Dislocation Majorana zero modes in perovskite oxide 2DEG

Much of the current experimental efforts for detecting Majorana zero modes have been centered on probing the boundary of quantum wires with strong spin-orbit coupling. The same type of Majorana zero mode can also be realized at crystalline dislocations in 2D superconductors with the nontrivial weak topological indices. Unlike at an Abrikosov vortex, at such a dislocation, there is no other low-lying midgap state than the Majorana zero mode so that it avoids usual complications encountered in experimental detections such as scanning tunneling microscope (STM) measurements. We will show that, using the anisotropic dispersion of the t2g orbitals of Ti or Ta atoms, such a weak topological superconductivity can be realized when the surface two-dimensional electronic gas (2DEG) of SrTiO3 or KTaO3 becomes superconducting, which can occur through either intrinsic pairing or proximity to existing s-wave superconductors.

where σ † c a r, , creates an electrons at site r with spin polarization σ = ↑ , ↓ and orbital a = x, y (representing d xz and d yz orbitals respectively). Because of the quasi-1D natures of d xz and d yz orbitals, |t| ≫ |t′ |. This simple model is sufficient to explain why we can reach the Lifshitz transition by lifting the Fermi level only in the order of the heavy-mass dispersion bandwidth 4t′ and the electron filling fraction only by ′ t t / ; this would involve raising the Fermi level by ~0.1 eV and adding 0.3 electrons per unit cell when compared to the KTaO 3 surface ARPES data 16 (we also note that superconductivity has been observed around this 2D electron density for the ionic liquid gated KTaO 3 24 ). Such shift in the Fermi level can be achieved by both the electrical gating and optically induced oxygen vacancies 15,16,38,39 ; more recently, a first-principle calculation showed how a large shift in chemical potential can occur a surface 2DEG from a cubic perovskite oxide heterostructure 40 . Meanwhile, the contribution from the d xy orbital is suppressed as it has the light-mass dispersion in both the x-and the y-direction.
We further need to consider the hybridization between the d xz and d yz orbitals in order to obtain from them two bands, one giving rise to the outer Fermi surface closer to the van Hove singularity at crystalline momentum points X = (π, 0) and Y = (0, π) and the other giving rise to the inner Fermi surface closer to the Γ point. Microscopically, the hybridization between d xz and d yz orbitals is mainly due to the on-site atomic spin-orbit coupling x y x y x y 2 2 2 2 It is clear that, for (t − t′ ) ≫ λ, t′ ′ , the orbital hybridization would have little effect near the X/Y points except for shifting the Lifshitz transition to µ λ = − − ′ + t t 4( ) 2 2 . Even with a large λ, as shown in Fig. 1(a) where we used t = 10t′ = 0.5 eV and λ = 0.26 eV, the former approximating the first principle calculation for KTaO 3 16,41,42 while the latter larger by roughly a factor of 2, the distinction between the heavy-mass and the light-mass bands remains sharp. Hence the physics near the (π, 0) point would be dominated by the d yz and near (0, π) by the d xz orbital.
Since the inversion symmetry is obviously broken in the surface 2DEG, the spin degeneracy at the Fermi surface should be generically split by the non-zero Rashba spin-orbit coupling. For our analysis, it will be sufficient to consider only the most generic Rashba term, which is orbital independent, . As the chemical potential μ moves, there is a Lifshitz transition at which the outer Fermi surface crosses the van Hove points at X and Y. As we approach the Lifshitz transition, the Scientific RepoRts | 6:25184 | DOI: 10.1038/srep25184 low-energy band structure near the (π, 0) point, which would mainly originate from the d yz orbital, can be given by the first-quantized Hamiltonian x y x y y x x y 0 0  Likewise the band structure in the vicinity of (0, π), which would originate mainly from the d xz orbital can be obtained by the π/2 rotation of the momentum and the spin in Eq. (4). It is also analogous to the Rashba wire that the spin degeneracy at (0, π) remains unbroken, which means that the Fermi surface splitting does not lead to two separated Lifshitz transitions.
