Topological superconductivity and unconventional pairing in oxide interfaces

To pinpoint the microscopic mechanism for superconductivity has proven to be one of the most outstanding challenges in the physics of correlated quantum matter. Thus far, the most direct evidence for an electronic pairing mechanism is the observation of a new symmetry of the order-parameter, as done in the cuprate high-temperature superconductors. Like distinctions based on the symmetry of a locally defined order-parameter, global, topological invariants allow for a sharp discrimination between states of matter that cannot be transformed into each other adiabatically. Here we propose an unconventional pairing state for the electron fluid in two-dimensional oxide interfaces and establish a direct link to the emergence of nontrivial topological invariants. Topological superconductivity and Majorana edge states can then be used to detect the microscopic origin for superconductivity. In addition, we show that also the density wave states that compete with superconductivity sensitively depend on the nature of the pairing interaction. Our conclusion is based on the special role played by the spin-orbit coupling and the shape of the Fermi surface in SrTiO$_{3}$/LaAlO$_{3}$-interfaces and closely related systems.

The two-dimensional electron fluid that forms 1 at the interface between the insulators SrTiO 3 and LaAlO 3 is an example of an engineered quantum system, where a new state of matter emerges as one combines the appropriate building blocks. The subsequent discovery of superconductivity 2 in the interface, along with the ability to control the ground state via applied electric fields 3 opened up intense research. The key open question is whether electronic correlations promote new states of matter, such as unconventional superconductivity or novel magnetic states [4][5][6][7][8] and how such phases are related to each other.
New states of matter can be sharply distinguished from conventional behavior when they break a symmetry or differ in their topology. The nontrivial consequences of the mapping from momentum space to the space of Hamiltonians, as found in topological insulators and superconductors, have recently had a major impact on solid state physics 9,10 . Here we propose a new electronic pairing mechanism for superconductivity in oxide interfaces that is due to the exchange of particle-hole excitations and that leads to topological superconductivity with Majorana bound states and related nontrivial topological aspects. Specifically, we find a time-reversal preserving topological superconductor that has attracted recent attention [11][12][13][14][15][16] . In contrast, conventional electron-phonon coupling in the same system would lead to a topologically trivial state. We also study competing states, expected to emerge nearby superconductivity in the phase diagram. For a conventional pairing mechanism we find charge density wave order, while an in-plane spin density wave with magnetic vortices competes with unconventional superconductivity.

I. INTERACTING LOW-ENERGY MODEL
The crucial states near the Fermi energy of the oxide interface are made up of titanium 3d xz and 3d yz orbitals [17][18][19] . The orientation of the electron clouds of the 3d-orbitals leads to a wave function overlap along the x-direction that is much larger for d xz states compared to d yz , and vice versa for the y-direction. Each orbital is then characterized by a light mass m l and a heavy mass m h , leading to the experimentally observed strongly anisotropic electronic structure 17,19 . For example, the energy of the d xz states can be described by where m h /m l 15 · · · 30. ε yz follows from Eq. (1) by interchanging k x and k y . In addition, the electronic properties of the polar interface between insulating oxides is strongly affected by the spin-orbit interaction. Due to the Dresselhaus-Rashba effect 20,21 , the electronic states experience a momentum dependent splitting and mixing of spin-states, naturally explaining magneto-transport experiments 22,23 . The effect might also be responsible for the observed phase separation in interfaces 24 . Focusing on the d xz and d yz states, the most general form up to linear order in momentum that is consistent with the C 4v -point group symmetry and time-reversal invariance is given by H so (k) = 1 2 λτ 2 σ 3 + α 0 τ 0 (k x σ 2 − k y σ 1 ) + α 1 τ 1 (k x σ 1 − k y σ 2 ) + α 3 τ 3 (k x σ 2 + k y σ 1 ) , where the Pauli matrices σ i and τ j (i, j = 0, . . . 3) act in spin and orbital space, respectively. Projecting out the d xy band that is closest in energy and shifted by 0 and including the atomic spin-orbit coupling H so = λL · s we FIG. 1. Part (a) shows the spectrum of the effective two-band Hamiltonian using the realistic parameter stated in the main text. In this paper, we restrict the analysis on the 4 most strongly nested subspaces (highlighted in red and blue). The orbital weight (color) and orientation of the spin (red arrow) are illustrated in (b) and (c) for the outer and inner Fermi surface, respectively. Note that, as a consequence of time-reversal and π-rotation symmetry about the z-axis, the spin has to lie in the xy-plane.
In Fig. 1(a) we show the bands that result from the combination of the anisotropic masses in Eq. (1) and the spin-orbit coupling (2). Two of the four bands are pushed to higher energies by the atomic spin-orbit coupling λτ 2 σ 3 and can thus be neglected for the following low-energy analysis as long as the chemical potential is tuned sufficiently far away from the bottom of these bands. The remaining two bands are split by the Dresselhaus-Rashba coupling and show strong nesting in the highlighted regions. We emphasize the similarity of the Fermi surface to the one reported in Ref. 19 for the surface states of SrTiO 3 . The nesting is a consequence of the mass anisotropy and becomes exact in the limit m l /m h → 0.
This allows us to use a low-energy theory that involves only the degrees of freedom in the vicinity of the most parallel slices of the Fermi surface. In total, there are four equivalent strongly nested subspaces that are related by the point group symmetries. Without loss of generality, let us focus on, e.g., the one indicated in red in Fig. 1(a). In this subset of momentum space, we introduce helicity creation and annihilation operators c (σ,j) and c † (σ,j) which diagonalize the quadratic part of the Hamiltonian. Here σ = ± refers to the sign of k x and j = 1 (j = 2) denotes the outer (inner) Fermi surface. To relate these operators to observables, Fig. 1(b) and (c) show the spin-orientation and the orbital weight of the states in the vicinity of the outer and inner Fermi surface, respectively.
There are two types of interaction processes allowed by momentum conservation which we will refer to as backscattering and forward scattering. The most general momentum independent backscattering term is given by where (σ = ±) We emphasize that from now on the Pauli matrices σ s , as in Eq. (4), do not describe the physical spin but rather act in the abstract isospin space of the local helicity operators. The momentum of the operator c σ,j is measured relative to the center σk j of the corresponding red region in Fig. 1(a). Using the phase convention for the eigenstates defined in the Supplementary Information, one finds that the π-rotation symmetry with respect to the z-axis implies that u has to be symmetric, u T = u. The remaining symmetries of the point group then fully determine the interaction in the other three most strongly nested subspaces. In addition, time-reversal symmetry imposes the constraint u s,s = 0 if either s = 2 or s = 2. Let us first assume that the cutoff Λ ⊥ for the momenta perpendicular to the Fermi surface can be chosen smaller than the distance between the inner and outer Fermi surface. This means that the red regions in Fig. 1(a) do not overlap. In this situation, momentum conservation rules out further interaction processes such that only u 00 , u 11 = −u 22 , u 33 and u 30 = u 03 can be non-zero. Here V and Λ denote the volume of the system and the cutoff of the nested subspaces tangential to the Fermi surface. Only in the regimes (I) and (II) the couplings diverge indicating that the system develops an instability. The red (blue) regions correspond to the bare couplings for a microscopically repulsive (attractive) interaction. The schematic phase diagrams taking into account finite mass anisotropies are shown in part (b). The non-flowing couplings determine the properties of the charge density wave as shown in (c) and (d) for the unconventional and conventional superconductor, respectively. The nomenclature of the phases is explained in the main text.
In case of forward scattering, where all four fermions have the same index σ, the combination of Fermi statistics and point symmetries leads to only one independent coupling constant.

