Topological superconductivity and unconventional pairing in oxide interfaces

# Topological superconductivity and unconventional pairing in oxide interfaces

Mathias S. Scheurer Institut für Theorie der Kondensierten Materie, Karlsruher Institut für Technologie, Karlsruhe, D-76131, Germany    Jörg Schmalian Institut für Theorie der Kondensierten Materie, Karlsruher Institut für Technologie, Karlsruhe, D-76131, Germany Institut für Festköperphysik, Karlsruher Institut für Technologie, Karlsruhe, D-76131, Germany
July 7, 2019
###### Abstract

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/LaAlO-interfaces and closely related systems.

The two-dimensional electron fluid that formsOhtomo2004 () at the interface between the insulators SrTiO and LaAlO 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 superconductivityReyren2007 () in the interface, along with the ability to control the ground state via applied electric fieldsCaviglia2008 () opened up intense research. The key open question is whether electronic correlations promote new states of matter, such as unconventional superconductivity or novel magnetic statesBrinkman2007 (); Li2011 (); Bert2011 (); Banerjee2013 (); Li2014 () 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 physicsHasanKane (); Qi (). 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 attentionZhangKane (); Keselman (); Nakosai (); Nakosai2 (); Deng (); Fu (). 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 and orbitalsSantander-Syro2011 (); Joshua2012 (); King2014 (). The orientation of the electron clouds of the -orbitals leads to a wave function overlap along the -direction that is much larger for states compared to , and vice versa for the -direction. Each orbital is then characterized by a light mass and a heavy mass , leading to the experimentally observed strongly anisotropic electronic structureSantander-Syro2011 (); King2014 (). For example, the energy of the states can be described by

 εxz(k)=k2x2ml+k2y2mh (1)

where . follows from Eq. (1) by interchanging and . 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 effectDresselhaus1955 (); Rashba1960 (), the electronic states experience a momentum dependent splitting and mixing of spin-states, naturally explaining magneto-transport experimentsCaviglia2010 (); BenShalom2010 (). The effect might also be responsible for the observed phase separation in interfacesBucheli2013 (). Focusing on the and states, the most general form up to linear order in momentum that is consistent with the -point group symmetry and time-reversal invariance is given by

 Hso(k)=12λτ2σ3+α0τ0(kxσ2−kyσ1)+α1τ1(kxσ1−kyσ2)+α3τ3(kxσ2+kyσ1), (2)

where the Pauli matrices and () act in spin and orbital space, respectively. Projecting out the band that is closest in energy and shifted by and including the atomic spin-orbit coupling we find . , , and were determined in first principles calculationsZhong2013 (). As and depend sensitively on details of the interface we use , estimated from magnetotransport experimentsCaviglia2010 ().

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 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. King2014, for the surface states of SrTiO. The nesting is a consequence of the mass anisotropy and becomes exact in the limit .

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 and which diagonalize the quadratic part of the Hamiltonian. Here refers to the sign of and () 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

 Hback=3∑s,s′=0∑qJ−s(q)us,s′J+s′(−q), (3)

where ()

 Jσs(q)=∑k,j,j′c†(σ,j)(k+q)(σs)j,j′c(σ,j′)(k). (4)

We emphasize that from now on the Pauli matrices , 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 is measured relative to the center 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 -rotation symmetry with respect to the -axis implies that has to be symmetric, . 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 if either or . Let us first assume that the cutoff 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 , , and can be non-zero.

In case of forward scattering, where all four fermions have the same index , 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) calculationShankar (), 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. Chubukov, ; Vafek, 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:

 ΔDWα,β (5a) ¯¯¯¯¯ΔSCα,β (5b)

where and are double indices comprising helicity and the Fermi surface sheet index . Near the Fermi surface, we linearize the band dispersion with denoting the component of the momentum perpendicular to the Fermi surface. For simplicity, let us first focus on the situation 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 .

If , only and 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 with . 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 and , 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 -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, results from conventional electron-phonon pairing, whereas is an unconventional superconductor, where particle-hole fluctuations effectively change the sign of .

