Landau levels, self-adjoint extensions and Hall conductivity on a cone

TL;DR

This paper studies the Landau levels and Hall conductivity of a 2D electron gas on a conical surface, analyzing how the cone's geometry and self-adjoint extensions influence the quantum Hall effect.

Contribution

It compares extrinsic and intrinsic approaches to quantum Hall effect on a cone, emphasizing the role of self-adjoint extensions and the conical tip's influence.

Findings

Hall conductivity is enhanced by the conical geometry.

Self-adjoint extensions significantly affect Landau levels.

The conical tip influences quantum dynamics and Hall effect.

Abstract

In this work we obtain the Landau levels and the Hall conductivity at zero temperature of a two-dimensional electron gas on a conical surface. We investigate the integer quantum Hall effect considering two different approaches. The first one is an extrinsic approach which employs an effective scalar potential that contains both the Gaussian and the mean curvature of the surface. The second one, an intrinsic approach where the Gaussian curvature is the sole term in the scalar curvature potential. From a theoretical point of view, the singular Gaussian curvature of the cone may affect the wave functions and the respective Landau levels. Since this problem requests {\it self-adjoint extensions}, we investigate how the conical tip could influence the integer quantum Hall effect, comparing with the case were the coupling between the wave functions and the conical tip is ignored. This last case corresponds to the so-called {\it Friedrichs extension}. In all cases, the Hall conductivity is enhanced by the conical geometry depending on the opening angle. There are a considerable number of theoretical papers concerned with the self-adjoint extensions on a cone and now we hope the work addressed here inspires experimental investigation on these questions about quantum dynamics on a cone.