II. PAIRING INSTABILITY AND TOPOLOGICAL SUPERCONDUCTIVITY
Having derived the interacting low-energy Hamiltonian, we can now deduce the associated instabilities. We perform a standard fermionic one-loop Wilson renormalization group (RG) calculation 26 , in which high-energy degrees of freedom are successively integrated out yielding an effective Hamiltonian with renormalized coupling constants. If, during this procedure, some of the couplings diverge, the system will develop an instability. Following Refs. 27 and 28 we identify the physical nature of this instability by determining the order parameter that has the highest transition temperature, allowing for all possible (momentum independent) particle-hole and particle-particle ordered states: where α and β are double indices comprising helicity σ = ± and the Fermi surface sheet index j = 1, 2.
Near the Fermi surface, we linearize the band dispersion (k) ±v j k ⊥ with k ⊥ denoting the component of the momentum perpendicular to the Fermi surface. For simplicity, let us first focus on the situation v 1 = v 2 which is quantitatively a good approximation even when the chemical potential gets closer to the bottom of these bands. Below, we will also discuss the more general case v 1 = v 2 .
If v 1 = v 2 , only u 11 and u 33 out of the five coupling constants flow as shown in Fig. 2(a). We find two regimes, denoted by (I) and (II), where the running couplings diverge. In both cases, the instability is of superconducting type characterized by the two non-zero anomalous expectation values ∆ SC (−,j),(+,j) with j = 1, 2. As expected, we only have intra-Fermi surface pairing, i.e. only Kramer partners are paired. In region (I), the superconducting order parameters of the nearby Fermi surfaces have opposite sign whereas in (II) the sign is the same. The corresponding superconducting states will be denoted by SC +− and SC ++ , respectively. In the region (III), none of the coupling constants diverge which means that, for sufficiently small bare couplings, the system will not develop any instability and, thus, reside in the metallic phase.
To unveil the microscopic pairing mechanism of the two superconducting states, we start from a repulsive Coulomb interaction between the d-orbitals and project onto the effective low-energy theory. This places us into region (I) of the RG flow in Fig. 2(a). In contrast, an attractive interaction due to electron-phonon coupling would lead to initial couplings in region (II). Consequently, SC ++ results from conventional electronphonon pairing, whereas SC +− is an unconventional superconductor, where particle-hole fluctuations effectively change the sign of u 33 .
Both SC +− and SC ++ respect time-reversal symmetry as far as the degrees of freedom of the nested subspaces are concerned. It is natural to assume that this holds for the entire Fermi surface and that, in addition, the system does not break the point symmetries relating the nested segments. In this case the gap is finite on the entire Fermi surface as seen in recent experiments 29 . Being fully gapped, it is natural to ask whether the time-reversal invariant two-dimensional superconductor (class 30 DIII) is topologically trivial or nontrivial 31 , which is of great interest as it strongly influences its physical properties. The most prominent fea-ture of a nontrivial topological superconductor is the appearance of spin-filtered counter propagating Majorana modes at its edge when surrounded by a trivial phase 32 . It has been shown 33 that the associated topological invariant N ∈ Z 2 is fully determined by the sign of the paring field on the Fermi surfaces. It holds where the product involves all Fermi surfaces, ψ j and ∆ j denote the wave function of the non-interacting part of the Hamiltonian and the pairing field at an arbitrary point on the jth Fermi surface. Furthermore, m j is the number of time-reversal invariant points enclosed by the jth Fermi surface and T is the unitary part of the timereversal operator, given by T = iτ 0 σ y in the basis of Eq. (2). As, in the present case, both Fermi surfaces enclose only one time-reversal invariant point, the superconductor is topological (trivial) if the sign of δ j is different (identical) on the two Fermi surfaces. Inserting the order parameters derived above, we obtain the pairing Hamiltonian with γ 0 = u 00 + 2u 11 + u 33 , γ 3 = 2u 30 for the superconductor SC ++ and γ 0 = 2u 30 , γ 3 = u 00 − 2u 11 + u 33 for the SC +− -state. Calculating δ j in Eq. (6), one finds (see Supplementary Information for details) that the superconductor is topological if |γ 0 | < |γ 3 | and trivial for the reversed inequality sign. At |γ 0 | = |γ 3 |, the gap closes as is characteristic for a topological phase transition. Recalling the flow depicted in Fig. 2(a), one immediately sees that SC ++ is trivial, whereas SC +− is a topological superconductor. Accordingly, the experimental observation of topological features of the superconducting state implies that the pairing mechanism must be unconventional as it is the case for SC +− . Vice versa, a trivial state is only consistent with conventional, electronphonon induced superconductivity. We emphasize the difference of this result to recent work [34][35][36][37] proposing the emergence of Majorana fermions in the heterostructure. In Refs. 34-37, Majorana physics is predicted to arise from the coexistence of magnetism and superconductivity. This means that (physical, spin-1/2) time-reversal symmetry is broken, whereas the SC +− -state respects time-reversal symmetry.