To realize isolated Majorana zero modes, the final necessary component of the band structure is the Zeeman field. Near the Lifshitz transition, there will be both higher density of oxygen vacancies near the surface as well as enhancement of the quasi-1D characteristics of the d xz,yz orbitals. Both can give rise to ferromagnetism: the former 39,43 because of the oxygen vacancy acting as the magnetic impurity 44 while the latter through the inter-orbital Hund's rule coupling 45 . Both of these effects should be amplified by the enhanced density of states near the van Hove singularity that occurs at the Lifshitz transition. We will consider the ferromagnetic ordering in the perpendicular direction as was observed in the experiment with the density of oxygen vacancy induced by circularly polarized light 39 ; we also note that, in case our 2DEG that arises from the heterostructure as described in 40 , we can also obtain the Zeeman field by either inserting a ferromagnetic layer between the 2DEG and the insulating substrate or using an appropriate ferromagnetic insulator as the substrate. Then, the ferromagnetism-induced Zeeman coupling τ = ∑ † K h c s c Z Z z r r r 0 shall split the Lifshitz transition into two separated ones, as shown in Fig. 1(b), giving rise to a finite range of μ for which there is a single hole pocket without spin degeneracy around the M = (π, π) point; for this plot we used t′ ′ = t′ = 0.05 eV with α 0 = 0.05 eV and h Z = 0.05 eV.
Dislocation Majorana zero mode in proximity induced superconductivity. For the superconducting state, we will first consider the case where the pairing is induced through proximity to the conventional s-wave superconductor. This will ensure the s-wave pairing in the oxide surface 2DEG. We also point out that inducing superconductivity through proximity effect can have the advantage of achieving superconductivity at higher temperature. To enhance the pairing gap magnitude, we would need strong tunneling between the superconductor and the oxide surface 2DEG. This can be achieved through using the higher-T c two-band superconductors such as FeSe [46][47][48] ; note that by symmetry, the single orbital superconductor is unlikely to have a strong tunneling to the both the d xz and d yz orbitals. Hence our heterostructure will consist of the capping two-band s-wave superconductor on the (001) surface of SrTiO 3 or KTaO 3 as shown in Fig. 2(a).
The combination of the Zeeman field h Z and the s-wave pairing gap |Δ s | in the oxide 2DEG near the Lifshitz transition can give rise to the non-trivial weak index, ν = (1, 1), i.e. non-trivial 1D topological invariants along k x,y = π. For instance, the following low-energy effective BdG Hamiltonian with k x = π is exactly equivalent to the Rashba-Zeeman wire superconducting state 9,49 . where μ α 's are Pauli matrices acting on the particle-hole Nambu space, δμ is the deviation of the chemical potential from the value at the Lifshitz transition for h Z = 0, and we use the basis − , . It is well known that this 1D BdG Hamiltonian is topologically equivalent to the Kitaev chain 50 (class D 51 ) when (a) shows the light-and heavy-mass band dispersion (in blue and red, respectively), along k y = π with (solid line) and without (dotted line) the orbital hybridization. (b) shows the lower band from the d xz /d yz orbitals after the orbital hybridization, with the spin degeneracy removed by the Rashba spin orbit coupling α 0 = 0.05eV and the perpendicular Zeeman field h Z = 0.05 eV. Note that the Lifshitz transition point is split, allowing a single hole pocket without spin degeneracy around (π, π).
Scientific RepoRts | 6:25184 | DOI: 10.1038/srep25184 The orbital hybridization will not affect the topological nature of the superconductivity as long as the s-wave pairing is intra-orbital. Given that the s-wave pairing has no spin dependence, we see that even if we take into account the band hybridization and write π = k k ( , ) x y  in the band basis, the Zeeman coupling and the s-wave pairing terms will remain unchanged, and hence so remain the condition for the topologically non-trivial superconductivity (see Supplementary Information for details).
Because of the nontrivial weak topological indices ν = (1, 1), unpaired Majorana zero modes occur at dislocations whose Burger's vector B in units of lattice spacings satisfies B ⋅ ν = 1 (mod 2), where mod 2 is from the Z 2 nature of weak topological indices in class D 52 . To confirm this, we have performed BdG calculations of the lattice models describing the 2DEG in proximity to a two-orbital s-wave superconductor (sSC). As shown in Fig. 2(b,c), we have obtained the Majorana zero mode at each dislocation from the numerical exact diagonalization of the real-space BdG Hamiltonian. Our calculation was done on a 240 × 240 unit cell with periodic boundary conditions. Two edge dislocations with the Burger's vector = ±B e x are placed by one half system size in the x-direction, with the links between the dislocations shifted as shown in Fig. 2(a) (see Methods for details on implementation). This oxide surface with the pair of dislocations is coupled by tunneling amplitude of t i = 0.05 eV to the s-wave superconductor. The sSC has the band structure well-matched with that of the oxide surface (see Methods for the band structure details) and the pairing gap of |Δ s | = 0.05 eV. Figure 2(b,c) shows the probability distribution of the dislocation zero energy states, showing sharp peak for both the oxide surface and the sSC, even though the latter does not have any dislocation.
