Energy spectrum for two-dimensional potentials in very high magnetic fields

TL;DR

This paper develops a supersymmetry-inspired method to analyze the energy spectrum of particles in two-dimensional potentials under very high magnetic fields, revealing a self-similar spectrum akin to the Hofstadter butterfly.

Contribution

It introduces a novel supersymmetry-based approach for solving the energy spectrum in 2D potentials at high magnetic fields, including exact solutions for periodic potentials.

Findings

Spectrum exhibits self-similarity similar to Hofstadter butterfly.

Exact solutions obtained for rational magnetic flux values.

Method applicable to arbitrary potentials in the lowest Landau level.

Abstract

A method, analogous to supersymmetry transformation in quantum mechanics, is developed for a particle in the lowest Landau level moving in an arbitrary potential. The method is applied to two-dimensional potentials formed by Dirac delta scattering centers. In the periodic case, the problem is solved exactly for rational values of the magnetic flux (in units of flux quantum) per unit cell. The spectrum is found to be self-similar, resembling the Hofstadter butterfly.