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We present a perturbative approach to the study of the Hofstadter model for when the amount of flux per plaquette is close to a rational fraction. Within this approximation, certain eigenstates of the system are shown to be multicomponent wave functions that connect smoothly to the Landau levels of the continuum. The perturbative corrections to these are higher Landau level contributions that break rotational invariance and allow the perturbed states to adopt the symmetry of the lattice. In the presence of interactions, this approach allows for the calculation of generalized Haldane pseudopotentials, and in turn, the many-body properties of the system. The method is sufficiently general that it can apply to a wide variety of lattices, interactions, and magnetic field strengths.
Harper et al. (Mon,) studied this question.