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The fractional quantum Hall effect (FQHE) is studied in the semiclassical limit in the framework of the Hofstadter model with a short-range interaction between fermions. In the mean-field approximation, the repulsion between fermions leads to a periodic potential. Numerical calculations show that in the case of the periodic potential with a period that is a multiple on 1 of the magnetic cell (is filling of a separated band) composite Hofstadter bands (HBs) are formed. The composite HBs are split into 1 subbands, which are separated by the Dirac points. The Chern number of -full filled composite HBs is equal to the Chern number C_ of the corresponding HB. The Chern number, equal to C_, corresponds to -filling of -composite HB. Thus, FQHE is realized by fractional filling of composite HBs.
Igor N. Karnaukhov (Thu,) studied this question.