Quantum Hall effect at half-filled Landau level: Pairing of composite fermions

Abstract

We discuss the possibility of the quantum Hall effect at half-filled Landau level in terms of the pairing of the composite fermions. In the absence of Coulomb energy, we show that the ground state of the system is described by the {\it p}-wave BCS pairing state of composite fermions. When the ratio $α\equiv (e^2/ε\ell_B)/ε_F$ ($\ell_B$ : the magnetic length, $ε_F$ : Fermi energy of the composite fermions) is larger than a critical value $α_c \sim 8.2$ the gap of the pairing state vanishes. However, $α$ remains less than $α_c$ if $\hbar ω_c \gg e^2/ε\ell_B$ holds. Then in this situation it is possible that the pairing state which results in the quantum Hall effect occurs. The effect of the real spin degrees of freedom and the Zeeman energy is also discussed.