Composite Hofstadter bands with Dirac fermion spectrum of fractional quantum Hall states

Abstract: 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 $\frac{1}{\nu}$ of the magnetic cell ($\nu$ is filling of a separated band) composite Hofstadter bands (HBs) are formed. The composite HBs are split into $\frac{1}{\nu}$ subbands, which are separated by the Dirac points. The Chern number of $\gamma$-full filled composite HBs is equal to the Chern number $C_{\gamma}$ of the corresponding HB. The Chern number, equal to $\nu C_{\gamma}$, corresponds to $\nu$-filling of $\gamma$-composite HB. Thus, FQHE is realized by fractional filling of composite HBs.