Nonequilibrium transport of helical Luttinger liquids through a quantum dot

We study a steady-state nonequilibrium transport between two interacting helical edge states of a two-dimensional topological insulator, described by helical Luttinger liquids, through a quantum dot. For a noninteracting dot, the current is obtained analytically by including the self-energy correction to the dot's Green function. For an interacting dot, we use the equation-of-motion method to study the influence of weak on-site Coulomb interaction on the transport. We find the metal-to-insulator quantum phase transition for attractive or repulsive interactions in the leads when the magnitude of the interaction strength characterized by a charge sector Luttinger parameter K goes beyond a critical value. The critical Luttinger parameter K-cr depends on the hopping strengths between the dot and the leads, as well as the energy level of the dot with respect to the Fermi levels of the leads, ranging from the weak-interaction regime for the dot level off-resonance to the strong-interaction regime for the dot in resonance with the equilibrium Fermi level. Near the transition, there are various singular behaviors of current noise, dot density of state, and the decoherence rate (inverse of lifetime) of the dot, which are briefly discussed.