III. COMPETING PHASES AND SPIN TEXTURES
Eventually, our RG flow will always favor a superconducting state. However, by successively reducing the characteristic energy scale, we are increasingly sensitive to details of the low-energy theory and, consequently, the fact that the nesting is not perfect for m l /m h > 0 becomes relevant. In this sense, any finite m l /m h introduces a cutoff to the flow. If the flow is cut off before the superconducting instabilities take place, other competing phases can emerge, as illustrated in Fig. 2(b). Depending on the values of the non-flowing coupling constants u 00 and u 30 , one can either find a charge density wave (CDW 12 ), three different spin density waves (SDW 11 , SDW 22 , SDW 12 ) or the corresponding superconducting states are dominant for arbitrary m l /m h as shown in Fig. 2(c) and (d). The superscripts in the density waves CDW ij and SDW ij refer to the particle-hole expectation value ∆ DW (−,i),(+,j) (and i ↔ j if i = j) that is non-zero in the respective phase. The difference between CDW 12 and SDW 12 is the relative sign of ∆ DW and ∆ DW (−,j),(+,i) , rendering the order parameter symmetric and antisymmetric under time-reversal in the former and in the latter case, respectively.
The spatial structure of the charge and spin density waves can easily be determined from the wave functions of the system and the order parameters ∆ DW α,β . As in the case of the superconducting order parameter, we assume that no additional point group symmetry is broken. In the case of the CDW 12 -phase, one then finds that the local charge density is given by where Q 12 = k 1 +k 2 is the associated nesting vector. The first contribution stems solely from the nested subspace, highlighted in red in Fig. 1(a) and the ellipsis stands for the terms emanating from the remaining three subspaces which are fully determined by the π/2-rotation and reflection symmetry at the xz-axis. The resulting charge profile is illustrated in Fig. 3(a). Note that the periodicity crucially depends on the ratio of the x-and y-component of the nesting vector Q 12 . Similarly, the spatial structure of the spin density waves SDW 12 and SDW 11 , SDW 22 can be calculated (for details see Supplementary Information) yielding the textures shown in Fig. 3(b) and (c), respectively. Here we have used that, in the red regions of Fig. 1(a), the spins are approximately aligned along the y-axis (see Fig. 1(b) and (c)). Within this approximation, the expectation value of the spin lies in the xy-plane in case of the spin density phase SDW 12 . The two-dimensional vector field is therefore a lattice of vortices both with positive and negative winding number. In the phases SDW 11 and SDW 22 , the spin is free to rotate in three dimensions. One finds a complicated periodic arrangement of isolated Skyrmions and Antiskyrmions as well as closely bound Skyrmion-Antiskyrmion pairs (see Fig. 3(d)). The emergence of a Skyrmion lattice, which leads to interesting physical effects (see e.g. Ref. 38), is consistent with recent work 8,39 pointing out that these magnetic topological defects naturally appear as solutions of the Ginzburg Landau equations for systems with spin-orbit interaction.
On top of that, the difference between the density wave phases in Fig. 2 conducting states SC +− and SC ++ can be exploited to gain information about the pairing mechanism in the heterostructure. As the orbital contribution to the magnetization is negligible for large mass anisotropies, the experimental observation of in-plane magnetization 5 is only consistent with the SDW 12 -state. This implies that the superconducting phase of SrTiO 3 /LaAlO 3 is supposed to be unconventional and topologically nontrivial.
As already stated above, we have also considered the case of different Fermi velocities, v 1 = v 2 (see Supplementary Information for more details of the analysis). Then all four backscattering couplings flow. Nonetheless, exactly as before, the leading instability is generically superconducting for sufficiently large mass anisotropies. However, in the present case, the anomalous expectation value ∆ SC (−,j),(+,j) is only finite on the Fermi surface with the larger Fermi velocity. Remarkably, we still find that the superconductor resulting from the conventional electron-phonon pairing mechanism is topologically triv-ial, whereas the unconventional superconductor is nontrivial. This proves that the correspondence between the pairing mechanism and the topological properties of the superconducting phases in the heterostructure holds irrespective of the values of the Fermi velocities. For completeness, we also considered the case of very weak spinorbit interaction where the energetic cutoff of the lowenergy model is much larger than the spin-orbit splitting. Then the red regions in Fig. 1(a) overlap pairwise and, consequently, momentum conservation is much less restrictive making more backscattering terms possible. Surprisingly, still in this situation, the observation of a topologically nontrivial superconducting phase is only consistent with the pairing mechanism being unconventional.
The phase diagram of the two-dimensional electron fluid that forms at the interface between the perovskite oxides LaAlO 3 and SrTiO 3 combines two fascinating notions of condensed matter physics: Topology and uncon-ventional superconductivity. We find that, very generically, the observation of signatures of topologically nontrivial superconductivity, such as the appearance of Majorana bond states, directly implies that the underlying pairing mechanism must be unconventional. In addition, the spin density wave phases competing with topological superconductivity show topological spatial textures as well. Depending on the value of the coupling constants, we find lattices of both Skyrmions and vortices.

