Boundary-induced Majorana coupling in a planar topological Josephson junction

TL;DR

This paper investigates how boundary effects and tunneling influence Majorana modes in a topological Josephson junction on a 3D topological insulator, revealing energy splitting mechanisms and magnetic field effects.

Contribution

It introduces a model for boundary-induced Majorana coupling in a planar topological Josephson junction, analyzing energy splitting due to tunneling effects both analytically and numerically.

Findings

Tunneling causes energy splitting of Majorana modes without wave function overlap.

Energy splitting varies with external magnetic field, affecting Fraunhofer patterns.

Analytical and numerical results agree in the weak tunneling limit.

Abstract

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.