Route to Topological Superconductivity via Magnetic Field Rotation

The verification of topological superconductivity has become a major experimental challenge. Apart from the very few spin-triplet superconductors with p-wave pairing symmetry, another candidate system is a conventional, two-dimensional (2D) s-wave superconductor in a magnetic field with a sufficiently strong Rashba spin-orbit coupling. Typically, the required magnetic field to convert the superconductor into a topologically non-trivial state is however by far larger than the upper critical field Hc2, which excludes its realization. In this article, we argue that this problem can be overcome by rotating the magnetic field into the superconducting plane. We explore the character of the superconducting state upon changing the strength and the orientation of the magnetic field and show that a topological state, established for a sufficiently strong out-of-plane magnetic field, indeed extends to an in-plane field orientation. We present a three-band model applicable to the superconducting interface between LaAlO3 and SrTiO3, which should fulfil the necessary conditions to realize a topological superconductor.

the minimal Zeeman splitting required to reach the topological phase is where  0 measures the distance of the band energy at k = 0 to the Fermi energy, and Δ is the SC energy gap 5 . The obstacle for realizing this topological state experimentally is to find a system which remains superconducting in the required high magnetic fields. Suggested model systems are, e.g., neutral ultra-cold atoms in an optical trap 5 , or heterostructures where Cooper pairs are induced through the proximity effect 2, [13][14][15][16] .
The problem of realizing the topological s-wave state has two distinct aspects: (i) μ B H t must be larger than Δ . While the presence of the Rashba SOC allows in principle s-wave superconductivity in a Zeeman field larger than Δ , the orbital critical field H c2 is typically much smaller. (ii) The superconductor must have ε 0 smaller than the Zeeman splitting. This requires a low band filling and, therefore, superconductivity must be stabilized by yet another band with larger filling. In this article, we address both of these aspects and demonstrate that the problems can be overcome in real solid-state systems.
A simple way to circumvent the orbital critical field H c2 is to rotate the magnetic field into the plane of the 2D superconductor. The in-plane field however leads to an unusual type of pairing. In the presence of Rashba SOC, an in-plane magnetic field shifts the Fermi surfaces out of the Brillouin-zone center (cf. Fig. 1), and the electron pairs thereby acquire a finite center-of-mass momentum (COMM) 11 . Edge states in an in-plane magnetic field have recently been investigated for p-wave superconductors, but with zero COMM 4,17 . As we show here, the inclusion of a finite COMM in such a field geometry is indispensable for the discussion of topology. Specifically, we analyze the topological properties of an s-wave superconductor under rotation of the magnetic field within a fully self-consistent treatment of the SC order parameter. It is verified that the topological state reached in out-of-plane fields indeed persists to in-plane field orientations, if the COMM is appropriately chosen to minimize the free energy. For in-plane fields the energy gap closes, accompanied by a topological transition. Nevertheless, chiral edge modes remain even for a regime with a closed gap.
We discuss the experimental realizability of a topological s-wave superconductor in a nearly in-plane magnetic field. As a candidate system, which can possibly fulfil the required conditions, we consider the metallic LaAlO 3 -SrTiO 3 (LAO-STO) interface 18,19 . For this system, several models are proposed for a topologically non-trivial superconducting state, which rely on an unconventional order parameter [20][21][22] . Assuming instead an s-wave pairing state, we demonstrate that a multi-band model involving the titanium t 2g orbitals allows for a topologically non-trivial superconductor in a realistic parameter regime for the LAO-STO interface. We suggest that it may be achieved with the currently used experimental setups.

Results
In order to investigate the magnetic-field dependence of an s-wave superconductor with Rashba SOC in transparently simple terms, we use a one-band tight-binding model on a square lattice in the x-y-plane at zero temperature. In our analysis of the topology upon rotating the magnetic field H into the plane, we include the Zeeman coupling of the electrons to the magnetic field, but neglect the orbital coupling. This approximation is well justified for the nearly in-plane field orientation on which we focus here; but orbital effects are necessarily important for the superconducting state in an out-of-plane magnetic field.
