Hierarchical Nature of the Quantum Hall Effects

I demonstrate that the wave function for a $\ensuremath{\nu}=n+\stackrel{\texttildelow{}}{\ensuremath{\nu}}$ quantum Hall state with Landau levels $0,1,\dots{},n\ensuremath{-}1$ filled and a filling fraction $\stackrel{\texttildelow{}}{\ensuremath{\nu}}$ quantum Hall state with $0<\stackrel{\texttildelow{}}{\ensuremath{\nu}}\ensuremath{\le}1$ in the $n$th Landau level can be obtained hierarchically from the $\ensuremath{\nu}=n$ state by introducing quasielectrons which are then projected into the (conjugate of the) $\stackrel{\texttildelow{}}{\ensuremath{\nu}}$ state. In particular, the $\stackrel{\texttildelow{}}{\ensuremath{\nu}}=1$ case produces the filled Landau level wave functions hierarchically, thus establishing the hierarchical nature of the integer quantum Hall states. It follows that the composite fermion description of fractional quantum Hall states fits within the hierarchy theory of the fractional quantum Hall effect. I also demonstrate this directly by generating the composite fermion ground-state wave functions via application of the hierarchy construction to fractional quantum Hall states, starting from the $\ensuremath{\nu}=1/m$ Laughlin states.