A. General symmetry analysis
The symmetry classification of the electron-electron interaction can be performed efficiently by introducing a specific phase convention for the local eigenbasis of the free Hamiltonian. Here we define this convention which will then be used to represent the point symmetries and time-reversal on the helicity operators c, c † . Finally, all possible momentum independent interaction terms within the most strongly nested subspaces (see Fig. 4(a)) will be derived. We consider all three relevant cases, non-overlapping low-energy subspaces with both identical and different Fermi velocities as well as quasi-degenerate Fermi surfaces (see Fig. 4(b)-(d)), simultaneously.

Phase convention and representation of symmetries
Using a path-integral representation, the quadratic part of the theory can be written as where k ≡ (ω n , k) and Ψ, Ψ are four-component Grassmann fields describing spinful Fermions in the two orbitals {xz, yz}. Furthermore, H is the Hamiltonian defined in the main text characterized by the anisotropic masses (1) and the spin-orbit coupling in Eq. (2).
We diagonalize S 0 by performing the unitary transformation where with φ α (k) denoting an eigenvector of H(k). As explained in the main text, we can restrict the analysis of instabilities to one of the most strongly nested subspaces. We choose the subspace highlighted in red in Fig. 4(a) and introduce helicity fields c (σ,j) andc (σ,j) in the local coordinate systems yielding after linearizing the spectrum. Here s 1 = +1, s 2 = −1 and η denotes the spin-orbit splitting in the case of quasidegenerate Fermi surfaces. For stronger spin-orbit coupling, where the four red regions in Fig. 4(a) are disjoint, one has η = 0 by construction. In Eq. (12) and in the following, we use the compact notation k ≡ (ω n , k , k ⊥ ) and where Λ ⊥ and Λ are the momentum cutoffs normal and tangential to the Fermi surface. If the Fermi velocities are identical, we will use the notation introduced in the main text where j = 1 (j = 2) refers to the outer (inner) Fermi surface. If this is not the case, it will be most convenient to label the fields such that v 1 > v 2 .
To make the helicity operators unique, we have to fix the phases of the eigenstates in Eq. (11). This is achieved by exploiting the invariance of the Hamiltonian under πrotation R c2 and time-reversal Θ. The former symmetry implies that and hence we can construct the eigenstates with negative k x from those with k x > 0 via Consecutive application of time-reversal and πrotation leads to the k-space local antiunitary symmetry of the Hamiltonian. If the Fermi surfaces in Fig. 4 are non-degenerate, we can adjust the phases of the eigenstates such that for k x > 0. From Eq. (15), it follows that Eq. (17) actually holds also for k x < 0. In addition, we have shown that Eq. (17) can still be satisfied if the Fermi surfaces are exactly degenerate.
Having fixed the phases of the local eigenstates, the representation of time-reversal and π-rotation symmetry on the helicity fields is well defined. Note that the remaining elements of the point group C 4v cannot be represented in the most strongly nested subspace as these operations act between different subspaces. For the very same reason, however, the remaining symmetries are also irrelevant when deriving the most general interaction within one the subspaces.
Time-reversal acts according to in the basis of Eq. (9) and, consequently, as in the local eigenbasis. Using Eqs. (15) and (17), we can write and, thus, conclude Similarly, for the π-rotation symmetry, one finds and the same forc.