The Rashba SOC and the Zeeman coupling to the magnetic field H are combined into Figure 1. Energy bands ξ + k (pink) and ξ − k (blue) and Fermi surfaces with α > 0 and magnetic filed |H| < H t . (a) For H = 0, the two bands touch at k = 0. (b,c) For |H| > 0, the band splitting at k = 0 is equal to the Zeeman splitting 2μ B |H|. The centers of the shifted Fermi surfaces in (c) are at the momenta momenta q + /2 = (q + /2, 0) and q − /2 = (q − /2, 0), respectively. Although q + ≈ − q − , their absolute values are in general different. In (c), the spins on the k x -axis orient according to the magnetic field rather than according to the SOC, if H y > α|sin k F |. Note that the Fermi energy is somewhat larger than |ε 0 | because of the SOC induced band splitting. with s = ± 1, the Bloch vector h k = αg k + μ B H, and g k = (sin k y , − sin k x , 0); σ is the vector with the Pauli matrices as components. The strength of the Rashba SOC α derives originally from the Dirac Hamiltonian, but may have other sources in multi-band systems (see section "Discussion"). Diagonalizing the kinetic energy together with  S gives the two chiral energy bands with the nearest-neighbor hopping amplitude t and the chemical potential μ (thus  µ = − − t 4 0 ). In these bands, the spin is either parallel or antiparallel to h k and has a component which rotates either counter-clockwise or clockwise upon circulating the Fermi surfaces (see Fig. 1).
Out-of-plane magnetic field. For an out-of-plane magnetic field with H x = H y = 0, the topological properties of the superconducting state are readily established (see e.g. Ref. 5). Its Hamiltonian and σ 0 is the 2 × 2 unit matrix; the pairing field Δ is calculated self-consistently from Eq. (7) (with q = 0). The four eigenenergies E k,n obtained from diagonalizing (2) are generally the solutions of a 4th order polynomial, but simplify to for k x = k y = 0, since g k=0 = 0. The number n labels the four combinations of the plus and minus signs. It follows that the energy gap closes at k = 0 for , which thereby allows for a topological transition 5,6 .
The topological character of the SC state is given by the TKNN number n n n z k k is the z-component of the Berry curvature 12 . The sum over n runs over the occupied bands E k,n < 0 and λ = 1, …, 4 labels the components of the eigenvectors u n (k) of the matrix H k . The number C is integer valued, if the occupied energy levels are separated by a finite gap from the unoccupied levels. The value of C and therefore the topology of the quantum state changes when the energy gap closes at |H z | = H t . For magnetic fields |H z | > H t , the energy gap opens again. This reopening of a gap, above the paramagnetic limiting field μ B |H z | = Δ , is tied to the presence of SOC, which protects the spin-singlet pairing channel. (In the presence of SOC, the spin susceptibility χ S of the spin-singlet superconductor remains finite down to T = 0. In particular, if Δ ≪ α, χ S is almost equal the Pauli susceptibility of the normal state. Therefore, the Clogston-Chandrasekhar paramagnetic limit = ∆/ H 2 CC 23 does also not apply.) Only the momenta k for which ξ ± k lie within a window Δ below the Fermi energy contribute to C. The sign of this contribution reflects the winding direction of the x-y components of the spin σ ( / ) , in momentum space (see Fig. 1). If |H z | < H t , the k-integrated Berry curvatures in the vicinity of the two normal-state Fermi surfaces cancel exactly [ Fig. 2(a)] and consequently C = 0.
The topological state emerging for |H z | > H t is of different nature in two distinct density regimes: (A) small electron density (μ < − 2t): the condition |H z | > H t leads to ξ > ∆ + k for all k and therefore the ξ + k -band is empty and does not contribute, i.e., the pink (positive) contributions to C in Fig. 2(a) vanish. Consequently, the superconducting state is characterized by =  C 1, depending on the sign of H z . This situation is realized for small band fillings. (B) densities near half-filling (|μ| < 2t): in this regime, two separate topological transitions are possible.
