Self-similar occurrence of massless Dirac particles in graphene under magnetic field

TL;DR

This paper develops an effective Hamiltonian approach to analytically describe the central Hofstadter band in graphene under magnetic fields, revealing the persistent presence of massless Dirac particles and their recursive Landau level structure.

Contribution

The work introduces a new effective Hamiltonian method that accurately captures the weak-field behavior of the Hofstadter butterfly in graphene, highlighting the universal occurrence of massless Dirac particles.

Findings

Massless Dirac particles exist within the central Hofstadter band at all magnetic flux levels.

The recursive pattern of the Hofstadter band indicates the presence of Dirac particles under arbitrary magnetic fields.

The width of the Hofstadter band varies with magnetic field, affecting experimental observability.

Abstract

Intricate interplay between the periodicity of the lattice structure and that of the cyclotron motion gives rise to a well-known self-similar fractal structure of the energy eigenvalue, known as the Hofstadter butterfly, for an electron moving in lattice under magnetic field. Evolving from the $n=0$ Landau level, the central band of the Hofstadter butterfly is especially interesting since it may hold a key to the mysteries of the fractional quantum Hall effect observed in graphene. While the entire Hofstadter butterfly can be in principle obtained by solving Harper's equations numerically, the weak-field limit, most relevant for experiment, is intractable due to the fact that the size of the Hamiltonian matrix, that needs to be diagonalized, diverges. In this paper, we develop an effective Hamiltonian method that can be used to provide an accurate analytic description of the central Hofstadter band in the weak-field regime. One of the most important discoveries obtained in this work is that massless Dirac particles always exist inside the central Hofstadter band no matter how small the magnetic flux may become. In other words, with its bandwidth broadened by the lattice effect, the $n=0$ Landau level contains massless Dirac particles within itself. In fact, by carefully analyzing the self-similar recursive pattern of the central Hofstadter band, we conclude that massless Dirac particles should occur under arbitrary magnetic field. As a corollary, the central Hofstadter band also contains a self-similar structure of recursive Landau levels associated with such massless Dirac particles. To assess the experimental feasibility of observing massless Dirac particles inside the central Hofstadter band, we compute the width of the central Hofstadter band as a function of magnetic field in the weak-field regime.