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

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.

arXiv comment17 pages, 9 figures

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Cited by in corpus (17)

• Cloning of Dirac fermions in graphene superlattices
• Creating anomalous Floquet Chern insulators with magnetic quantum walks
• Emergence of massless Dirac fermions in graphene's Hofstadter butterfly at switches of the quantum Hall phase connectivity
• Long-range ballistic transport of Brown-Zak fermions in graphene superlattices
• Gap Structure of the Hofstadter System of Interacting Dirac Fermions in Graphene
• Periodic Landau gauge and Quantum Hall effect in twisted bilayer graphene
• Hofstadter butterflies and magnetically induced band gap quenching in graphene antidot lattices
• Classification of Topological Phase Transitions and van Hove Singularity Steering Mechanism in Graphene Superlattices
• Magnetic vortex liquid and anomalous thermal Hall conductivity in frustrated magnets with bond-dependent interactions
• Replica Higher-Order Topology of Hofstadter Butterflies in Twisted Bilayer Graphene
• Fractal defect states in the Hofstadter butterfly
• Stability of zero energy Dirac touchings in the honeycomb Hofstadter problem
• Driven Hofstadter Butterflies and Related Topological Invariants
• Translational symmetry breaking and the disintegration of the Hofstadter butterfly
• Wannier Diagram and Brown-Zak Fermions of Graphene on Hexagonal Boron-Nitride
• SU(3) fermions in a three-band Graphene-like model
• Topological constraints on the electronic band structure of hexagonal lattice in a magnetic field