Abstract
Two-dimensional topological superconductivity has attracted great interest due to the emergence of Majorana modes bound to vortices and propagating along edges. However, due to its rare appearance in natural compounds, experimental realizations rely on a delicate artificial engineering involving materials with helical states, magnetic fields, and conventional superconductors. Here we introduce an alternative path using a class of three-dimensional antiferromagnets to engineer a two-dimensional topological superconductor. Our proposal exploits the appearance of solitonic states at the interface between a topologically trivial antiferromagnet and a conventional superconductor, which realize a topological superconducting phase when their spectrum is gapped by intrinsic spin-orbit coupling. We show that these interfacial states do not require fine-tuning, but are protected by asymptotic boundary conditions.
| Original language | English |
|---|---|
| Article number | 037002 |
| Journal | Physical Review Letters |
| Volume | 121 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 17 Jul 2018 |
| MoE publication type | A1 Journal article-refereed |
Funding
Lado J. L. Sigrist M. Institute for Theoretical Physics , ETH Zurich, 8093 Zurich, Switzerland 17 July 2018 20 July 2018 121 3 037002 8 March 2018 © 2018 American Physical Society 2018 American Physical Society Two-dimensional topological superconductivity has attracted great interest due to the emergence of Majorana modes bound to vortices and propagating along edges. However, due to its rare appearance in natural compounds, experimental realizations rely on a delicate artificial engineering involving materials with helical states, magnetic fields, and conventional superconductors. Here we introduce an alternative path using a class of three-dimensional antiferromagnets to engineer a two-dimensional topological superconductor. Our proposal exploits the appearance of solitonic states at the interface between a topologically trivial antiferromagnet and a conventional superconductor, which realize a topological superconducting phase when their spectrum is gapped by intrinsic spin-orbit coupling. We show that these interfacial states do not require fine-tuning, but are protected by asymptotic boundary conditions. Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung 10.13039/501100001711 163186 Japan Society for the Promotion of Science 10.13039/501100001691 Topological matter represents one of the most intriguing frameworks to realize unconventional physics, due to d - 1 -dimensional excitations originating from topological properties of the d -dimensional systems [1–3] . This allows us to realize electronic spectra in solid-state platforms, which resemble those found for some elementary particles in high-energy physics and exhibit often an even larger variety. Topologically nontrivial band structures give rise to chiral modes in Chern insulators [4] , helical modes in quantum spin Hall insulators [5] , and Majorana modes in topological superconductors [6–8] . In particular, Majorana zero-energy modes in one-dimensional (1D) topological superconductors have fostered intense research efforts both in their detection [9–11] and manipulation [12–15] , motivated by their potential for topological quantum computing [16,17] . Yet, one of the biggest challenges is that nature lacks materials with 1D topological superconductivity, and thus the only hope for its experimental realization relies on an artificial engineering in nanostructures [18–25] . Two-dimensional (2D) topological superconductors share the exciting phenomena as their 1D counterparts, while providing additional flexibility. On the one hand, Majorana bound states can be found in vortex cores [26–28] , which display properties of interest for topological quantum computing [29] . On the other hand, propagating excitations at edges may allow the exploration of the physics of supersymmetry associated with interacting Majorana fermions [30–33] . However, natural 