  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 SNS
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 between 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.
