Landau levels on the 2D torus: a numerical approach

TL;DR

This paper introduces a numerical method to compute the spectrum of a particle on a 2D torus under magnetic and conservative forces, achieving high accuracy and revealing new fine structures in Landau levels.

Contribution

The paper presents a novel numerical approach using Fourier transforms to accurately compute Landau levels on a 2D torus with complex magnetic fields.

Findings

Accurate computation of Landau levels with twelve-digit precision.

Reproduction of Landau levels under uniform magnetic fields.

Discovery of a new fine structure within Landau levels for sinusoidal magnetic fields.

Abstract

A numerical method is presented which allows to compute the spectrum of the Schroedinger operator for a particle constrained on a two dimensional flat torus under the combined action of a transverse magnetic field and any conservative force. The method employs a fast Fourier transform to accurately represent the momentum variables and takes into account the twisted boundary conditions required by the presence of the magnetic field. An accuracy of twelve digits is attained even with coarse grids. Landau levels are reproduced in the case of a uniform field satisfying Dirac's condition. A new fine structure of levels within the single Landau level is formed when the field has a sinusoidal component with period commensurable to the integer magnetic charge.