both configurations. Under these assumptions, one can generally describe the isotropic order parameter with spin-singlet and triplet components as

(11) |

with and standing for the orbital index, and having for each channel three possible orbital flavors. Furthermore, due to the selected tetragonal crystal symmetry, one can achieve three different types of inter-orbital pairings. The spin-singlet configurations are orbital triplets and can be described by a symmetric superposition of opposite spin states in different orbitals. On the other hand, spin-triplet components can be expressed by means of the following -vectors,

with indicating the spin-triplet configuration built up with and -orbitals. In general, independently of the orbital mixing, spin-triplet pairing can be expressed in a matrix form as

(12) |

where the -vector components are related to the pairing order parameter with zero spin projection along the corresponding symmetry axis. The three components , and are expressed in terms of the equal spin , and the anti-aligned spin gap functions. Then, since the components of the -vector are associated to the zero spin projection of spin-triplet configuration, if the -vector points along a given direction, the parallel spin configurations lie in the plane perpendicular to the -vector orientation. In the presence of time-reversal symmetry, the superconducting order parameter should satisfy the following relation

(13) | ||||

(14) |

with the appropriate choice of the U(1) gauge. In addition, the pairing order parameter has four-fold rotational symmetry and mirror reflection symmetry with respect to the and planes as dictated by the point group . Thus, it has to be transformed according to the following relations:

where equals to for A representation, for B representation, and for E representation. Such symmetry properties are very important for singling out the possible solutions of the Bogoliubov-de Gennes equation. The energy gap functions are then explicitly constructed by taking into account the corresponding irreducible representations. For the one-dimensional representations, we have that the state is given by

while for representation,

and for representation,

and for representation,

Finally, for E representation, there are doubly degenerate mirror-even and mirror-odd solutions,

where and denote arbitrary constants for the linear superposition. As a consequence of the symmetry constraint and of the inter-orbital structure of the order parameter, different types of isotropic spin-triplet and singlet-triplet mixed configurations can be obtained. Equal spin-triplet and opposite spin-triplet pairings are mixed in representation. On the other hand, in , equal spin-triplet and spin-singlet pairing are mixed. For and representations, only equal spin-triplet pairing is allowed and all types of pairings can be realized in E representation. It is worth noting that , and E have pairing between all the orbitals in the - and - channels, while those that are odd under mirror reflection, i.e. and , can make electron pairing only in the - channel, that is by mixing the and /-orbitals as shown in Table 1. Such symmetry constraint is important when searching the ground-state configuration.

## Iii Energy gap equation and phase diagram

In order to investigate which one of the possible symmetry-allowed solutions is more stable energetically, we solve the Eliashberg equation within the mean field approximation by taking into account the multi-orbital effects near the transition temperature. The linearized Eliashberg equation within the weak coupling approximation is given by

(15) | ||||

(16) | |||

Here, , , and denote the spin states and , , and stand for the orbital indices. is the anomalous Green’s function. Since we assume an isotropic Cooper pairing, which is -independent, the summation over momentum and Matsubara frequency in Eq. (III) gets simplified. Finally, the problem is reduced to the diagonalization of matrix. We then study the relative stability of the irreducible representations as listed in Table 1. The analysis of the energetically most favourable superconducting states is performed as a function of , assuming that and for a given temperature .

Fig. 3 shows the superconducting phase diagram for representative amplitudes of the spin-orbit coupling, , and inversion asymmetry interaction, , while varying both the chemical potential and the ratio of the pairing couplings . Due to the inequivalent mixing of the orbitals in the paired configurations, it is plausible to expect a significant competition between the various symmetry allowed states and that such interplay is sensitive not only to the pairing orbital anisotropy but also to the structure and the number of Fermi surfaces. A direct observation is that for larger than the phase is stabilized with respect to the because it contains a channel of spin-triplet pairing in the - sector which is absent in . However, such simple deduction does not provide a direct explanation of the reason why the phase wins the competition with other superconducting phases, e.g. and E phases, which also can gain condensation energy by pairing electrons in the - sector. Since a different type of -vector orientation enters into the and E configurations while in the state the - channel has a spin-singlet pairing, one can deduce that the interplay between the spin of the Copper pairs and that of the single electron states close to the Fermi level is relevant to single out the most favorable superconducting phase.

