Fractional quantum Hall effect in a periodic potential

The fractional quantum Hall effect in a periodic potential or modulation of the magnetic field is studied by symmetry, topological, and Chern-Simons field-theoretic methods. With periodic boundary conditions, the Hall conductance in a finite system is known to be a fraction whose denominator is the degeneracy of the ground state. We show that in a finite system, translational symmetry predicts a degeneracy that varies periodically with system size and equals 1 for certain commensurate cases which we argue are physically representative. However, this analysis may overlook gaps due to finite-size effects that vanish in the thermodynamic limit. This possibility is addressed using a fermionic Chern-Simons field theory in the mean-field approximation. In addition to solutions describing the usual Laughlin or Jain states whose properties are only weakly modified by the periodic background, we also find solutions whose existence depends on the presence of the background. In these incompressible states, the Hall conductance is a fraction not equal to the filling factor, and its denominator is the same as that of the fractional charge and statistics of the elementary quasiparticle excitations.