2D topological superconductors are rather elusive [34] , rendering artificial engineering of 2D topological superconductors an important milestone, very much like their 1D counterparts. This further motivated extensions of the original mechanisms for 1D topological superconductivity to two dimensions, based on topological insulators [8] , Yu-Shiba-Rusinov lattices [25] , or 2D electron gases with Rashba spin-orbit coupling [35] . In this Letter, we introduce an alternative route to create topological superconductivity, exploiting an interface between two bulk ordered phases. Our proposal consists of a heterostructure formed by an insulating bulk antiferromagnet and a conventional bulk superconductor [Fig. 1(a) ]. Individually, both systems have an excitation gap, both in the bulk as well as at the surface. However, for a special class of antiferromagnetic insulators, as we will discuss below, protected gapless Andreev bound states emerge at the interface between the two 3D systems. These states are mathematically similar to the Jackiw-Rebbi soliton [36] , so that interfacial modes exist irrespective of the magnitudes and spatial profiles of the two electronic orders. Furthermore, once intrinsic spin-orbit coupling is introduced, the interface states open a gap, giving rise to a topological superconducting state [Fig. 1(b) ]. Therefore, this mechanism shows that antiferromagnetic insulators, commonly overlooked, are potential candidates to engineer topological superconductors. 1 10.1103/PhysRevLett.121.037002.f1 FIG. 1. (a) Schematic graph of a three-dimensional superconducting-antiferromagnet heterostructure, where a two-dimensional topological superconductor emerges at the interface (b). Two branches of subgap quasiparticle excitations appear at the interface, which have zero-energy modes in the absence of spin-orbit coupling (c) and are gapped for nonvanishing spin-orbit coupling (d). In the latter case, the system is topological with a Chern number C = 2 leading to two chiral edge modes (e). This topological phase is robust and exists in a wide parameter range besides a gapless and a trivial superconducting phase (f). For (c),(d), we consider a sharp interface ( W = 0 ), with t ′ = t , Δ 0 = 0.4 t , and m 0 = 0.7 t . The key ingredient for our proposal is the existence of Dirac lines [37–45] , lines of points in the Brillouin zone where the low energy model is a Dirac equation, in the nonmagnetic state of the antiferromagnet. There is no specific requirement for the superconductor, apart from having a conventional s -wave Cooper pairing. For the sake of concreteness, we start by introducing a minimal model that exemplifies such a phenomenology. For this purpose, we take an antiferromagnetic diamond lattice with lattice constant a , which can be viewed as a three-dimensional analog of the antiferromagnetic honeycomb lattice [46] . Such a structure would be the minimal model for an antiferromagnetic spinel X Y 2 Z 4 , with the magnetic ions sitting in the X sites [47–51] . In order to describe the antiferromagnet-superconductor heterostructure, we propose a Hamiltonian consisting of electron hopping H kin , antiferromagnetic ordering H AF , superconducting s -wave pairing H SC , and spin-orbit coupling H SOC [52] : H ^ = H ^ kin + H ^ AF + H ^ SC + H ^ SOC with H ^ kin = ∑ ⟨ i j ⟩ , s t i j c i , s † c j , s - ∑ i , s μ ( z i ) c i , s † c i , s , H ^ AF = ∑ i , s , s ′ m ( z i ) τ z i , i σ z s , s ′ c i , s † c i , s ′ , H ^ SC = ∑ i Δ ( z i ) [ c i , ↓ c i , ↑ + c i , ↑ † c i , ↓ † ] , H ^ SOC = ∑ ⟪ i j ⟫ i Λ σ → s , s ′ · ( r → i l × r → l j ) c i , s † c j , s ′ . (1) The parameters are chosen so that the Hamiltonian describes an insulating antiferromagnet for z < 0 , with magnetization perpendicular to the interface, and a conventional superconductor for z > 0 . In this way, the electronic spectra of the previous Hamiltonian has an antiferromagnetic gap for z = - ∞ and a superconducting gap for z = + ∞ . We may take m ( z ) = m 0 [ 1 - tanh ( z / W ) ] / 