The commensurability condition and fractional quantum Hall effect hierarchy in higher Landau levels

The odd structure of the fractional filling hierarchy, which is referred to as the fractional quantum Hall effect, is studied in higher Landau levels using the commensurability condition. The hierarchy of fillings that are derived in this manner is consistent with the experimental observations in the first three Landau levels. The relative poverty of the fractional structure in higher Landau levels compared with the lowest Landau level is explained using commensurability topological arguments. The commensurability criterion for correlated states specific for higher Landau levels (with $n\geq 1$), including also the paired states at half fillings of the spin-subbands of these levels, is formulated.