This wave function profile suggests that the STM would be a good experimental probe on our dislocation Majorana zero mode 53 . When the STM tip is brought to the sSC as shown in Fig. 2(a), the local differential conductance where the u i , v i are the electron and hole components of the i-th energy eigenstates, up to replacing the delta function by a Lorentzian with the width given by the STM energy resolution, which is chosen to be 0.1 meV for Fig. 2(d). We therefore predict that the STM will see a sharp zero bias anomaly when it is brought to the point on the sSC that is right over the dislocation, the point a of Fig. 2(d). This anomaly is unambiguously separated from the signal of other low lying states, which has a minimum energy of the induced oxide bulk pairing gap ~1.2 meV. This is because the Majorana zero mode is the only midgap state localized at the dislocation, unlike at the Abrikosov vortex where other low energy ( where Δ is the pairing gap and E F the Fermi energy) bound states are present. Hence the zero bias anomaly in the crystalline dislocation can be regarded as more unambiguous signature of the Majorana zero mode than that of the Abrikosov vortex 54 .
Dislocation Majorana zero mode in intrinsic superconductivity. We now consider the case of intrinsic superconductivity in the oxide 2DEG without proximity to conventional superconductors. When the oxide 2DEG becomes superconducting at this electron density, there arises possibility of a protected Kramer's doublet of Majorana zero modes at each dislocation when no Zeeman field is applied. Due to the Rashba spin-orbit coupling, the intrinsic superconductivity should generically have on the Fermi surface a mixture of the s-wave pairing and the p-wave pairing, the latter with momentum-dependent Cooper pair spin state (this feature is independent of the debate on whether the pairing symmetry of the intrinsic superconductivity will follow 25 that of the doped bulk tBdG x z y x y x s t x y 0 will be that of a time-reversal invariant 1D topological superconductor in class DIII 52,57 when δμ > 0 and |Δ s | < |Δ t ||sink y | is satisfied at the Fermi surfaces so that the gaps at the two Fermi surfaces have the opposite signs. In that case, there exists of a branch of helical Majorana edge state around k x,y = π. This means that, when we use the argument of the previous section with the additional constraint of the time-reversal symmetry, there should be a Kramer's doublet of Majorana zero modes at a dislocation with the Burger's vector of =B x or ŷ. Such a Majorana zero mode doublet has been shown to be topologically protected as long as the time-reversal symmetry is preserved 52,57,58 , i.e. the Zeeman field is zero. When the Zeeman field is non-zero, there can be a "re-entrant" unpaired Majorana zero modes at the dislocation. This is because Eq. (7) with the addition of the Zeeman field Given that gap closing cannot be avoided at the topological quantum phase transition, Fig. 3(a) gives us the complete topological phase diagram for the intrinsic superconductivity at a fixed δμ = 0.02 eV. In this plot, the tuning parameter η is introduced to determine the relative strength of the p-wave and s-wave pairings, i.e. (we have set |Δ s0 | = 0.04 eV, |Δ t0 | = 0.08 eV and h Z0 = 0.01 eV; a phase difference between the s-wave and the p-wave pairings was introduced, in order to have the pairing terms break the time-reversal symmetry when h Z ≠ 0). For h Z ≠ 0, as shown in Fig. 3(a), it is always possible to adiabatically tune η from 0 to 1 without gap closing, while for any value of η, one cannot increase h Z without closing the bulk gap at some point. We therefore conclude that the non-trivial high (trivial low) h Z phase at η = 1 is topologically equivalent to that of η = 0, the purely s-wave pairing case we considered in the previous subsection. However, if we restrict ourselves to the case with time-reversal symmetry, i.e. h Z = 0, Fig. 3(a) shows there is a gap closing around η = 0.7, consistent with the fact that the time-reversal invariant p-wave pairing (η = 0) is topologically distinct from the pure s-wave pairing (η = 1). At η = 0.4 and h Z = 0, the two zero energy states localized at the two edge dislocations with exactly identical probability distribution, as shown in Fig. 3(b), indicates the existence of the Kramer's doublet of Majorana zero modes at each dislocation. This confirms that the existence of the Kramer's doublet of Majorana zero modes at each dislocation characterizes this time-reversal invariant topologically non-trivial phase. This means that with an STM with an s-wave superconducting tip over this dislocation, we should be able to observe time-reversal anomaly 58 . As in the case of the proximity induced superconductivity, these dislocation Majorana zero modes, as shown by Fig. 3(c), are the only subgap modes of the system.

