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Non-equilibrium transport properties of charge and spin sector of two edgesof a quantum spin Hall insulator are investigated theoretically in afour-terminal configuration. A simple duality relation between charge and spinsector is found for two helical Tomonaga Luttinger liquids (hTTLs) connected tonon-interacting electron reservoirs. If the hTLLs on opposite edges are coupledlocally or non-locally, the mixing between them yields interesting physicswhere spin information can be easily detected by a charge measurement and viceversa. Particularly, we show how a pure spin density in the absence of chargecurrent can be generated in a setup that contains two hTLL and one spinfulTomonaga Luttinger liquid in between.

Charge-spin duality in non-equilibrium transport of helical liquids