Parity dependent Josephson current through a helical Luttinger liquid

We consider a superconductor-two dimensional topological insulator-superconductor junction and study how the 2 pi- and 4 pi- periodic Josephson currents are affected by the electron-electron interaction. In the long-junction limit the supercurrent can be evaluated by modeling the system as a helical Luttinger liquid coupled to superconducting reservoirs. After having introduced bosonization in the presence of the parity constraint we turn to consider the limit of perfect and poor interfaces. For transparent interfaces, where perfect Andreev reflections occur at the boundaries, the Josephson current is marginally affected by the interaction. However, if strong magnetic scatterers are present in the weak link, the situation changes dramatically. Here Coulomb interaction plays a crucial role both in low and high temperature regimes. Furthermore, a phase-shift of Josephson current can be induced by changing the direction of the magnetization of the impurity.