The boundary between the and phase exhibits a sudden variation when one tunes the chemical potential across the value for which the number of Fermi surfaces changes. Such abrupt transition is however plausible when passing through a Lifshitz point in the electronic structure of the normal state because other pairing channels get activated at the Fermi level. Along this line, the role of the electron filling is also quite important and sets the competition between the energetically most stable phases. Indeed, one can notice that the () phase is stabilized for higher (lower) and lower (higher) . Furthermore, we find that, in the case of two Fermi surfaces, state is further stabilized by decreasing the chemical potential and moving to a regime of extreme low concentration. On the other hand, a transition to the phase is achieved by electron doping. In the doping regime of four bands at the Fermi level, the - boundary evolves approximately as a linear function of . This implies that tends to be less stable and a higher ratio is needed to achieve such configuration at a given chemical potential. Finally, approaching the doping regime of six Fermi surfaces, the - boundary becomes independent of the amplitude of . It is remarkable how the doping can substantially alter the competition between the and phases thus manifesting the intricate consequences of the spin-orbital character of the electronic structure close to the Fermi level.

To explicitly and quantitatively demonstrate the energy competition among all the symmetry allowed phases, one can follow the behavior of the eigenvalues of the linearized Eliashberg equations as a function of the ratio (Fig. 4).

Figs. 4(a)(b)(c) show the eigenvalues of the Eliashberg matrix equation for all the irreducible representations as a function of when the number of Fermi surfaces is (a) two (), (b) four () and (c) six () as indicated by the dotted lines in Fig. 3. With the increase of , the magnitude of the eigenvalues of the irreducible representations including the - channel, i.e. , , and E representations, increases in all the cases with two, four and six Fermi surfaces. On the other hand, the eigenvalues of the other representation are independent of since is irrelevant for this pairing channel. When the number of Fermi surfaces is two, representation is the most dominant pairing for all . Although the magnitude of the eigenvalues for and representations also increase with , these solutions never become dominant as compared with the state. When the number of Fermi lines is four or six, the eigenvalue of phase is larger than that of representation for lower .

## Iv Topological properties and energy excitation spectrum in the bulk and at the edge

In the previous section, we confirmed that both and pairing can be energetically stabilized in a large region of the parameters space. Then, it is relevant to further consider the nature of the electronic structure of these superconducting phases in order to provide key elements and indications to be employed for the detection of the most favorable inter-orbital superconductivity. The analysis is based on the solution of the Bogoliubov-de Gennes (BdG) equation for the evaluation of the low-energy spectral excitations both in the bulk and at the edge of the superconductor for both and . The matrix Hamiltonian in momentum space is given by

(17) |

with being the normal state Hamiltonian.

### iv.1 Bulk energy spectrum and topological superconductivity

In order to determine the excitation spectrum, we solve the BdG equations for both the and configurations. For convenience we introduce the gap amplitude and we set the components of the -vectors to be

(18) |

for and

(19) |

for state. Here, the parameter is set as a scale of energy.

We start focusing on the doping regime of four bands at the Fermi level. In this case, the state has a fully gapped electronic structure for all the bands at the Fermi level as demonstrated by the inspection of the in-plane angular dependence of the gap magnitude (see Figs. 5(b), (c), (d) and (e)).

On the other hand, representation has point nodes in the -M direction as shown in Fig. 6. It is interesting to further investigate the nature of the nodal phase by asking whether the existence of the nodes is related to a non-vanishing topological invariant. Since the model Hamiltonian owes particle-hole and time-reversal symmetry, one can define a chiral operator as a product of the particle-hole and time-reversal operators. Since the chiral symmetry operator anticommutes with , by employing a unitary transformation rotating the basis in the eigenbasis of , the Hamiltonian can be put in an off-diagonal form with antidiagonal blocks. Hence, the determinant of each block can be put in a complex polar form and, as long as the eigenvalues are non-zero, it can be used to obtain a winding number by evaluating its trajectory in the complex plane. On a general ground, we point out that the number of singularities in the phase of the determinant is a topological invariant Tewari and Sau (2012) because it cannot change without the amplitude going to zero, thus implying a gap closing and a topological phase transition. For this symmetry class, then, one can associate and determine the winding number around each node by following, for instance, the approach already applied successfully in the Refs. Yada et al. (2011); Sato et al. (2011); Brydon et al. (2011).

