Robustness of Majorana Fermion induced Fractional Josephson Effect

It is shown in previous works that the coupling between two Majorana end states in superconducting quantum wires leads to fractional Josephson effect. However, in realistic experimental conditions, multiple bands of the wires are occupied and the Majorana end states are accompanied by other fermionic end states. This raises the question concerning the robustness of fractional Josephson effect in these situations. In this work, we show that the absence of the avoided energy crossing which gives rise to the fractional Josephson effect is robust, even when the Majorana fermions are coupled with arbitrary strengths to other fermions. Moreover, we calculate the temperature dependence of the fractional Josephson current and show that it is suppressed by thermal excitations to the other fermion bound states.