Manipulating the exchange phase of Majoranas

The non-Abelian exchange statistics of Majorana zero modes make them interesting for technological and fundamental reasons. But unlike their non-Abelian contribution upon an exchange (braid), the π/4 Abelian contribution has not been widely recognized. We interpret this phase as an Aharonov-Casher phase from a vortex encircling an e/4 charge associated with the Fermi sea rather than the chargeless Majorana modes. We calculate this fractional charge across different systems and dimensions, revealing that in 2D it co-localizes with the Majorana zero mode at the vortex core, while in 3D, they localize at opposite ends of the vortex, isolating the Abelian and non-Abelian contributions. Our results suggest vortex interference experiments could detect the separate effects of an isolated e/4 fractional charge and the Majorana zero mode.