Symmetry analysis of the interaction
Now we will derive the most general momentum independent interaction of the low-energy theory consistent with the symmetries of the system. Let us write where the Greek letters are double indices comprising σ = ± and j = 1, 2. The tensor W has to satisfy due to Hermiticity and, as a consequence of Fermi statistics, can be chosen such that It turns out that the dimensionless parameterization, with is very convenient for the following analysis. In Eq. (27), we have already taken into account Eq. (25) and that only forward scattering (described by V ) and backscattering (W ) are allowed by momentum conservation, which is directly clear from Fig. 4. Throughout this work, we assume that Umklapp processes are not possible. Due to Fermi statistics, the forward scattering tensors must have the form whereas the backscattering tensor has 16 degrees of freedom, which we parametrize according to The Hermiticity constraint in Eq. (24) implies that g 0 (σ), g ss ∈ R. Note that g ss ∝ u ss with u ss used in the main text to define the backscattering terms.
Next, let us derive the constraints resulting from πrotation symmetry. Demanding that Eq. (23) be invariant under Eq. (22), we find The former conditions means that, as expected, forward scattering is identical for the patches centered around k j and −k j . Consequently, all forward scattering processes are characterized by one coupling constant g 0 ≡ g 0 (+) = g 0 (−). Applying the expansion (29), the second constraint is equivalent to g T = g as stated in the main text.
Similarly, to make the interaction time-reversal symmetric, we require invariance of Eq. (23) under Eq. (21). Again using the parameterization (27), we find that V is not further restricted, whereas the backscattering tensor has to satisfy In the representation (29) this is equivalent to demanding g s,s = 0 if either s = 2 or s = 2.
Consequently, in the limit of weak spin-orbit interaction, where 0 < η v j Λ ⊥ and the red regions in Fig. 4 overlap pairwise, the backscattering tensor is given by Eq. (29) with g =    g 00 g 10 0 g 30 g 10 g 11 0 g 31 0 0 g 22 0 g 30 g 31 0 g 33 However, if the four most strongly nested subspaces are disjoint, momentum conservation rules out further backscattering terms. Writing down all interaction terms that are consistent with momentum conservation and expanding them in Pauli matrices as in Eq. (29), one finds that only g 00 , g 11 = −g 22 , g 33 , g 21 = g 12 , g 30 and g 03 can be finite. Comparison with Eq. (32) then yields FIG. 5. Diagrams to be evaluated for the RG. Closed loops involve integration over fast modes only.

B. Wilson RG
In this part, we provide more details of the RG calculation and discuss the flow equations for all three regimes in Fig. 4(b)-(d).

Generic form of the RG equations
In the Wilson approach, applied to Fermions with a finite Fermi surface in Ref. 26, fast modes with momenta Λ ⊥ e −∆l < k ⊥ < Λ ⊥ , ∆l > 0, are integrated out yielding, after proper rescaling, an effective action with renormalized parameters. The quadratic part of the action simply splits into the contributions from the fast and slow modes, whereas the interaction leads to nontrivial terms in the effective action that can only be treated perturbatively.
The corresponding one-loop contributions are shown diagrammatically in Fig. 5. The tadpole diagram, Fig. 5(a), represents the impact of the interaction on the bands of the system. Here and in the following, we will neglect this contribution to the RG flow, since, by definition, we assume that all possible interaction effects on the chemical potential and on the spin-orbit coupling have already been accounted for by S 0 .
The other two diagrams, Fig. 5(b) and (c), are usually referred to as ZS and BCS, respectively, and lead to the corrections and of the interaction tensor W. In Eqs. (34) and (35), we have introduced the Green's function of fast modes. Note that ∆ BCS W αβ γδ and ∆ ZS W αβ γδ have been symmetrized to satisfy Eq. (25). Evaluating the shell integrals asymptotically in the limit ∆l → 0 and using the dimensionless parameterization (26), one finds the tensor valued RG equation dω αβ γδ (l) dl = 1 − δ σµ,σν ω αβ µν (l)ω µν γδ (l) p=+,− 1 x jµ (1 + p κ jµ ) + x jν (1 + p κ jν ) where x j := v j /v 1 and κ j := s j η/(Λ ⊥ v j ) have been defined. From Eqs. (27) and (37), it is already clear that i.e., irrespective of the Fermi velocities and the strength of the spin-orbit coupling, the forward scattering terms are not renormalized.
To simplify the following analysis, let us set η → 0 in the flow equation (37). Note that this rules out only the intermediate regime where the energetic cutoff is of the same order as the spin-orbit splitting η, since, for stronger spin-orbit interaction, we have η = 0 by construction (see Fig. 4(b) and (c)). Inserting (σ α , σ β , σ γ , σ δ ) = (−, +, +, −) in Eq. (37) of the backscattering tensor. Here the contribution of the first and second line emanate from the ZS and BCS diagram, respectively. Next, we will restate Eq. (39) in terms of the coupling constants g ss for the two cases of large spin-orbit coupling and quasi-degenerate Fermi surfaces.

