Revisiting 2π phase slip suppression in topological Josephson junctions.*

Current state of the art devices to detect and manipulate Majorana fermions commonly consist of networks of Majorana wires. We study a key ingredient of these networks - a topological Josephson junction in the presence of charging energy. The phase dependent tunneling contains 4π periodic single particle tunneling, allowed due to the Majorana modes at the edges of the junction, and the usual 2π periodic Josephson tunneling. For small values of the charging energy, the low energy physics of conventional Josephson junctions is described by 2π phase slips. In a topological junction, we expect 2π phase slips to be suppressed. However, we find that if the ratio between the strengths of the Majorana assisted tunneling and the Josephson tunneling is small — as is likely to be the case for many setups — 2π phase slips may still occur. We provide effective descriptions of the system in terms of 2π and 4π phase slips valid throughout the parameter space. Finally, we discuss the implications of our results on the dissipative phase transitions expected in this system and in assessing the viability of certain Majorana schemes.