Both and 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 experimentsRichter (). Being fully gapped, it is natural to ask whether the time-reversal invariant two-dimensional superconductor (classAltlandZirnbauer () DIII) is topologically trivial or nontrivialSchnyder (), which is of great interest as it strongly influences its physical properties. The most prominent feature of a nontrivial topological superconductor is the appearance of spin-filtered counter propagating Majorana modes at its edge when surrounded by a trivial phaseBernevigsBook (). It has been shownZhang () that the associated topological invariant is fully determined by the sign of the paring field on the Fermi surfaces. It holds

 (6)

where the product involves all Fermi surfaces, and denote the wave function of the non-interacting part of the Hamiltonian and the pairing field at an arbitrary point on the th Fermi surface. Furthermore, is the number of time-reversal invariant points enclosed by the th Fermi surface and is the unitary part of the time-reversal operator, given by 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 is different (identical) on the two Fermi surfaces. Inserting the order parameters derived above, we obtain the pairing Hamiltonian

 Hpair=Δ∑k,j,j′ c(+,j)(k)(γ0σ0+γ3σ3)j,j′c(−,j′)(−k) +H.c. (7)

with , for the superconductor and , for the -state. Calculating in Eq. (6), one finds (see Supplementary Information for details) that the superconductor is topological if and trivial for the reversed inequality sign. At , the gap closes as is characteristic for a topological phase transition. Recalling the flow depicted in Fig. 2(a), one immediately sees that is trivial, whereas 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 . Vice versa, a trivial state is only consistent with conventional, electron-phonon induced superconductivity.

We emphasize the difference of this result to recent workMohanta (); Fidkowski1 (); Fidkowski2 (); Kim () proposing the emergence of Majorana fermions in the heterostructure. In Refs. Mohanta, ; Fidkowski1, ; Fidkowski2, ; Kim, , Majorana physics is predicted to arise from the coexistence of magnetism and superconductivity. This means that (physical, spin-) time-reversal symmetry is broken, whereas the -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 becomes relevant. In this sense, any finite 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 and , one can either find a charge density wave (), three different spin density waves (, , ) or the corresponding superconducting states are dominant for arbitrary as shown in Fig. 2(c) and (d). The superscripts in the density waves and refer to the particle-hole expectation value (and if ) that is non-zero in the respective phase. The difference between and is the relative sign of and , 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 . As in the case of the superconducting order parameter, we assume that no additional point group symmetry is broken. In the case of the -phase, one then finds that the local charge density is given by

 ρ(x)∝cos(Q12⋅x)+…, (8)

where 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 -rotation and reflection symmetry at the -axis. The resulting charge profile is illustrated in Fig. 3(a). Note that the periodicity crucially depends on the ratio of the - and -component of the nesting vector .

Similarly, the spatial structure of the spin density waves and , 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 -axis (see Fig. 1(b) and (c)). Within this approximation, the expectation value of the spin lies in the -plane in case of the spin density phase . The two-dimensional vector field is therefore a lattice of vortices both with positive and negative winding number. In the phases and , 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. Rosch, ), is consistent with recent workLi2014 (); Garaud () 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(c) and (d) neighboring the superconducting states and 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 magnetizationLi2011 () is only consistent with the -state. This implies that the superconducting phase of SrTiO/LaAlO is supposed to be unconventional and topologically nontrivial.

As already stated above, we have also considered the case of different Fermi velocities, (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 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 trivial, 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 spin-orbit interaction where the energetic cutoff of the low-energy 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 and SrTiO combines two fascinating notions of condensed matter physics: Topology and unconventional 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.

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Acknowledgements. – We are grateful for discussions with S. Beyl, A. V. Chubukov, A. M. Finkel’stein, E. J. König, D. Mendler, and A. D. Mirlin. We acknowledge financial support by the Deutsche Forschungsgemeinschaft through grant SCHM 1031/4-1.

## Appendix A Supplementary information

### a.1 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 , . 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.

#### a.1.1 Phase convention and representation of symmetries

Using a path-integral representation, the quadratic part of the theory can be written as

 S0=T∑ωn∑k¯¯¯¯Ψα(k)[−iωδα,β+Hα,β(k)]Ψβ(k), (9)

where and , are four-component Grassmann fields describing spinful Fermions in the two orbitals . Furthermore, is the Hamiltonian defined in the main text characterized by the anisotropic masses (1) and the spin-orbit coupling in Eq. (2).

