Three-phase Majorana zero modes at tiny magnetic fields

Proposals for realizing Majorana fermions in condensed-matter systems typically rely on magnetic fields, which degrade the proximitizing superconductor and plague the Majoranas' detection. We propose an alternative scheme to realize Majoranas based only on phase-biased superconductors. The phases (at least three of them) can be biased by a tiny magnetic field threading macroscopic superconducting loops, focusing and enhancing the effect of the magnetic field onto the junction, or by supercurrents. We show how a combination of the superconducting phase winding and the spin-orbit phase induced in closed loops (Aharonov-Casher effect) facilitates a topological superconducting state with Majorana end states. We demonstrate this scheme with an analytically tractable model as well as simulations of realistic setups comprising only conventional materials.