  Applying a unified approach, we study the integer quantum Hall effect (IQHE)
and fractional quantum Hall effect (FQHE) in the Hofstadter model with short
range interactions between fermions. An effective field, that takes into
account the interaction between fermions, is determined by both amplitude and
phase. Its amplitude is proportional to the interaction strength, the phase
corresponds to the minimum energy. In fact, the problem is reduced to the
Harper equation with two different scales: the first is a magnetic scale with
the cell size corresponding to a unit quantum magnetic flux, the second scale
determines the inhomogeneity of the effective field, forms the steady fine
structure of the Hofstadter spectrum and leads to the realization of fractional
quantum Hall states. In a sample of finite size with open boundary conditions,
the fine structure of the Hofstadter spectrum consists of the Dirac branches of
the fermion excitations and includes the fine structure of the edge chiral
modes. The Chern numbers of the topological Hofstadter bands are conserved
during the formation of their fine structure. The edge modes are formed into
the Hofstadter bands. They connect the nearest-neighbor subbands and determine
the conductance for the fractional filling.