At a magnetic field B t , the character of the ξ + k -band changes from particle-to hole-like and thereby reverses the sign of Ω (k) in the vicinity of the corresponding Fermi surface [ Fig. 2(b)]. Therefore, a topological transition to C = ± 2 occurs, depending on the sign of H z , with both, the ξ − k -and the ξ + k -band, partially occupied. A realization of this superconducting state close to half filling is however unlikely due to possibly competing orders. For an even larger magnetic field μ B |H z | > H t , the ξ + k -band is again lost, and C changes to = ± 1. The topological properties of both cases, (A) and (B), correspond to those described in ref. 4 in the context of spin-triplet superconductors.
A special characteristic feature of s-wave superconductivity in the presence of Rashba SOC is that the magnetic field induces an inter-band pairing contribution where a quasi-particle of the ξ + k -band is paired with one from the ξ − k -band at opposite momentum. This pairing contribution induces interior energy gaps above and below the Fermi energy (clearly visible e.g. in the spectra presented below, cf. ref. 24). A more detailed discussion of the relation between intra-and inter-band pairing is given in the Supplementary Informations. The SC ground state with an in-plane magnetic field component therefore contains in general two order parameters ∆ + q and ∆ − q . These enter the generalized on-site pairing term as

In
The singlet order parameter for COMM q is calculated self-consistently from where V is the strength of the pairing-interaction. With increasing in-plane magnetic-field strength | | H , the difference |q + − q − | grows. Such a finite COMM state is spatially non-uniform 28 with lines of zero pair density, similar to the SC state proposed by Larkin and Ovchinnikov for a singlet superconductor in a strong Zeeman field 29 . Characteristic for this state is a mixing of intra-and inter-band pairing and the absence of a full energy gap (see Supplementary Informations) 11,28 .
For | | ≠ H 0, the topological characterization is more involved. In situation (B), close to half-filling, both bands ξ + k and ξ − k are partially occupied and consequently two COMMs q ± appear. Therefore no full energy gap is present, which implies that C is not an integer and therefore unsuitable to characterize the topology (nevertheless, edge modes may still occur, see Supplementary Informations). For this reason, we focus below on situation (A) when only the lower band ξ − k is occupied and therefore only electron pairs with COMM q − form. Such a state is spatially uniform and similar to the state introduced by Fulde and Ferrell 30 , but it carries a finite charge current perpendicular to H . If also H z ≠ 0, it exhibits a full energy gap and therefore C is integer-valued, except for a limited crossover region < < + H H H discussed below. Equation (3) and therefore the definition of H t is valid as well in the presence of an in-plane magnetic field component whereby H z is replaced by |H|.
If |H| > H t , one finds that α|g k | < μ B |H| for all occupied momenta k in the ξ − k -band. Furthermore, if H is strictly in-plane, say along the y-axis, h k is parallel to H on the k x -axis and therefore the spins are parallel (ξ − k -band) or anti-parallel (ξ + k -band) to H as well [ Fig. 1(c)]. As a consequence, no intra-band pairing in the singlet channel is possible for k y = 0, i.e., the intra-band energy gap closes at the two Fermi points with k y = 0. This gap closing for an in-plane field orientation corresponds to a topological transition from C = − 1 to C = 1.
The topological phase of situation (A) with an in-plane magnetic field component is described by the same Hamiltonian as in Eq. (2) replacing Δ by ∆ − q . In the following we show that the topological state found for a sufficiently strong out-of-plane magnetic field can persist when the field is rotated -even down to an in-plane field orientation with arbitrarily small H z .

Phase diagram and edge states.
