Notes on the Squashed Sphere Lowest Landau Level

TL;DR

This paper investigates how the geometry of a squashed sphere affects Landau levels by analyzing a quantum particle's behavior on an oblate Haldane sphere, revealing increased localization with spheroid eccentricity.

Contribution

It provides a detailed numerical study of Landau levels on a squashed sphere, highlighting the impact of surface geometry on quantum localization phenomena.

Findings

Particle localization increases between equator and poles with spheroid eccentricity

Conventional Landau level features persist despite surface squashing

Geometry influences quantum states on curved surfaces

Abstract

In a recent article, we were motivated by the question of whether any of the remarkable condensed matter phenomena, such as the quantum Hall effect (QHE), the Integer quantum Hall effect (IQHE) etc., could potentially be observed in the extreme astrophysical environments of neutron stars. As a prequel to that work, and with the aim of understanding better the role of the geometry of the conducting surface on the structure of Landau levels, in this article we study the quantum dynamics of a quantum particle on a squashed sphere. More specifically, we study the dynamics of a single particle on an oblate squashed Haldane sphere i.e. a 2-sphere enclosing a single magnetic monopole at its center. While several features of the conventional Haldane sphere persist, by numerically solving the Schrodinger equation in this background, we find that the particle becomes increasingly localised in a band between the equator and the poles, with a corresponding increase of the eccentricity of the spheroid.