2 the antiferromagnetic order parameter, Δ ( z ) = Δ 0 [ 1 + tanh ( z / W ) ] / 2 the superconducting order parameter, and μ ( z ) = μ 0 [ 1 + sgn ( z ) ] / 2 the chemical potential fixing half filling on the antiferromagnetic side. The parameter W controls the smoothness of the change between the two orders, which in the limit W → 0 becomes sharp. Spin-orbit coupling enters as a next-nearest-neighbor hopping [52] between sites i and j , and r → i l ( r → l j ) is the vector between nearest neighbors i ( j ) and l . We denote τ → and σ → the Pauli matrices for the sublattice ( A and B ) and the spin, respectively. The heterostructure within the fcc lattice is chosen so that the interface (perpendicular to the z axis) consists only of sites belonging to one of the two sublattices, i.e., a zigzaglike interface. Using the standard fcc lattice vectors a → 1 , a → 2 , a → 3 , we can also define the interface plane by two of them, say a → 1 and a → 2 , such that the z axis is parallel to a → 1 × a → 2 . The first interesting finding is that, in the absence of spin-orbit coupling ( Λ = 0 ), the spectrum of the combined structure develops gapless quasiparticle excitations at the interface [Fig. 1(c) ]. These gapless Andreev modes are protected against different choices of the interface profile for the antiferromagnetic order, the superconducting order, and the chemical potential. Because of their robustness and structure shown below, we refer to these protected Andreev modes as solitonic states. Switching on spin-orbit coupling ( Λ ≠ 0 ) leads to a fully gapped spectrum for the solitonic states [Fig. 1(d) ]. The second remarkable observation is the appearance of the topological Chern invariant C = 2 for the gapped system, indicating the presence of two propagating Majorana modes at the edges of the interface [Fig. 1(e) ]. This chiral state relies on the broken time reversal symmetry due to the antiferromagnetic order. The emergence of this topological insulating state by combining two topologically trivial insulating systems is the main finding of our Letter. This topological superconducting state is robust upon changing parameters [Fig. 1(f) ], raising two questions. First, why does the interface between the two topologically trivial gapped materials show robust zero-energy modes? Second, why does including a small spin-orbit coupling give rise to a topological superconducting state? We first address the origin of the gapless interface states, starting with the Bloch Hamiltonian for the pristine diamond lattice H ^ kin = ∑ k → , s f ( k → ) c A , k → , s † c B , k → , s + c . c . , where f ( k → ) = t [ 1 + e i k → · a → 1 + e i k → · a → 2 ] + t ′ e i k → · a → 3 , where a → 1 , a → 2 , a → 3 are the lattice vectors of the fcc lattice and t ′ = t corresponds to the cubic symmetry. The spectrum possesses lines in k space where the valence and conduction band touch. The projected two-dimensional Brillouin zone perpendicular to the z axis is hexagonal with the Γ point (line) in the center and the K and K ′ points (lines) at the boundary. Depending on the ratio t ′ / t , one Dirac line forms around the Γ point or two disconnected Dirac lines form around K and K ′ points [Figs. 2(a) and 2(b) ] [42] . Focusing on such a Dirac line, we can formulate an effective low-energy Hamiltonian H ^ D = ∑ k → , s ( p z - i p r ) c A , k → , s † c B , k → , s + c . c . We use that the momentum p → is tied to the reference frame of the line, such that p ϕ is tangential to the line, p r is perpendicular to the line, and the z axis and p z are perpendicular to the two other components, slightly tilted with respect to the z axis. This low-energy model allows us to study the interface between the superconductor and antiferromagnet analytically. Using the spatially dependent order parameters Δ ( z ) and m ( z ) as introduced above, the effective Hamiltonian takes the form H ^ = ∑ i , j , s [ τ x i , j p z + τ y i , j p r ] c i , k → ∥ , s † c j , k → ∥ s + ∑ i , s m ( z ) τ z i , i σ z s , s c i , k → ∥ , s † c i , k → ∥ s + ∑ i Δ ( z ) c i , - k → ∥ , ↑ c i , k → ∥ , ↓ + c . c . , (2) where k → ∥ is the conserved Bloch momentum parallel to the interface, i , j sum runs over the two sites A , B , and μ = 0 . 