We diagonalize by performing the unitary transformation

 Ψα(k)=Uα,α′(k)fα′(k),¯¯¯¯Ψα(k)=U∗α,α′(k)¯fα′(k), (10)

where

 U(k)=[ϕ1(k),ϕ2(k),ϕ3(k),ϕ4(k)] (11)

with denoting an eigenvector of . 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 and in the local coordinate systems yielding

 S0=∫k¯c(σ,j)(k)[−iωn+σvjk⊥+sjη]c(σ,j)(k) (12)

after linearizing the spectrum. Here , and denotes the spin-orbit splitting in the case of quasi-degenerate Fermi surfaces. For stronger spin-orbit coupling, where the four red regions in Fig. 4(a) are disjoint, one has by construction. In Eq. (12) and in the following, we use the compact notation and

 ∫k⋯=∫∞−∞dω2π∫Λ∥−Λ∥dk∥2π∫Λ⊥−Λ⊥dk⊥2π…, (13)

where and 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 () refers to the outer (inner) Fermi surface. If this is not the case, it will be most convenient to label the fields such that .

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 -rotation and time-reversal . The former symmetry implies that

 H(k)=Rc2ΨH(−k)Rc2Ψ†,Rc2Ψ=iσz, (14)

and hence we can construct the eigenstates with negative from those with via

 ϕα(−k):=Rc2Ψ†ϕα(k),kx>0. (15)

Consecutive application of time-reversal and -rotation leads to the -space local antiunitary symmetry

 H(k)=σxH∗(k)σx (16)

of the Hamiltonian. If the Fermi surfaces in Fig. 4 are non-degenerate, we can adjust the phases of the eigenstates such that

 ϕα(k)=σxϕ∗α(k) (17)

for . From Eq. (15), it follows that Eq. (17) actually holds also for . 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 -rotation symmetry on the helicity fields is well defined. Note that the remaining elements of the point group 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

 Ψα(ω,k) \lx@stackrelΘ⟶(iσy)α,β¯¯¯¯Ψβ(ω,−k), (18a) ¯¯¯¯Ψα(ω,k) \lx@stackrelΘ⟶Ψβ(ω,−k)(iσy)β,α (18b)

in the basis of Eq. (9) and, consequently, as

 fα(k) \lx@stackrelΘ⟶i[U†(k)σyU∗(−k)]α,α′¯fα′(ω,−k) (19a) ¯fα(k) \lx@stackrelΘ⟶ifα′(ω,−k)[(U†(k)σyU∗(−k))†]α′,α (19b)

in the local eigenbasis. Using Eqs. (15) and (17), we can write

 U†(k)σyU∗(−k) =U†(k)σy[ϕ∗1(−k),…] =U†(k)σyσx[ϕ1(−k),…] =−sign(kx)U†(k)σzσz[ϕ1(k),…] =−sign(kx)\mathbbm1 (20)

and, thus, conclude

 c(±,j)(ω,k∥,k⊥) \lx@stackrelΘ⟶∓i¯c(∓,j)(ω,−k∥,−k⊥), (21a) ¯c(±,j)(ω,k∥,k⊥) \lx@stackrelΘ⟶∓ic(∓,j)(ω,−k∥,−k⊥). (21b)

Similarly, for the -rotation symmetry, one finds

 c(±,jα)(ω,k∥,k⊥) \lx@stackrelRc2⟶∓c(∓,jα)(ω,−k∥,−k⊥) (22)

and the same for .

#### a.1.2 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

 Sint=∫k1,k2,k3,k4¯cα(k4)¯cβ(k3)cγ(k2)cδ(k1)Wαβγδ×δ(k1+k2−k3−k4), (23)

where the Greek letters are double indices comprising and . The tensor has to satisfy

 Wαβγδ=(Wδγβα)∗ (24)

due to Hermiticity and, as a consequence of Fermi statistics, can be chosen such that

 Wαβγδ=−Wβαγδ=−Wαβδγ. (25)