We start the analysis of the SC state with the discussion of the self-consistent solutions of the SC order parameter. Figure 3 shows the magnetic-field dependence of ∆ − q for different angles θ of the field direction. The Rashba SOC ensures the presence of a finite in-plane spin component which allows for singlet pairing. Therefore the Zeeman coupling to a field in z-direction (θ = 0° and q − = 0) cannot wipe out superconductivity completely (pink curve) when orbital depairing is not included. A finite in-plane field component leads instead to a finite critical magnetic field H c (θ), above which there are no solutions for ∆ − q . An interesting result is the somewhat larger value for Δ q in an in-plane magnetic field θ | | < ( ) H H c than in an out-of-plane magnetic field of the same magnitude. Consequently, the magnetic field H t (θ) at which the energy gap closes, grows with increasing angle θ and is maximal for an in-plane direction. Likewise, the field θ ( ) + H t above which the energy gap opens again and the topological state emerges, is maximal for θ = 90°, whereas The resulting phase diagram for different magnetic-field orientations is qualitatively drawn in Fig. 4. The topologically trivial SC state (C = 0) is bounded by the ellipse given by H t (θ), which itself is within the slightly larger ellipse formed by θ ( ) + H t . The white regime in between separates the state with C = 0 from the states with C = ± 1. In this crossover region the energy gap is closed and C is not an integer. A further topological transition occurs for the in-plane field orientation θ = 90° and > + H H t : if θ sweeps through 90°, the out-of-plane field component H z changes sign and, accordingly, C changes from − 1 to 1. As discussed above, the energy gap is closed as well along this transition line.
The importance of finite-COMM pairing for the topological properties of the SC state is illustrated using the energy spectra shown in Fig. 5. These spectra are calculated for a stripe geometry with open boundary conditions in y-direction, which allows for in-gap edge modes (drawn in red and green). We choose the in-plane magnetic-field component in the y-direction and thereby obtain a shift of the Fermi surfaces out of the Brillouin-zone center in k x -direction. Therefore a COMM q − = (q, 0) (q ≥ 0 for H y > 0) has to be taken into account for pairing in the ξ − k -band. The free energy of the SC state is minimized for the smallest q which is still large enough to avoid an indirect closing of the energy gap (see Fig. 5(a-c) Fig. 5(c)], the energy gap closes at two k x -points, and a topological transition occurs. The magnetic field strength, above which a finite q − is required (dashed line in Fig. 4), depends on the angle θ. In the limit θ → 90°, it is necessarily smaller than H t and approaches H t for vanishing SOC. For every change in the field orientation the COMM q − has to be recalculated self-consistently. Since the calculations are performed on a finite lattice, q − evolves necessarily in discrete steps, determined by the system size, upon sweeping the angle θ. Up to this unavoidable discreteness, the onset of a finite q − is a smooth transition, across which the energy gap evolves continuously.
