Majorana ϕ0-junction in a disordered spin-orbit coupling nanowire with tilted magnetic field

Majorana Josephson junctions in nanowire systems exhibit a pseudo-4π period current-phase relation in the clean limit. In this work, we study how this current-phase relation responds to a tilted magnetic field in a disordered Majorana Josephson junction within the Bogoliubov-de Gennes approach. We show that the tilted magnetic field induces a ϕ0 phase shift to the current-phase relation. Most importantly, we find that this ϕ0-junction behavior is robust even in the presence of disorders.