Gate controlled spin-density wave and chiral FFLO superconducting phases in interacting helical liquids

We explore the phases exhibited by an interacting helical liquid in the presence of finite chemical potential (applied gate voltage) and spin imbalance (applied magnetic field). We find that the helical nature gives rise to quantum orders that are expected to be absent in nonchiral one-dimensional electronic systems. For repulsive interactions, the ordered state has an oscillatory spin texture whose ordering wave vector 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 ground state that is not condensed in the absence of interactions and is superconducting for attractive interactions. This state is characterized by Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-type oscillations where the Cooper pairs obtain a finite center-of-mass momentum. These phases can be realized on the edge of two-dimensional systems exhibiting the quantum spin Hall effect or on dislocation lines in weak topological insulators.