Discussion
We have shown in this paper how isolated dislocation Majorana zero mode can arise from both the proximity induced and intrinsic superconductivity in the oxide 2DEG. Its existence can be considered the most pertinent criterion for the topologically non-trivial superconductivity in the oxide 2DEG, and it can be experimentally detected through STM. The crucial requirement for achieving such superconductivity is that the oxide 2DEG needs to be close to the Lifshitz transition.
The key difference between the proximity-induced and the intrinsic oxide 2DEG superconductivity is that the Zeeman field is a necessary condition for the non-trivial topology in the former but not for the latter. The physical consequence is that for the intrinsic superconductivity in the absence of the Zeeman field, the dislocation can host a Kramer's doublet of Majorana zero modes; this is not possible if the superconductivity is induced through proximity to an s-wave superconductor. By contrast, in the presence of nonzero Zeeman field, the only possible protected midgap state on a dislocation is a single Majorana zero mode regardless of the origin of superconductivity.
While our intrinsic superconductivity with the non-trivial weak index at the zero Zeeman field has the essentially same pairing symmetry as the topological superconductivity investigated by Scheurer and Schmalian 27 , these states are topologically distinct. From Eq.(7), our non-trivial phase requires δμ > 0 while that of Scheurer and Schmalian requires δμ < 0 with the Fermi surfaces enclosing the Γ point, and with this pairing symmetry the gap closing around δμ = 0 cannot be avoided. This reflects the fact that, with the reflection symmetry, the topological invariant of the DIII class in 2D can be  rather than  2 59,60 . The existence (absence) of the dislocation Majorana doublet for δμ > 0 (δμ < 0) can be regarded as a physical manifestation of this topological distinction. We leave to future work what type of interaction would favor this pairing symmetry near the Lifshitz transition.
Lastly, we want to point out that it is generically easy to change the topology of the superconducting state of the (001) perovskite oxide 2DEGs. This is because the universal anisotropic band structure makes it easy to access the van Hove singularity through gating and optically inducing oxygen vacancies. While there have been previous works on the physical realization of the the dislocation Majorana zero mode 34-37 , they have not provided easy means to alter the weak indices of the superconducting states. Therefore we conclude that not only is the dislocation Majorana zero mode the most robust topological feature of the oxide 2DEG superconductor but also that the oxide 2DEG superconductor is the particularly suitable system for realizing the dislocation Majorana zero mode.

Weak indices and dislocation Majorana zero modes.
It is possible in a 2D superconductor on a square lattice to consider the 1D topological invariants defined along k x,y = π, which are known as the weak indices [61][62][63] . In general, the weak index ν i can be defined for each time-reversal invariant momentum G i /2 (which makes G i a reciprocal lattice vector) as a topological invariant of the manifold perpendicular to G i but contains G i /2, and hence the weak indices can be written as a single vector ν = ∑ i ν i G i , where G i is the unit vector parallel to G i . The C 4v symmetry of of our 2DEG means that its ν will have only a single independent component ν and therefore can be written as ν = ν(1, 1). The weak indices is clearly topologically protected when the system has crystalline symmetry, the topological crystalline insulators 64 being one class of examples. In this paper, we will focus on its manifestation through the Majorana zero mode localized at its crystalline topological defect -the edge dislocation [32][33][34][35] .
We first note that the non-trivial weak indices in a superconductor imply the existence of a branch of Majorana edge modes around k edge = π. Since restricting ourselves to the k x = π manifold means converting the 2D mean-field Hamiltonian H BdG (k x , k y ) into the 1D Hamiltonian H BdG (k x = π, k y ), the non-trivial weak index means that, for the simplest case of the class D, where the time-reversal symmetry is broken, a single protected Majorana zero mode exists at k x = π for the edge running in the x-direction. This is possible only if there is a branch of chiral Majorana edge state centered around k x = π. Note that the existence of this branch of the edge state is determined by the projection of ν to the time-reversal invariant momentum (π, 0).
