Towards Scalable Braiding: Topological Superconductivity Unlocked under Arbitrary Magnetic Field Directions in Curved Planar Josephson Junctions
The non-Abelian statistics of Majorana zero modes (MZMs) are central to fault-tolerant topological quantum computing. Planar Josephson junctions provide a particularly versatile platform for realizing robust topological superconductivity hosting MZMs over a broad parameter space. However, it is generally believed that such topological superconductivity is restricted to a narrow range of in-plane magnetic field orientations, posing a major obstacle to scalable and noncollinear junction-network architectures. Here, we uncover that the apparent suppression of MZMs under misaligned fields does not arise from the destruction of topological superconductivity itself, but instead originates from emergent shifted bulk states at other momenta that obscure the global excitation gap and MZMs. By introducing spatial modulations along the junction to scatter and gap out these bulk states, we restore a global topological gap and recover MZMs for arbitrary in-plane magnetic field orientations. Remarkably, such modulations can be naturally realized by transforming a straight junction into a curved geometry, rendering the topological gap robust against field misalignment and enabling MZMs survival in complex junction networks. Building on this robustness, we propose a scalable protocol for MZMs braiding and fusion using gate or superconducting-phase control, opening new routes toward scalable topological quantum computing.