  Topological superconductivity in quasi-one-dimensional systems is a novel
phase of matter with possible implications for quantum computation. Despite
years of effort, a definitive signature of this phase in experiments is still
debated. A major cause of this ambiguity is the side effects of applying a
magnetic field: induced in-gap states, vortices, and alignment issues. Here we
propose a planar semiconductor-superconductor heterostructure as a platform for
realizing topological superconductivity without applying a magnetic field to
the 2D electron gas hosting the topological state. Time-reversal symmetry is
broken only by phase-biasing the proximitizing superconductors, which can be
achieved using extremely small fluxes or bias currents far from the
quasi-one-dimensional channel. Our platform is based on interference between
this phase biasing and the phase arising from strong spin-orbit coupling in
closed electron trajectories. The principle is demonstrated analytically using
a simple model, and then shown numerically for realistic devices. We show a
robust topological phase diagram, as well as explicit wavefunctions of Majorana
zero modes. We discuss experimental issues regarding the practical
implementation of our proposal, establishing it as an accessible scheme with
contemporary experimental techniques.