The chiral, particle-hole and time-reversal operators are expressed as

(20) | ||||

(21) | ||||

(22) |

Here, , and denote the unit matrix, the identity matrix in the particle-hole space and the orbital space, respectively. Since we consider time-reversal symmetric pairings, the chiral operator anticommutes with the Hamiltonian,

(23) |

One can then introduce a unitary matrix which diagonalizes the chiral operator ,

(24) | ||||

(25) |

In this basis the BdG Hamiltonian is block antidiagonalized by ,

(26) | |||

(27) |

Then, the determinant of matrix block can be put in a complex polar form and, as long as the eigenvalues are non-zero, it can be used to obtain the winding number by evaluating its trajectory in the complex plane as

(28) | ||||

in Eq. (28) is a closed line contour which encloses a given node as schematically shown in Fig. 7(b). From the explicit calculation, we find that the amplitude of is (see Figs. 6(a) and 7(a)).

We generally find that from two to six point nodes can occur along the -M direction and their number is related to that of the Fermi surfaces. Interestingly, the position of the point nodes is not fixed and pinned to the lines of the Fermi surface in the normal state. In general their position along the diagonal of the Brillouin zone depends on the amplitude and indirectly on the values of the spin-orbit and inversion asymmetry couplings. Then, two adjacent point nodes with opposite winding numbers can, in principle, be moved until they merge and then disappear by opening a gap in the excitation spectrum. This behavior is generally demonstrated in Fig. 8. A phase diagram can be determined in terms of the amplitude and the chemical potential . The nodal superconductor can undergo different types of Lifshitz transitions and in general those occurring in the normal state are not linked to the nodal merging in the superconducting phase. Indeed, one of the characteristic feature of the nodal superconductor is that by changing the filling, through , one can drive a transition from two to four and six point nodes independently of the number of bands crossing the Fermi level in the normal state. It is rather the strength of to play an important role in tuning the nodal superconductor. An increase of tends to reduce the number of nodes until a fully gapped phase appears. Since the critical lines are sensitive to the spin-orbit and inversion asymmetry couplings, one can get line crossings which allow to have multiple merging of nodes such that the superconductor can even undergo a direct transition from six to two at (Figs. 8(e) and (d)) or from four to zero point nodes, as for instance nearby the crossing between the blue and orange lines at in the Fig. 8.

Since the position of the point nodes are fixed the each Fermi surface in the limit of small and their distances in the Brillouin zone increases with the level splitting by and , a larger is required to annihilate the point nodes when both and grow in amplitude as demonstrated by the shift of the green and blue critical lines in Figs. 8(a), (b) and (c) for different values of , and Figs. 8(d), (b) and (e) in terms of . When considering these results in the context of two-dimensional superconductors which emerge at the surface or interface of band insulators we observe that the achieved topological transitions can be driven by gate voltage and temperature, since and are tunable by electric fields and the amplitude of can be controlled by the temperature and the electric field as well.

### iv.2 Local density of states at the edge of the superconductor

Having established that the nodes in the are protected by a non-vanishing winding number, one can expect that flat zero energy surface Andreev bound states (SABS) occur at the boundary of the superconductor.

In this subsection, we investigate the SABS and the local density of states (LDOS) for two different terminations of the two-dimensional superconductor, i.e. (100) and (110) oriented edges. We start by discussing the LDOS for the (100) and (110) edges at representative values of , and , and by varying the chemical potential in order to compare the cases with a different number of point nodes in the bulk energy spectrum at , and as shown in Figs. 9(a), (b) and (c), respectively.

As expected, the momentum resolved LDOS indicate that zero energy SABS can be observed but only for specific orientations of the edge. Indeed, as reported in Figs. 9(d)-(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 is directly related to the presence of a non-trivial winding number that is protecting the point nodes. For the (110) edge, isolated point nodes exist in the surface Brillouin zone and they have winding numbers with opposite sign. Then, the ZESABS, which connect the nodes with a positive and negative winding number, emerge in the gap. On the other hand, when considering the (100) oriented termination, the winding numbers for positive and negative are completely opposite in sign and they cancel each other when projected on the (100) surface Brillouin zone. Thus, flat zero energy states cannot occur for the (100) edge. Nevertheless, helical edge modes are observed inside the energy gap as demonstrated in Fig. 9(d). This is because the Majorana edge modes with positive and negative chirality can couple, get split and acquire a dispersion. The differences in the edge ABS also manifest in the momemtum integrated LDOS. For the (110) edge, due to the presence of the ZESABS, the LDOS shows pronounced zero energy peaks (see dash-dotted line in Figs. 9(j), (k) and (l)). On the other hand, for the (100) boundary, they lead to a broad peak or exhibit many narrow spectral structures reflecting the complex dispersion of the edge states.