2 10.1103/PhysRevLett.121.037002.f2 FIG. 2. Two structures of Dirac lines projected in the interface plane, either around the valleys K ( K ′ ) (a) and around Γ (b). The in-plane momentum is expressed in polar form by p r and ϕ , where p r denotes the radial distance to the Dirac line and ϕ parametrizes the angle. For p r = 0 , each point in the Dirac line will give rise to the states of Eq. (3) , located at the interface as shown in (c). The two interface states Ψ 1 † and Ψ 2 † depend on the radial momentum p r and angle ϕ and disperse linearly near the Dirac line (d). The Hamiltonian (2) defines a system that is inhomogeneous along the z direction, where for z → - ∞ the Hamiltonian is purely antiferromagnetic and for z → ∞ purely superconducting. Remarkably, for p r = 0 and a profile fulfilling these asymptotic conditions, two solitonic zero-energy Andreev modes exist localized at the interface [Fig. 2(c) ], with the following ansatz [9,36] Ψ α , k → ∥ † ( z ) = g ( z ) [ c A , k → ∥ , ↑ † - c A , - k → ∥ , ↓ + ( - 1 ) α i ( c B , k → ∥ , ↑ † + c B , - k → ∥ , ↓ ) ] , (3) where g ( z ) = C exp [ ∫ 0 z [ m ( z ′ ) - Δ ( z ′ ) ] d z ′ ] , C is the normalization constant, and α = 1 , 2 is the branch index. Note that, although these states are pinned to zero energy, they are not Majorana modes. Furthermore, such states will also exist in the more generic case | Δ ( z → ∞ ) | > | m ( z → ∞ ) | and | m ( z → - ∞ ) | > | Δ ( z → - ∞ ) | . Away from p r = 0 , the two solitonic wave functions have a finite energy dispersion in the direction of the radial momentum p r [Fig. 2(d) ], yielding the effective Hamiltonian H ^ = - ∑ α ( - 1 ) α v r p r Ψ α , k → ∥ † Ψ α , k → ∥ . The existence of these states for each point of the Dirac line implies that the zero mode surface of the heterostructure reflects the original Dirac lines of the antiferromagnet. Thus, any change of the Dirac line structure would be reflected in these zero-energy interface modes, as shown in Figs. 3(a) and 3(b) . 3 10.1103/PhysRevLett.121.037002.f3 FIG. 3. (a), (b) Zero-energy modes ω ( k → ∥ ) = 0 obtained for the Hamiltonian of Eq. (1) in the absence of spin orbit. The transition between the two states (a),(b) is controlled by the ratio t ′ / t ; i.e., one could induce (a) with tensile ( t ′ / t = 0.7 ) and (b) with compressive uniaxial strain ( t ′ / t = 1.5 ). Upon introduction of spin-orbit coupling, the topological gap opens up, generating the Berry curvature Ω localized around the former zero modes (c),(d). The two gapped spectra have a Chern number of C = 2 for (c) and C = - 1 for (d). In a next step, we introduce the intrinsic spin-orbit coupling, equivalent to a momentum and sublattice dependent exchange field. In the vicinity of the Dirac lines, it takes the effective form H ^ SOC ∝ Λ τ z i , i [ - sin ( ϕ ) σ x s , s ′ + cos ( ϕ ) σ y s , s ′ ] c i , k → ∥ , s † c i , k → ∥ , s ′ , where ϕ denotes the position on the Dirac line as shown in Figs. 2(a) and 2(b) . We note that, for the different situations of the Dirac lines, the SOC takes a vortexlike profile, but with opposite vorticities around Γ and K , K ′ . By projecting the spin-orbit coupling term onto the solitonic basis, we arrive at the following low-energy Hamiltonian, H ( p r , ϕ ) = ( Ψ 1 , k → ∥ † Ψ 2 , k → ∥ † ) ( v r p r - i λ e i ϕ i λ e - i ϕ - v r p r ) ( Ψ 1 , k → ∥ Ψ 2 , k → ∥ ) , (4) where λ ( Λ , μ , W ) = ± | ⟨ Ψ 1 | H SOC | Ψ 2 ⟩ | ∝ Λ generates a gap in the spectra and finite Berry curvature where the zero modes were located [Figs. 3(c) and 3(d) ]. The gap is linear in the spin-orbit coupling and depends on the chemical potential and the profile width W [Fig. 4(a) ]. This Hamiltonian has the structure of a chiral p -wave superconductor, since the spin-orbit coupling λ takes the form of a chiral gap function. In this way, the superconducting phase in the interface acquires chirality with a nonvanishing Chern number, if λ ≠ 0 . 