Two-Dimensional Topological Superconductivity with Antiferromagnetic Insulators

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.

Figure 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 $\mathcal{C}=2$ 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 $t^{\prime }=t$, ${\mathrm{\Delta }}_0=0.4t$, and $m_0=0.7t$.

Figure 2

Two structures of Dirac lines projected in the interface plane, either around the valleys $K$ ($K^{\prime }$) (a) and around $\mathrm{\Gamma }$ (b). The in-plane momentum is expressed in polar form by $p_r$ and $\varphi$, where $p_r$ denotes the radial distance to the Dirac line and $\varphi$ 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 ${\mathrm{\Psi }}_1^{†}$ and ${\mathrm{\Psi }}_2^{†}$ depend on the radial momentum $p_r$ and angle $\varphi$ and disperse linearly near the Dirac line (d).

Figure 3

(a), (b) Zero-energy modes $\omega ({\overrightarrow{k}}_{\parallel })=0$ 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 $t^{\prime }/t$; i.e., one could induce (a) with tensile ($t^{\prime }/t=0.7$) and (b) with compressive uniaxial strain ($t^{\prime }/t=1.5$). Upon introduction of spin-orbit coupling, the topological gap opens up, generating the Berry curvature $\mathrm{\Omega }$ localized around the former zero modes (c),(d). The two gapped spectra have a Chern number of $\mathcal{C}=2$ for (c) and $\mathcal{C}=-1$ for (d).

Figure 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 $t^{\prime }/t$ in the antiferromagnet drives a topological phase transition between two topological states, indicated by the gap closing at $t^{\prime }/t\approx 1.1$ independent of spin-orbit orbit coupling. (c) Phase diagram for the topological phases at $\mathrm{\Lambda }=0.05t$, ${\mathrm{\Delta }}_0=0.4t$, and $m_0=0.7t$ as a function of the superconducting chemical potential $\mu$ and $t^{\prime }/t$, which shows extended gapped regions with $\mathcal{C}=2$, $\mathcal{C}=-1$, and a gapless state for large $\mu$. The topological superconducting state also arises in a saw-shaped interface (d), suggesting that a perfect interface is not a necessary requirement.