Josephson π state in superconductor-Luttinger liquid hybrid systems

Josephson current through a Luttinger liquid (LL) under a magnetic field is theoretically studied. We derive an analytical expression of the Josephson current for clean interfaces, by using the quasiclassical Green's function and the functional bosonization procedure. We show that critical currents can be renormalized by electron-electron interactions at perfect transparency when LL is adiabatically connected with superconductors. We also find that a generation of pstate, due to a spin-dependent energy shift in Andreev bound states (ABS), is prohibited even at zero temperature when the strength of repulsive interactions reaches some critical value. The suppression of π state is caused by the low energy fluctuations propagating in LL, making the Zeeman splitting in ABS blurred.