Finally, we discuss the dependence of LDOS at zero energy, i.e. . For the (110) edge, the zero energy peak mainly originates from the zero energy flat band. The height of the zero energy peak can be then characterized by (i) the strength of the localization of the edge state and (ii) the total length of the ZESABS within the surface Brillouin zone. The strength of the localization is defined by the inverse of the localization length and . In other words, the peak height generally increases with . On the other hand, as shown in Fig. 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 surfaces configuration. In this case, the total length of the zero energy flat band is roughly estimated as for and zero for where is the Fermi surface splitting along the -M direction and is a critical value above which the point-nodes disappear. Then, the height of the zero energy peak is proportional to for and vanishes for . This is a nonmonotonic dome-shaped behaviour of the ZELDOS as a function of . The explicit profile can be seen in Fig. 10 at and . For , the point nodes still exist in this parameter regime and the height of the zero energy peak develops with . Thus, we have that the ZESABS get strongly renormalized and they are tunable by a variation of the electron filling () and amplitude of the order parameter as shown in Fig. 8.

## V Discussion and summary

We investigated and determined the possible superconducting phases arising from inter-orbital pairing in an electronic environment marked by spin-orbit coupling and inversion symmetry breaking while focusing on momentum independent paired configurations. One remarkable aspect is that, although the inversion symmetry is absent, one can have symmetry allowed solutions that avoid mixing of spin-triplet and spin-singlet configurations. Importantly, states with only spin-triplet pairing can be stabilized in a large portion of the phase diagram.

Within those spin-triplet superconducting states, we unveil an unconventional type of topological phase in two-dimensional superconductors which arises from the interplay of spin-orbit coupling and orbitally-driven inversion-symmetry breaking. Since for this kind of model systems the atomic physics plays a relevant role and inevitably tends to yield orbital entanglement close to the Fermi level, we assume that local inter-orbital pairing is the dominant attractive interaction. As already mentioned, this type of pairing in the presence of inversion symmetry breaking allows to have solutions that do not mix spin-singlet and triplet configurations. The orbital-singlet/spin-triplet superconducting phase can have a topological nature with distinctive spin-orbital fingerprints in the low energy excitations spectra which make it fundamentally different from the topological configuration that is usually obtained in single band non-centrosymmetric superconductors. Here, a remarkable finding is that, contrary to a common view of an isotropic pairing structure leading to a fully gapped spectrum, a nodal superconductivity can be achieved when considering isotropic spin-triplet pairing. Although in a different context, we notice that akin paths for the generation of an anomalous nodal-line superconductor can be also encountered when local spin-singlet pairing occur in antiferromagnetic semimetals Brzezicki and Cuoco (2018).

In the present study, for a given symmetry, the superconducting phase can exhibit point nodes that are protected by a non-vanishing winding number. The most striking feature of the disclosed topological superconductivity is expressed by its being prone to both topological and Lifshitz-type transitions upon different driving mechanisms and interactions, e.g. when tuning the strength of intrinsic spin-orbit and orbital couplings or by varying doping and the amplitude of order parameter (e.g. by temperature). The essence of such topologically and electronically tunable superconductivity phase is encoded in the fundamental observation of having a control of the nodes position in the Brillouin zone. Indeed, the location of the point nodes is not determined by the symmetry of the order parameter in the momentum space, contrary to the single band non-centrosymmetric system, but rather it is a non-trivial consequence of the interplay between spin-triplet pairing and the spin-orbital character of the electronic structure. In particular, their position and existence in the Brillouin zone can be manipulated, through various types of Lifshitz transitions, if one varies the chemical potential, the amplitude of the spin-triplet order parameter, the inversion symmetry breaking term, and the atomic spin-orbit coupling. While electron doping can induce a change in the number of Fermi surfaces, such electronic transition is not always accompanied by a variation in the number of nodes within the superconducting state. This behavior allows to explore different physical scenarios which single out notable experimental paths for the detection of the targeted topological phase. Due to the strong sensitivity of the topological and Lifshitz transitions with respect to the strength of the superconducting order parameter, one can foresee the possibility to observe an extraordinary reconstruction of the superconducting state both in the bulk and at the edge by employing the temperature to drive the pairing order parameter to a vanishing value (i.e. at the critical temperature) starting from a given strength at zero temperature. Then, a substantial thermal reorganization of the superconducting phase can be obtained. While a variation of the number of nodes in the low energy excitations spectra can result difficult to be extracted by thermodynamic bulk measurements, we find that the electronic structure at the edge of the superconductor generally undergoes a dramatic reconstruction which manifests into a non-monotonous behaviour of the zero bias conductance or in an unconventional thermal dependence of the in-gap states. Another important detection scheme of the examined spin-triplet superconductivity emerges when considering its sensitivity to the doping or to the strength of the inversion symmetry breaking coupling which can be accessed by applying an electrostatic gating or pressure. Such gate/distortive control can find interesting applications especially when considering two-dimensional electron gas systems.

