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

TL;DR

This paper explores the pairing of composite fermions in half-filled Landau levels, proposing a p-wave BCS pairing state as a mechanism for the quantum Hall effect, influenced by Coulomb interactions and spin effects.

Contribution

It introduces a model where composite fermions form a p-wave pairing state at half-filled Landau levels, highlighting conditions for the quantum Hall effect based on Coulomb energy ratios.

Findings

Pairing state described by p-wave BCS state in absence of Coulomb energy.

Critical Coulomb energy ratio $oldsymbol{eta_c oughly 8.2}$ determines pairing gap vanishing.

Quantum Hall effect possible when magnetic energy dominates Coulomb interactions.

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 $\alpha \equiv (e^2/\epsilon \ell_B)/\epsilon_F$ ($\ell_B$ : the magnetic length, $\epsilon_F$ : Fermi energy of the composite fermions) is larger than a critical value $\alpha_c \sim 8.2$ the gap of the pairing state vanishes. However, $\alpha$ remains less than $\alpha_c$ if $\hbar \omega_c \gg e^2/\epsilon \ell_B$ 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.