One-dimensional topological superconductivity based entirely on phase control

Topological superconductivity in one dimension requires time-reversal symmetry breaking, but at the same time, it is hindered by external magnetic fields. We offer a general prescription for inducing topological superconductivity in planar superconductor-normal-superconductor-normal-superconductor (SNSNS) Josephson junctions without applying any magnetic fields on the junctions. Our platform relies on two key ingredients: the three parallel superconductors form two superconductor-normal-superconductor junctions with phase winding, and the Fermi velocities for the two spin branches transverse to the junction must be different from one another. The two phase differences among the three superconductors define a parameter plane which includes large topological regions. We analytically derive the critical curves where the topological phase transitions occur and corroborate the result with a numerical calculation based on a tight-binding model. We further propose material platforms with unequal Fermi velocities, establishing the experimental feasibility of our approach.