Another interesting feature of the multiple-nodes topological superconducting phase is given by the strong sensitivity of the edge states to the geometric termination as demonstrated in Fig. 9. This is indeed a consequence of the presence of nodes with an opposite sign winding number within the Brillouin zone. Hence, when considering the electronic transport along a profile which is averaging different terminations it is natural to expect multiple in-gap features.

Due to the multi-orbital character of the superconducting state, we expect that non-trivial odd-in-time pair amplitudes are also generated. In particular, we predict that both local odd-in-time spin-singlet and triplet states can be obtained in the bulk and at the edge. The local spin-singlet odd-in-time pair correlations are an exquisite consequence of the multi-orbital superconducting phase. Accessing the nature of their competition/cooperation and its connection to the nodal superconducting phase is a general and relevant problem in relation to the generation, manipulation and control of odd-in-time pair amplitudes.

Finally, we point out that the examined model Hamiltonian is generally applicable to two-dimensional layered materials, in the low/intermediate doping regime, having -bands at the Fermi level and subjected to both atomic spin-orbit coupling and inversion symmetry breaking, for instance due to lattice distortions and bond bending. Many candidate material cases can be encountered in the family of transition metal oxides. There, unconventional low-dimensional quantum liquids with low electron density can be obtained by engineering a two-dimensional electron gas (2DEG) at polar-non polar interfaces between two band insulators, on the surface of band insulators (i.e. (STO)) or by designing single monolayer heterostructures, ultra-thin films or superlattices. A paradigmatic case of superconducting 2DEG is provided by the (LAO/STO) heterostructure Cen et al. (2008); Caviglia et al. (2008); Schneider et al. (2009). Recent experimental observations by tunnelling spectroscopy have pointed out that the superconducting state can be unconventional due to the occurrence of in-gap states with peaks at zero and finite energies Kuerten et al. (2017). Although these peaks may be associated to a variety of concomitant physical mechanisms, e.g. surface Andreev bound states Buchholtz and Zwicknagl (1981); Hara and Nagai (1986); Hu (1994); Tanaka and Kashiwaya (1995); Kashiwaya and Tanaka (2000); Löfwander et al. (2001), anomalous proximity effect by odd-frequency spin-triplet pairing Tanaka and Kashiwaya (2004); Tanaka et al. (2005a); Asano et al. (2006); Tanaka et al. (2005b); Tanaka and Golubov (2007); Tanaka et al. (2007a, b), bound states due to the presence of magnetic impurities Ebisu et al. (2015), its nature can provide key info about the pairing symmetry of the superconductor. Also the observation of Josephson currents Stornaiuolo et al. (2017) across a constriction in the 2DEG confirms a fundamental unconventional nature of the superconducting state Tanaka and Kashiwaya (1996); Barash et al. (1996); Tanaka and Kashiwaya (1997). A common aspect emerging from the two different spectroscopic probes is that the superconducting state seems to have a multi-component character. Although it is not easy to disentangle the various contributions which may affect the superconducting phase in the 2DEG, we speculate that the proposed topological phase can be also included within the possible candidates for addressing the puzzling properties of the oxide interface superconductivity.

## Vi Acknowledgments

This work was supported by a JSPS KAKENHI (Grants No. No. JP15H05853 and JPH06136, JP15H03686), and "Oxide super-spin" core-to-Core program.

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