Large spin-orbit coupling
To begin with the former, we insert the parameterization (29) using g ss as given in Eq. (33) into Eq. (39) and find .
Setting v 1 → v 2 in Eq. (40), one obtains whereas g 00 and g 30 do not flow. This is the limit that has been discussed in detail in the main text. The resulting flow is shown in Fig. 2(a).
If v 1 = v 2 , all four backscattering coupling constants flow. The projection of the RG flow onto the g 11 -g 33plane is illustrated in Fig. 6(a). We observe that the structure of the flow diagram is very similar to Fig. 2(a) and that the three regions (I), (II) and (III) can still be identified. Note that g 00 , g 30 and g 33 can only diverge if g 11 diverges as well which is easily seen from Eq. (40). Hence, none of the couplings diverges in region (III).

Quasi-degenerate Fermi surfaces
Finally, we also discuss the situation of very weak spinorbit coupling where more backscattering terms are possible. To simplify the following analysis, we introduce new Fermion operators c andc via which renders the theory invariant except for a change of the coupling matrix g ss . One can show that, upon properly choosing α, the coupling matrix g in Eq. (32) can be brought into the reduced form g =    g 00 g 10 0 g 30 g 10 g 11 0 0 0 0 g 22 0 Using this interaction matrix in Eq. (39), we find (neglecting the primes for notational simplicity) FIG. 6. RG flow in the cases that have not been discussed in the main text. Part (a) shows the projection of the flow for v2/v1 = 0.4 using g00 = 1 and g30 = 0.1. For g00 < 0, the projection has essentially the same structure. In (b) and (c), the reduced flow in cases of quasi-degenerate Fermi surfaces is shown for g33 > 0 and g33 < 0, respectively. In all three plots, the red (blue) regions correspond to an initially repulsive (attractive) interaction thus identifying the unconventional (conventional) superconductor.
Consequently, the coupling constants diverge at all four stable fixed points denoted by (I)-(IV) in Fig. 6(b) and (c).

Microscopic interaction
A important part of our analysis is the identification of the pairing mechanisms in the different superconductors. For this purpose, we include matrix elements of the electron-electron interaction between the relevant d xz and d yz orbitals yielding both an intra-(U ) and interorbital (U ) Hubbard interaction, a Hund's coupling (J H ) term as well as pair-hopping (J ). In addition, we use J = J and U = U + 2J valid for the usual Coulomb interaction, but our results do not crucially depend on this assumption.
Projecting the interaction onto the low-energy theory, we find, using the model defined in the main text, in case of disjoint support in momentum space and g 00 g 11 −g 22 g 33 , |g 00 | |g 10 |, |g 30 |, |g 31 | (47) for near-spin degeneracy. In this way, we can estimate the initial conditions for the RG flow both for a microscopically repulsive (g 00 > 0) and for an electron-phonon induced, attractive (g 00 < 0) interaction. The two scenarios correspond, respectively, to the red and blue shaded regions of the flow diagrams in Fig. 2(a) and Fig. 6.

C. Mean-field equations and instabilities
Now we want to investigate which instabilities are associated with the divergences in the RG flow. Following Refs. 27 and 28, we analyze the mean-field equations with the renormalized couplings for any instability possible at finite temperature. The leading instability is the one with the highest transition temperature.
Let us assume spatial and temporal homogeneity of the particle-hole, and the particle-particle, mean-field parameters. The corresponding linearized self-consistency equations read (50) and for the density wave and superconducting order parameters, respectively. Here G denotes the non-interacting Green's function as given in Eq. (36) without the momentum constraints. Again focusing on the limit η → 0, we find As we are interested in the limit T v 2 Λ ⊥ , we only keep the leading (log-divergent) terms in the mean-field equations. Therefore, both the first term in Eq. (50) and the cases with σ α = σ β in Eq. (52) are subdominant. One then finds and where we have introduced In Eq. (54b), it has been exploited that ∆ SC α,β is antisymmetric such that it is sufficient to consider (σ α , σ β ) = (−, +). We see that, both for the density wave and for the superconducting channel, solely order parameters with σ α = −σ β are relevant. In addition, only the backscattering tensor W enters, whereas forward scattering, V , does not play any role at all.
Next, let us expand the density wave order parameters, and the anomalous expectation values, in Pauli matrices to rewrite Eq. (54) more explicitly for the three different scenarios shown in Fig. 4(b)-(d).

