Hysteresis in the I-V characteristics of overdamped topological Josephson junctions*

We consider two one-dimensional topological superconductors with four Majorana modes gamma_{1,2,3,4}, where gamma_2 and gamma_3 are coupled by an overdamped Josephson junction and thus form a two-level system with their energy difference dependent on the Josephson phase shift phi. We find a hysteresis in the I-V characteristics which is the consequence of the Landau-Zener effect at finite V where phi oscillates, while this hysteresis disappears in the topological trivial regime due to absence of the two energy levels. This hysteresis is not blurred in presence of other channels for tunneling of Cooper-pairs in the Josephson junction, thus serving as a useful signature for the experimental search of Majorana modes.