Fig. 1: Linear charge Hall effect under time reversal (TR) symmetry in a non-isolated system.
From: Time-reversal even charge hall effect from twisted interface coupling
a In the Hall conductivity σH, the TR odd part by definition requires TR symmetry breaking, while the TR even part vanishes by Onsager relation. The red, green, and blue arrows denote the three vectors in jH = σH × E. b Constraint by Onsager relation can be lifted in a non-isolated 2D system (blue surface) by coupling to an environment (black shading). c TR-even Hall current from twisted interfacial coupling with an environmental layer (gray surface), whereas a counterflow Hall current is expected in the latter, by Onsager relation on the whole structure: system (Sys) + environment (Env). Green shaded area denotes the interlayer hopping between the Brillouin zone (BZ) with twist angle θ. d The TR-even Hall voltage (VH) due to charge accumulation at the sample edges (red and black + / − ) can be detected with a layer-resolved measurement. Black arrows denote source and drain current j.