4 10.1103/PhysRevLett.121.037002.f4 FIG. 4. (a) The gap of the topological phase as a function of spin-orbit coupling calculated for the full model in Eq. (1) shows a linear scaling in different regimes. (b) Varying t ′ / t in the antiferromagnet drives a topological phase transition between two topological states, indicated by the gap closing at t ′ / t ≈ 1.1 independent of spin-orbit orbit coupling. (c) Phase diagram for the topological phases at Λ = 0.05 t , Δ 0 = 0.4 t , and m 0 = 0.7 t as a function of the superconducting chemical potential μ and t ′ / t , which shows extended gapped regions with C = 2 , C = - 1 , and a gapless state for large μ . The topological superconducting state also arises in a saw-shaped interface (d), suggesting that a perfect interface is not a necessary requirement. The Lifshitz transition found in the paramagnetic phase of the antiferromagnetic side by varying the hopping ratio t ′ / t [Figs. 2(a) and 2(b) ] has a final consequence for the gapped interface modes: this Lifshitz transition introduces a topological transition for the superconducting phase of the heterostructure. For t ′ < t , each of the two Dirac lines contributes through a single phase winding, adding together to a Chern number C = 2 [Fig. 3(c) ], while for t ′ > t , there is only a single Dirac line, winding in the opposite orientation around Γ , leading to C = - 1 for the interface superconductor [Fig. 3(d) ]. Because of corrections to the low-energy model, the topological phase transition found by exactly solving the model does not coincide perfectly with the bulk Lifshitz transition, but happens at t ′ slightly higher than t , as visible in Fig. 4(b) , which shows a gap closing at this transition point. This specific transition point depends on the chemical potential μ as shown in Fig. 4(c) , depicting a phase diagram with two topological phase transitions, from a gapless superconductor to the topological sector C = 2 and then C = - 1 . Our calculations demonstrate that the topological phases are robust, and their existence does not depend on details of the electronic structure of the superconductor, but is determined by the topology of the Dirac lines of the magnetic side. Since the symmetric case t ′ / t = 1 belongs to the sector C = 2 , the sector C = - 1 could be reached through uniaxial strain perpendicular to the interface, increasing t ′ / t . A last important issue, especially for future experimental realizations, is whether topological phases are sensitive to the quality of the interface. To test this, we now consider a saw-shaped interface, i.e., a tilted interface orientation yielding a periodicity ( 3 , 1 ) × ( 3 , 1 ) of the original unit cell. We observe that even for this “imperfect” heterostructure the interface develops a topological phase with C = - 1 for t ′ = t [Fig. 4(d) ]. The intervalley scattering induced by the interface supercell shifts the system to the sector C = - 1 . Similar results are obtained for other interface orientations, with the exception of the armchair interface, where the two sublattice sites appear in equal number at the interface. This result demonstrates that the topological phase can be ascribed to the robustness of the parent solitonic states and generically requires an imbalance between the two sublattice sites. Using a minimal model, we have shown how to engineer topological superconductivity connecting an insulating antiferromagnet with a conventional superconductor. While we use a single-orbital model, multiorbital extensions of Eq. (1) could, for example, capture the physics of antiferromagnetic spinels, such as CoAl 2 O 4 , which realizes an insulating antiferromagnetic diamond lattice [48] . However, it is unclear so far whether this material generates in the paramagnetic state the necessary Dirac lines (see Supplemental Material [53] ). 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