From fractional boundary charges to quantized Hall conductance

We study the fractional boundary charges (FBCs) occurring in nanowires in the presence of periodically modulated chemical potentials and connect them to the FBCs occurring in a two-dimensional electron gas in the presence of a perpendicular magnetic field in the integer quantum Hall effect (QHE) regime. First, we show that in nanowires the FBCs take fractional values and change linearly as a function of phase offset of the modulated chemical potential. This linear slope takes quantized values determined by the period of the modulation and depends only on the number of the filled bands. Next, we establish a mapping from the one-dimensional system to the QHE setup, where we again focus on the properties of the FBCs. By considering a cylinder topology with an external flux similar to the Laughlin construction, we find that the slope of the FBCs as function of flux is linear and assumes universal quantized values, also in the presence of arbitrary disorder. We establish that the quantized slopes give rise to the quantization of the Hall conductance. Importantly, the approach via FBCs is valid for arbitrary flux values and disorder. The slope of the FBCs plays the role of a topological invariant for clean and disordered QHE systems. Our predictions for the FBCs can be tested experimentally in nanowires and in Corbino disk geometries in the integer QHE regime.