Instabilities for identical velocities and disjoint momentum spaces
By definition, we have x j = 1 in this case and hence Inserting the coupling matrix (33) into Eq. (54), we find the mean-field equations summarized in Table I.
To discuss the implications of this result, let us first assume that the couplings diverge before the RG flow is cut off due to the finite curvature of the Fermi surface. In regime (I) of Fig. 2(a), the couplings behave asymptotically as g 11 ∼ −g 33 → ∞, whereas g 00 and g 30 stay finite. As is easily seen from the mean-field equations, the leading instability is, in this case, characterized by ∆ SC (−,jα),(+,j β ) ∝ (σ 3 ) jα,j β . Thus, the system resides in the SC +− -state. Correspondingly, in regime (II), we have g 11 ∼ g 33 → −∞ and hence ∆ SC (−,jα),(+,j β ) ∝ (σ 0 ) jα,j β , i.e. SC ++ , dominates. To derive the subleading instabilities, we have investigated the flow of all mean-field equations in Table I according to Eq. (41) and analyzed which of the order parameters is dominant before superconductivity eventually wins. Since, at that point, g 11 and g 33 are still finite, the result also depends on the value of the non-flowing coupling constants. The associated instabilities, that compete with SC +− and SC ++ , are shown in Fig. 2(c) and (d) of the main text.
yielding the mean-field equations presented in Table II. Note that, as a consequence of the asymmetry between the Fermions from the inner and outer Fermi surfaces, the instabilities SDW 11 and SC 11 dominate over SDW 22 and SC 22 , respectively. Using the RG flow in Eq. (40), one can easily determine the phase diagram. Remarkably, it turns out that superconductivity will still be the leading instability if the RG is not cut off before the backscattering coupling constants diverge. The difference, compared to the situation with identical velocities,  I. Mean-field equations for the density wave, Eq. (56), and superconducting order parameters, Eq. (57), in case of disjoint regions in momentum space and identical velocities. The plus (minus) sign in the column of R c2 and Θ means that the corresponding order parameter is symmetric (antisymmetric) under π-rotation and time-reversal, respectively. The mean-field equations with j = 1 and j = 2 are degenerate.
Mean-field equations (j = 1, 2) Order parameter R c2 Θ Phase Mean-field equations (j = 1, 2) Order parameter R c2 Θ Phase is that the order parameter of the resulting superconductor (SC 11 ) is both for the conventional and for the unconventional pairing scenario. We emphasize that the resulting meanfield theories in the associated blue and red part of the flow in Fig. 6(a) are not identical as the coupling constants are different. In Sec. IV D 2, we show that the superconductors even differ in their topology.

Quasi-degenerate Fermi surfaces
Since v 1 = v 2 in the limit of very weak spin-orbit splitting, L jα,j β is again given by Eq. (58). Using the reduced coupling matrix (43), we find the mean-field equations of Table III, where primes have been neglected for notational simplicity.
As the Fermi surfaces are quasi-degenerate, also superconductors with off-diagonal order parameters, ∆ SC (−,jα),(+,j β ) = 0 for j α = j β , are possible. Recall from Fig. 6(b), (c) and from Eq. (45) that there are four stable fixed points at which the couplings diverge. From the mean-field equations, the leading instability associated with the divergences at the four fixed points is readily found and summarized in Table IV. Interestingly, as long as curvature corrections of the Fermi surface are negligible, superconductivity generically wins even in the present case with the largest number of independent coupling constants.

D. Detailed calculation of the invariants
Since, in none of the superconducting phases derived above, time-reversal symmetry Θ is spontaneously broken by the strongly nested parts of the Fermi surface, it is reasonable to assume that this holds for the entire Fermi surface, if m l /m h is sufficiently small. Similarly, we only know that the superconducting order parameter is finite in the nested parts of the Fermi surfaces. Since the four equivalent strongly nested subspaces have been treated independently, our analysis does not tell TABLE III. Mean-field equations in case of overlapping regions in momentum space. The transformation behavior of the order parameters under R c2 and Θ can be found in Table I.

Mean-field equations (j = 1, 2)
Order parameter whether the point symmetries relating these subspaces are spontaneously broken and whether the order parameter changes sign along the Fermi surface. This crucially depends on the non-singular interaction channels between the nested subspaces. As the values of the corresponding coupling constants are a priori unknown, let us assume that the order parameter is finite on the entire Fermi surface which is motivated by the experimental analysis 29 of the superconducting gap of the system. As Θ 2 = −1, the superconductor belongs to class DIII and is characterized by a Z 2 topological invariant in two spatial dimensions. Here we present the calculation of the invariants in much more detail and for all three regimes of the system.

