  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.
