Duality between normal and superconducting junctions of multiple quantum wires

Abstract

We study junctions of single-channel spinless Luttinger liquids using bosonization. We generalize earlier studies by allowing the junction to be superconducting and find new charge nonconserving low-energy fixed points. We establish the existence of g↔1/g duality (where g is the Luttinger liquid parameter) between the charge-conserving (normal) junction and the charge nonconserving (superconducting) junction by evaluating and comparing the scaling dimensions of various operators around the fixed points in both the normal and superconducting sectors of the theory. For the most general two-wire junction, we show that there are two conformally invariant one-parameter families of fixed points which are also connected by a duality transformation. We also show that the stable fixed point for the two-wire superconducting junction corresponds to the situation where the crossed Andreev reflection (an incoming electron is transmitted as an outgoing hole) is perfect between the wires. For the three-wire junction, we study, in particular, the superconducting analogs of the chiral D_P and the disconnected fixed points obtained earlier in the literature in the context of charge-conserving three-wire junctions. We show that these fixed points can be stabilized for g < 1 (repulsive electrons) within the superconducting sector of the theory which makes them experimentally relevant.