Identical velocities
Again, let us start with the simplest situation of disjoint support in momentum space and identical velocities. As shown above, the resulting superconductors are c ∈ C, with j = 0 and j = 3 in the conventional and unconventional pairing scenario, respectively. Treating the interaction (23) at mean-field level and inserting Eq. (61), one finds where the mean-field parameter has been introduced. In the present case, with g as given in Eq. (33), we find where γ 0 = g 00 + 2g 11 + g 33 , j = 0, Before calculating the Z 2 -invariant, it is instructive to first investigate the excitation spectrum of the effective one-dimensional system. For this purpose, we introduce Nambu spinors to write the mean-field action in quadratic form, where Diagonalizing H(k) readily yields the four bands characterizing excitations in the superconducting phase. Obviously, the gap closes when i.e., if there are topologically distinct phases, they have to be separated by a manifold where Eq. (71) holds.
To calculate the invariant, we have to relate the effective one-dimensional theory to the full mean-field Hamiltonian, defined on the entire two-dimensional Brillouin zone. Suppose that the free Hamiltonian H has been diagonalized by applying the transformation (10). Here we use the convention that the eigenfunctions φ α are sorted for every k such that the energy increases with α. Furthermore, the phases are fixed by demanding that Eqs. (15) and (17)  Here denotes the strongly nested domain in the vicinity of k j (see Fig. 4(a)) and the ellipsis stands for the pairing terms in the remainder of the Brillouin zone. Using Eq. (10) to rewrite the f -operators in terms of the Ψ-fields, we find By construction, H is time-reversal invariant, which means that with K denoting complex conjugation and arbitrary ϕ ∈ R. It is straightforward to show that time-reversal invariance of the pairing term is equivalent to e 2iϕ σ y ∆ † (k)σ y = −∆(k).
Using the phase conventions (15) and (17) and writing c = |c|e iρ , ρ ∈ R, one finds that Eq. (77) is satisfied if Now we are prepared to calculate the Z 2 -invariant. According to Ref. 33, one simply has to evaluate the matrix elements where q j is an arbitrary point on the jth Fermi surface. As long as the gap of the superconducting system does not close, the sign of δ j is constant on the entire Fermi surface 33 and we are free to choose q j = k j . From Eq. (75), one then finds again exploiting Eqs. (15) and (17) First, note that the phase boundary |γ 0 | = |γ 3 | is in accordance with the analysis of the excitation spectrum (70) of the one-dimensional description. Secondly, let us rewrite the condition for a topologically nontrivial phase considering the conventional and the unconventional superconductor separately. In the latter case, the system is topological if and the RG flow (see region (I) in Fig. 2(a)) leads to the asymptotic behavior g 11 ∼ −g 33 → ∞, whereas g 00 and g 30 do not flow at all. Consequently, the unconventional superconductor SC +− is topologically nontrivial. In case of conventional pairing, though, the condition for N = −1 reads |g 00 + 2g 11 + g 33 | < |2g 30 | and the RG flow (regime (II) in Fig. 2(a)) behaves asymptotically as g 11 ∼ g 33 → −∞. Therefore, the conventional SC ++ -phase is trivial.

Quasi-degenerate Fermi surfaces
Finally, we also discuss the situation of very weak spinorbit coupling (0 < η v j Λ ⊥ ), where four distinct superconductors are possible as summarized in Table IV. From Eq. (64) and recalling the reduced backscattering coupling matrix (43), we find the mean-field parameters in Table V associated with the four stable fixed points in Fig. 6(b) and (c).
To investigate the topological properties of these states, note that, for any finite η, we can still sort the eigenfunctions by energy and then apply the phase conventions in Eqs. (15) and (17). Similarly to the above analysis, one can then rewrite the matrix elements φ j (k 1 )|θ H K∆ † (k 1 )|φ j (k 1 ) = ∓8|c| (γ 0 σ 0 + γ · σ) , (87) j = 1, 2, in terms of the mean-field parameters γ s . As in Eq. (80), the two possible signs correspond to the two possible choices of ϕ in the representation (76) of timereversal.
For the procedure of Ref. 33 for calculating the topological invariant to work, it is essential that the matrix elements in Eq. (87) between different Fermi surfaces can be neglected. For the superconductors (II) and (IV) this is indeed valid since, in the first case, m is exactly diagonal, and, in the latter, m becomes asymptotically diagonal when the couplings diverge. From Eq. (87), it is readily seen that the condition for having a topologically nontrivial superconductor is again given by |γ 0 | < |γ 3 |. Using the mean-field parameters defined in Table V and recalling Eq. (45), one finds that the superconductors associated with the fixed points (II) and (IV) are topological and trivial, respectively.
Consequently, even in the scenario of quasi-degenerate Fermi surfaces, only two fully gapped superconductors are possible: The unconventional superconductor (red region in Fig. 6(b), flowing to (II) in Fig. 6(c)) has a nontrivial Z 2 -invariant, whereas the conventionally paired state (blue region in Fig. 6(c)) is trivial.

E. Spatial structure of the density waves
In the following, we present more details about how the density wave profiles in Fig. 3 have been derived. The charge (s = 0) and spin (s = 1, 2, 3) expectation value is given by where τ denotes (imaginary) time, Ψ are the four component fields as in Eq. (9) and σ s act in spin-space. Ap-