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Gate controlled Spin-Density Wave and Chiral FFLO Superconducting phases in interacting Quantum Spin Hall edge states

We explore the phases exhibited by an interacting quantum spin Hall edge state in the presence of finite chemical potential (applied gate voltage) and spin imbalance (applied magnetic field). We find that the helical nature of the edge state gives rise to orders that are expected to be absent in non-chiral one-dimensional electronic systems. For repulsive interactions, the ordered state has an oscillatory spin texture whose ordering wavevector is controlled by the chemical potential. We analyze the manner in which a magnetic impurity provides signatures of such oscillations. We find that finite spin imbalance favors a finite current carrying groundstate that is not condensed in the absence of interactions and is superconducting for attractive interactions. This state is characterized by FFLO-type oscillations where the Cooper pairs obtain a finite center of mass momentum.