A single Majorana zero mode exists at the edge dislocation when there is a chiral Majorana edge state centered around k edge = π. To see how this arises, note that the dislocation can be created by severing all links, both through hopping and interaction, between two halves (y < 0 and y > 0) and then non-trivially re-connect the two halves to introduce the edge dislocation, with the x < 0 part glued back according to the original links but the x > 0 part has all the links altered by translating the sites of the y > 0 half by a lattice constant along the x-direction, which sets the Burger's vector of this dislocation to be =B x 30,32 . Now when this system was cut, there would have been Majorana edge states along the x-direction for both y < 0 and y > 0 with the opposite chirality. Hence when the system is glued back along the original links, the tunneling between the two edges would lead to the backscattering that gaps out these edge modes, with the mass gap being proportional to the tunneling amplitude. However, when the dislocation described above is introduced, there will be a qualitative effect on the tunneling between the k x = π edge state. This is because the k x = π edge mode wave function reverses its sign when we translate by one lattice site along the x-direction, the relative sign of the k x = π edge modes for y < 0 and y > 0 edges will change its sign at the dislocation. That means that if a dislocation is introduced when we glue back with only infinitesimally weak coupling across y = 0, the effective low energy action along y = 0 for the k x = π edge modes would be where the upper and the lower component correspond to the upper and the lower edge and m 0 is proportional to the tunneling amplitude for the k x = π modes; this action is well-known for having a single Majorana zero mode at our dislocation x = 0: By contrast, the existence of the k x = 0 branch is irrelevant as its tunneling amplitude does not change sign at the dislocation. Since the Majorana zero mode is protected as long as it remains separated from other Majorana zero mode, the Majorana zero mode that arose at the infinitesimal coupling across the y = 0 cut will persist when the coupling across y = 0 is increased to the bulk values. In general, the condition for the existence of the protected Majorana zero mode is ν ⋅ B = 1 (mod 2).
We can similarly show the existence of a Kramer's doublet of Majorana zero modes at the edge dislocation when there is a helical Majorana edge state centered around k edge = π. The key point here is that dislocation involves no time-reversal symmetry breaking and therefore, in the `cut and paste' picture, the Kramer's doublet needs to be maintained even with the inter-edge backscattering. Therefore, when ĩ s y is the intra-edge time-reversal operation, the effective low energy action for the k x = π helical edge mode would be This action gives us two Majorana zero modes, Real space Hamiltonian with dislocation. We need to have the real-space BdG Hamiltonian in order to obtain the dislocation Majorana zero mode through exact diagonalization. We first note that the terms in our real-space Hamiltonian could be divided into three groups, the first being the onsite term, In the real space, the dislocation point serves as a starting point for a branch cut along which the nearest-neighbor Hamiltonian ˆĤ r ( ) e i is applied on a next-nearest neighbor link. Our square lattice N x = 240 by N y = 240 latitce has a periodic boundary condition to both direction. In order to have a dislocations at (N x /4, N y /2 + 1) with the Burger's vector = +B e x and another dislocation at (3N x /4, N y /2) with = −B e x , we apply ˆĤ e y on the links connecting (n, N y /2) and (n + 1, N y /2 + 1) for N x /4 ≤ n < 3N x /4, +ˆK e e x y on the link connecting (n, N y /2) and (n + 2, N y /2 + 1) for N x /4 ≤ n < 3N x /4 − 1, and −ˆK e e x y on the link connecting (n, N y /2) and (n, N y /2 + 1); meanwhile between the two nearest neighbor pairs (N x /4, N y /2) and (N x /4, N y /2 + 1), (3N x /4, N y /2) and (3N x /4, N y /2 + 1) and also between the two next-nearest neighbor pairs (N x /4, N y /2 + 1) and (N x /4 + 1, N y /2), (3N x /4 − 1, N y /2 + 1) and (3N x /4, N y /2), all hoppings and pairings are set to zero. Note that for the case of proximity-induced superconductivity, the s-wave superconductor remains completely free of crystalline defects. While we set some of the parameters to be rather large for the sake of convenience in the numerical calculation, such choice does not affect the topological properties of the system. For instance, λ = 0.26 eV is about factor of 2 larger than the estimated value for the tantalum atom, while α 0 = 0.05 eV is several times larger than the estimated value from the magnetoconductivity measurement 65 . These choices are intended to increase the bulk energy gap so that our lattice size is sufficient to see a localized dislocation zero mode. This increase in the bulk energy gap occurs away from the BZ boundary k x,y = π, e.g. the larger α 0 increases the energy gap along k x = ± k y , while the larger λ lifts the higher d xz/yz band away from the Fermi level. Such changes do not affect weak indices, which are the 1D topological invariant along k x,y = π. Concerning real materials and experiments, as long as the system is in the topological regime and the induced bulk gap is large enough, i.e. much larger than the STM resolution, the dislocation Majorana zero mode and the zero energy anomaly can be detected clearly as shown in Fig. 2(c).
Lastly, we point out that with our p-wave pairing in Fig. 3(a) for h Z = 0 allows for a finite range of η for which there are nodal quasiparticles. While it is possible in principle to come up with a p-wave pairing for which the energy gap closes for only a single value of η, such p-wave pairing should have constant magnitude over the entire Fermi surface, which in general is not possible with our nearest-neighbor pairing.