Effective mass of the composite fermions and energy gaps of quantum Hall states
The effective mass of the quasi-particles in the fermion-Chern-Simons description of the quantum Hall state at half-filling is computed for electron-electron interactions $V(r)\sim r^{x-2}$, for $0<x<3/2$, following the previous work of Stern and Halperin, Phys. Rev. B {\bf 52}, 5890 (1995). The energy gap of quantum Hall states with filling factors $ν=\frac{p}{2p+1}$ for $p\gg 1$ can then be obtained either from the effective mass at half-filling, as proposed by Halperin, Lee and Read, Phys. Rev. B {\bf 47}, 7312 (1993), or evaluated directly from the self-energy of the system in presence of the residual magnetic field; both results are shown to agree as $p\to \infty$. The energy gap is then given by a self-consistent equation, which asymptotic solution for $p\gg 1$ and short-range interactions is $E_g(p)\sim (2p+1)^{-\frac{3-x}{2}}$, in agreement with previous results by Kim, Lee and Wen, Phys. Rev. B {\bf 52}, 17275 (1995). The power law for the energy gap seems to be {\it exact} to all orders in the perturbative expansion. Moreover, the energy gap for systems with Coulomb interaction is recovered in the limit $x\to 1$.