It turns out that the dimensionless parameterization,

 ωαβγδ=Λ∥2π2v1Wαβγδ (26)

with

 ω(σα,jα)(σβ,jβ)(σγ,jγ)(σδ,jδ)==⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩Vjα,jβjγ,jδ(σ),σα=σβ=σγ=σδ=σ,Wjα,jβjγ,jδ,(σα,σβ,σγ,σδ)=(−,+,+,−),Wjβ,jαjδ,jγ,(σα,σβ,σγ,σδ)=(+,−,−,+),−Wjβ,jαjγ,jδ,(σα,σβ,σγ,σδ)=(+,−,+,−),−Wjα,jβjδ,jγ,(σα,σβ,σγ,σδ)=(−,+,−,+),0,otherwise, (27)

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 ) and backscattering () 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

 Vjα,jβjγ,jδ(σ)=g0(σ)[δjα,jδδjβ,jγ−δjα,jγδjβ,jδ], (28)

whereas the backscattering tensor has degrees of freedom, which we parametrize according to

 Wjα,jβjγ,jδ=3∑s,s′=0gss′(σs)jα,jδ(σs′)jβ,jγ. (29)

The Hermiticity constraint in Eq. (24) implies that . Note that with used in the main text to define the backscattering terms.

Next, let us derive the constraints resulting from -rotation symmetry. Demanding that Eq. (23) be invariant under Eq. (22), we find

 Vjα,jβjγ,jδ(+) \lx@stackrel!=Vjα,jβjγ,jδ(−), (30a) Wjβ,jαjδ,jγ \lx@stackrel!=Wjα,jβjγ,jδ. (30b)

The former conditions means that, as expected, forward scattering is identical for the patches centered around and . Consequently, all forward scattering processes are characterized by one coupling constant . Applying the expansion (29), the second constraint is equivalent to 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 is not further restricted, whereas the backscattering tensor has to satisfy

 Wjδ,jγjβ,jα\lx@stackrel!=Wjα,jβjγ,jδ. (31)

In the representation (29) this is equivalent to demanding if either or .

Consequently, in the limit of weak spin-orbit interaction, where and the red regions in Fig. 4 overlap pairwise, the backscattering tensor is given by Eq. (29) with

 g=⎛⎜ ⎜ ⎜⎝g00g100g30g10g110g3100g220g30g310g33⎞⎟ ⎟ ⎟⎠. (32)

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 , , , , and can be finite. Comparison with Eq. (32) then yields

 g=⎛⎜ ⎜ ⎜⎝g0000g300g110000−g110g3000g33⎞⎟ ⎟ ⎟⎠. (33)

### a.2 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).

#### a.2.1 Generic form of the RG equations

In the Wilson approach, applied to Fermions with a finite Fermi surface in Ref. Shankar, , fast modes with momenta , , 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 .

The other two diagrams, Fig. 5(b) and (c), are usually referred to as ZS and BCS, respectively, and lead to the corrections

 ΔBCSWαβγδ=2WαβμνWμνγδ∫kG>μ(k)G>ν(k1+k2−k) (34)

and

 (35)

of the interaction tensor . In Eqs. (34) and (35), we have introduced the Green’s function

 G>α(k)=θ(|k⊥|−Λ⊥e−Δl)θ(Λ⊥−|k⊥|)iω−σαvjαk⊥−sjη (36)

of fast modes. Note that and have been symmetrized to satisfy Eq. (25). Evaluating the shell integrals asymptotically in the limit and using the dimensionless parameterization (26), one finds the tensor valued RG equation

 dωαβγδ(l)dl=(1−δσμ,σν)[ωαβμν(l)ωμνγδ(l)∑p=+,−1xjμ(1+pκjμ)+xjν(1+pκjν)+2(ωανγμ(l)ωμβνδ(l)−ωβνγμ(l)ωμανδ(l))∑p=+,−1xjμ(1+pκjμ)+xjν(1−pκjν)], (37)

where and have been defined. From Eqs. (27) and (37), it is already clear that

 dω(σ,jα)(σ,jβ)(σ,jγ)(σ,jδ)dl=dVjα,jβjγ,jδ(σ)dl=0, (38)

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 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 , since, for stronger spin-orbit interaction, we have by construction (see Fig. 4(b) and (c)). Inserting in Eq. (37) then yields, after some algebra, the flow equation

 dWjα,jβjγ,jδdl=∑jμ,jν4xjμ+xjν(Wjα,jνjγ,jμWjμ,jβjν,jδ−Wjα,jβjν,jμWjμ,jνjγ,jδ) (39)

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 for the two cases of large spin-orbit coupling and quasi-degenerate Fermi surfaces.

