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We demonstrate how the boundary-driven reconstruction of the superconductingorder parameter can be employed to manipulate the zero-energy Majorana boundstates (MBSs) occurring in a topological Josephson junction. We focus on aninterface of two p-wave superconductors, which are described by a spin-vectororder parameter $\bf{d}$. Apart from the sensitivity of $\bf{d}$ to externalZeeman/exchange fields, here, we show that the orientation of $\bf{d}$throughout the junction can be controlled by electrically gating the weak link.The remarkable local character of this knob is a manifestation of the edgereconstruction of the order parameter, which takes place whenever different$\bf{d}$-vector configurations in each superconductor compete and are close inenergy. As a consequence, the spin-dependent superconducting-phase differenceacross the junction is switchable from $0$ to $\pi$. Moreover, in the regimewhere multiple edge MBSs occur for each superconductor, the Andreev-bound-state(ABS) spectra can be twisted by the application of either a charge- orspin-phase difference across the interface, and give rise to a rich diversityof nonstandard ABS dispersions. Interestingly, some of these dispersions showband crossings protected by fermion parity, despite their $2\pi$-periodiccharacter. These crossings additionally unlock the possibility of nontrivialtopology in synthetic spaces, when considering networks of such 1D junctions.Lastly, the interface MBSs induce a distinct elecronic spin polarization nearthe junction, which possesses a characteristic spatial pattern that allows thedetection of MBSs using spin-polarized scanning tunneling microscopy. Thesefindings unveil novel paths to mechanisms for ABS engineering and single-outsignatures relevant for the experimental detection and manipulation of MBSs.

Magnetoelectrically-Tunable Andreev-Bound-State Spectra and Spin Polarization in P-Wave Josephson Junctions