Boundary-induced Majorana coupling in a planar topological Josephson junction

Understanding environmental effects in a topological Josephson junction is vital for identifying signatures of Majorana modes. We consider a planar Josephson junction formed on the surface of a three-dimensional topological insulator, which possesses Majorana modes inside the junction and boundary modes outside. We find that tunneling between the inner and outer modes gives rise to effective coupling between the inner Majorana modes and hence induces energy splitting of their states even in the absence of the direct spatial overlap of their wave functions. The energy splitting is obtained analytically in the weak tunneling limit and is numerically investigated for an arbitrary tunneling strength. We discuss in detail the evolution of the energy splitting with an external perpendicular magnetic field and its effect on the shape of the Fraunhofer pattern.