#### a.2.2 Large spin-orbit coupling

To begin with the former, we insert the parameterization (29) using as given in Eq. (33) into Eq. (39) and find

 dg00dl =−2g211(v1−v2)2v2(v1+v2), (40a) dg30dl =−2g211v1−v2v2, (40b) dg11dl=−2g11v2(v1+v2)[g33(v21+v22+6v1v2)+g00(v1−v2)2−2g30(v21−v22)], (40c) dg33dl =−2g211v21+v22+6v1v2v2(v1+v2). (40d)

Setting in Eq. (40), one obtains

 dg11dl=−8g11g33,dg33dl=−8g211, (41)

whereas and 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 , all four backscattering coupling constants flow. The projection of the RG flow onto the --plane 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 , and can only diverge if diverges as well which is easily seen from Eq. (40). Hence, none of the couplings diverges in region (III).

#### a.2.3 Quasi-degenerate Fermi surfaces

Finally, we also discuss the situation of very weak spin-orbit coupling where more backscattering terms are possible. To simplify the following analysis, we introduce new Fermion operators and via

 c(k) =(eiα2⋅σ00eiα2⋅σ)c′(k),α∈\mathbbmR3 (42a) ¯c(k) =¯c′(k)(e−iα2⋅σ00e−iα2⋅σ), (42b)

which renders the theory invariant except for a change of the coupling matrix . One can show that, upon properly choosing , the coupling matrix in Eq. (32) can be brought into the reduced form

 g′=⎛⎜ ⎜ ⎜ ⎜⎝g00g′100g′30g′10g′110000g′220g′3000g′33⎞⎟ ⎟ ⎟ ⎟⎠. (43)

Using this interaction matrix in Eq. (39), we find (neglecting the primes for notational simplicity)

 dg11dl =8g22g33, (44a) dg22dl =8g11g33, (44b) dg33dl =8g11g22, (44c) dgp,p′dl =0,otherwise. (44d)

The resulting flow of the ratio of the coupling constants is illustrated in Fig. 6(b) and (c) for different signs of . The reduced flow has four fixed points with which are stable if and only if . Right at the fixed points, Eq. (44) is solved by

 g11(l)=p1g(l),g22(l)=p2g(l),g33(l)=g(l) (45a) with g(l)=g(0)1−8lp1p2g(0). (45b)

Consequently, the coupling constants diverge at all four stable fixed points denoted by (I)-(IV) in Fig. 6(b) and (c).

#### a.2.4 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 and orbitals yielding both an intra- () and inter-orbital () Hubbard interaction, a Hund’s coupling () term as well as pair-hopping (). In addition, we use and 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,

 g00≃g11≃g33,|g00|≫|g30| (46)

in case of disjoint support in momentum space and

 g00≃g11≃−g22≃g33,|g00|≫|g10|,|g30|,|g31| (47)

for near-spin degeneracy. In this way, we can estimate the initial conditions for the RG flow both for a microscopically repulsive () and for an electron-phonon induced, attractive () interaction. The two scenarios correspond, respectively, to the red and blue shaded regions of the flow diagrams in Fig. 2(a) and Fig. 6.

### a.3 Mean-field equations and instabilities

Now we want to investigate which instabilities are associated with the divergences in the RG flow. Following Refs. Chubukov, and Vafek, , 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,

 (48)

and the particle-particle,

 (49)

mean-field parameters. The corresponding linearized self-consistency equations read

 ΔDWα,β∼∫kGα(k)δα,β+2ΔDWα′,β′Wα′βαβ′∫kGα(k)Gβ(k) (50)

and

 ¯¯¯¯¯ΔSCα,β∼2Wα′β′αβ¯¯¯¯¯ΔSCβ′,α′∫kGα(k)Gβ(k) (51)

for the density wave and superconducting order parameters, respectively. Here denotes the non-interacting Green’s function as given in Eq. (36) without the momentum constraints. Again focusing on the limit , we find

 ∫kGα(k)Gβ(k)=−Λ∥(2π)2v↑ (52) ×⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩f(vjαΛ⊥/T)+f(vjβΛ⊥/T)xjα+xjβ,σα=−σβ,f(vjαΛ⊥/T)−f(vjβΛ⊥/T)xjα−xjβ,