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We propose a scheme to detect the Majorana bound states (MBSs) by athermodynamically stable D.C. Josephson current with $4\pi$-periodicity in thesuperconducting phase difference, which is distinct from the previous A.C.$4\pi$-periodicity found in topological superconducting Josephson junctions.The scheme, consisting of a quantum dot coupled to two s-wave superconductingleads and a floating topological superconductor supporting two MBSs at itsends, only exploits the interplay of a local Zeeman field and the exotichelical and self-Hermitian properties of MBSs, without requiring theconservation of fermion parity and not relying on the zero-energy property ofMBSs. Our D.C. $4\pi$-periodicity is thus robust against the overlap betweenthe two MBSs and various system parameters, including the local Coulombinteraction, the tunneling asymmetry, and the width of superconducting gap,which facilitates experimentally detection of the MBSs.

Detection of Majorana bound states by thermodynamically stable $4\pi$-periodic D.C. Josephson current