Fractional Josephson Effect with and without Majorana Zero Modes

It is known that the low-energy physics of the Josephson effect in the presence of Majorana zero modes exhibits a $4\pi$ periodicity as the Aharonov-Bohm flux varies in contrast to the $2\pi$ Josephson periodicity in usual superconducting junctions. We study this fractional Josephson effect in 1D topological superconductors in Majorana nanowire systems by focusing on the features of the phase-energy relations in a superconducting semiconductor nanowire with spin-orbital coupling by including different factors operational in experimental systems, such as short wire length, suppression of superconducting gap, and the presence of an Andreev bound state. We show that even in the absence of Majorana zero modes, some non-topological physical effects can manifest a $4\pi$ periodicity of the phase-energy relation in the Josephson junction, thus providing a false positive signal for fractional Josephson effect with no underlying Majorana zero modes. Furthermore, we consider several scenarios of inhomogeneous chemical potential distributions in the superconducting nanowire leading to four Majorana bound states and construct the effective four Majorana model to correctly describe the low-energy theory of the Josephson effect. In this setup, multiple Majorana zero modes can also have the $4\pi$ fractional Josephson effect, although the underlying physics arises from Andreev bound states since two close by Majorana bound states effectively form Andreev bound states. Our work demonstrates that the mere observation of a fractional Josephson effect simulating $4\pi$ periodicity cannot, by itself, be taken as the definitive evidence for topological superconductivity. This finding has important implications for the ongoing search for non-Abelian Majorana zero modes and efforts for developing topological qubits.