Boundary conditions for the quantum Hall effect

TL;DR

This paper develops a self-consistent model of the integer quantum Hall effect on an infinite strip, analyzing how boundary conditions and finite-size effects influence the Hall conductivity and revealing new boundary-induced states.

Contribution

It introduces a boundary condition framework that uncovers novel boundary states and refines understanding of Hall conductivity quantization in finite geometries.

Findings

Robin boundary conditions create unique boundary states.

Finite-size effects modify the quantization pattern.

High electric fields can cause breakdown of the quantum Hall effect.

Abstract

We formulate a self-consistent model of the integer quantum Hall effect on an infinite strip, using boundary conditions to investigate the influence of finite-size effects on the Hall conductivity. By exploiting the translation symmetry along the strip, we determine both the general spectral properties of the system for a large class of boundary conditions respecting such symmetry, and the full spectrum for (fibered) Robin boundary conditions. In particular, we find that the latter introduce a new kind of states with no classical analogues, and add a finer structure to the quantization pattern of the Hall conductivity. Moreover, our model also predicts the breakdown of the quantum Hall effect at high values of the applied electric field.