Charge-spin duality in nonequilibrium transport of helical liquids

Nonequilibrium transport properties of the charge and spin sectors of two edges of a quantum spin Hall insulator are investigated theoretically in a four-terminal configuration. A simple duality relation between charge and spin sector is found for two helical Tomonaga Luttinger liquids (hTLLs) connected to noninteracting electron reservoirs. If the hTLLs on opposite edges are coupled locally or nonlocally, the mixing between them yields interesting physics where spin information can be easily detected by a charge measurement and vice versa. Particularly, we show how a pure spin density in the absence of charge current can be generated in a setup that contains two hTLL and one spinful Tomonaga Luttinger liquid in between.