Current-phase relation in a topological Josephson junction: Andreev bands versus scattering states

We consider a long topological Josephson junction formed on a conducting two-dimensional (2D) surface ofa 3D topological insulator (TI). The superconducting correlations are proximity-induced bys-wave supercon-ductors covering the surface. The 1D spacing between the coverings is either unfilled or filled by a magneticinsulator. Generally, the Josephson current mediated by the TI surface is determined by scattering modes aswell as by the states localized around the junction (Andreev bound statesorAndreev bands). We find out thatit is crucial to take into account both contributions to determine the current-phase relation of the topologicalJosephson junction. We analyze the dependence of the Josephson current on the thickness of the junction as wellas the deviations from the sinusoidal shape of the current-phase relatio