Geometric phase in p-n junctions of helical edge states

The quantum spin Hall effect is endowed with topologically protected edge modes with a gapless Dirac spectrum. Applying a magnetic field locally along the edge leads to a gapped edge spectrum with the opposite parity for winding of spin texture for conduction and valence bands. Using Pancharatnam's prescription for the geometric phase it is shown that mismatch of this parity across a p-n junction, which could be engineered into the edge by electrical gate induced doping, leads to a phase dependence in the two-terminal conductance which is quantized to either zero or pi. It is further shown that application of a nonuniform magnetic field across the junction could lead to a nonquantized value of this geometric phase which is tunable between zero and pi. A current asymmetry measurement which is shown to be robust against electron-electron interactions is proposed to infer the appearance of this Pancharatnam's geometric phase in transport across such junctions.