Eventually, Fig. 5 also shows the evolution of the in-gap edge modes (green and red lines) under the rotation of the magnetic field. (Note that for θ > 0, the direction of the in-plane field component relative to the boundary is important. If the in-plane field is orthogonal to the boundary (here: y-direction), the k x -dispersion of the edge mode is unperturbed and remains gapless. If there is a field component parallel to the boundary (in x-direction), the edge-mode disperses in y-direction as well. Since the dispersion in y-direction is quantized through the edges, the edge modes acquire an energy gap around the Fermi energy (also observed for the edge modes in chiral p-wave superconductors 4,17 ). This mesoscopic energy gap is of the size of the normal-state level spacing around E F and vanishes like 1/M in the thermodynamic limit, where M is the number of lattice sites in y-direction.) Starting from θ = 0°, the energy difference ξ ξ − − − − k k grows for increasing θ and thus the dispersion of the two opposite edge modes becomes asymmetric [ Fig. 5(b)]. Upon approaching θ = 90°, the energy gap closes at two k x points [ Fig. 5(c)]. Consequently, q must be chosen to ensure that these closing points are located at the Fermi energy in order to prevent the energy bands above and below from overlapping. In this situation, the two edge modes are degenerate. Indeed, the two modes carry edge currents flowing in the same direction opposite to the flow of the bulk current. These modes are similar to the edge modes found for p-wave superconductors in an in-plane magnetic field 17,31,32 , except for the presence of a finite COMM pairing due to the Rashba SOC. . At H y = H t (90°), the energy gap closes at k x = 0 [ Fig. 5(d)]. However, the minimum of the ξ + k -band is at a momentum k x < 0 and somewhat below the Fermi energy (indicated by the left black arrow), whereas the maximum of its mirrored hole-band is at a momentum k x > 0 somewhat above the Fermi energy (right black arrow). Thus, the ξ + k -band and its mirrored hole band ξ − − + For magnetic fields > + H H y t , all states of the ξ + k -band are above the Fermi energy. In this regime an infinitesimally small out-of-plane magnetic-field component H z is sufficient to remove the two gap-closing points and ensure well defined TKNN numbers C = ± 1. Eventually, superconductivity breaks down at H y = H c (θ): Above this critical magnetic field, the two gap-closing points move into the continuum of the energy bands above and below the Fermi energy. The upper and lower bands therefore overlap and the self-consistent solution for the SC order parameter is lost. Although θ ( ) + H t depends only weakly on the SOC strength α, the critical magnetic field H c (θ) grows with increasing α. In order to obtain (°) > (°)

Discussion
How can the above topologically non-trivial SC state be realized in a solid-state system? We infer that an ideal candidate system would consist of several partially filled energy bands with a sizable Rashba SOC. Such a model 33,34 was proposed e.g. to describe the physics of the conducting interface between LaAlO 3 and SrTiO 3 18,19 .
The condition  µ > + ∆ H B 0 2 2 for C ≠ 0 implies that the lower limit for the Zeeman splitting is given by Δ . The corresponding magnetic field is typically larger than the upper critical field H c2 above which orbital depairing destroys superconductivity. Therefore the topological state is not accessible with a magnetic field oriented along the z-axis. The topological state can be reached only for a nearly in-plane field orientation with H z < H c2 but > + H H t . This excludes the situation (B) with a close to half-filled band, because of the presence of two different COMMs q ± as discussed above. The alternative situation, on the other hand, requires that  µ > H B 0 , which is close to the Fermi energy E F in a one-band model. For a band filling large enough to allow for a SC state (for a reasonable interaction strength V), E F for this partially filled band must be at least several meV. The magnetic field required to overcome this energy would be far too large for experimental realizations.
In a multi-band setup,  0 refers to the energy of the degeneracy point of a spin-orbit coupled doublet hosting the possibly topological state (see Fig. 6), relative to E F . The necessity of stabilizing a superconducting state with  → 0 0 requires the presence of at least two bands at E F : A lower band provides the electron density for a sufficient gain of condensation energy in the superconducting state, whereas a second band has a minimum close enough to the Fermi energy so that it can be emptied or partially filled through an external control parameter.
A model which implements these features, and additionally also a strong Rashba-like SOC, recently emerged from the theoretical description of the LAO-STO interface. At this interface, the intrinsic electrostatic potential in LAO induces a nearly 2D electron liquid, which resides mainly in the titanium 3d t 2g orbitals of the first TiO 2 layer 19,35 . The band-structure of this oxide interface provides a prototype for a class of similar interface systems that we introduce here by constructing a tight-binding Hamiltonian yz . Furthermore, the in-plane d xy orbital is lowered in energy by Δ 0 relative to the out-of-plane d xz and d yz orbitals because of the symmetry breaking interface.
The spin-orbit coupling on the Ti atoms is described by , where the angular momentum operator L for l = 2 is represented in the {d xz , d yz , d xy } basis 33 . This term intermixes the t 2g orbitals and generates three doublets; the upper two doublets are split by 2Δ SO (see Fig. 6). In