Bardeen-Cooper-Schrieffer pairing of composite fermions

Abstract

Topological pairing of composite fermions has led to remarkable ideas, such as excitations obeying non-Abelian braid statistics and topological quantum computation. We construct a $p$-wave paired Bardeen-Cooper-Schrieffer (BCS) wave function for composite fermions in the torus geometry, which is a convenient geometry for formulating momentum space pairing as well as for revealing the underlying composite-fermion Fermi sea. Following the standard BCS approach, we minimize the Coulomb interaction energy at half filling in the lowest and the second Landau levels, which correspond to filling factors $\ensuremath{\nu}=1/2$ and $\ensuremath{\nu}=5/2$ in GaAs quantum wells, by optimizing two variational parameters that are analogous to the gap and the Debye cutoff energy of the BCS theory. Our results show no evidence for pairing at $\ensuremath{\nu}=1/2$ but a clear evidence for pairing at $\ensuremath{\nu}=5/2$. To a good approximation, the highest overlap between the exact Coulomb ground state at $\ensuremath{\nu}=5/2$ and the BCS state is obtained for parameters that minimize the energy of the latter, thereby providing support for the physics of composite-fermion pairing as the mechanism for the $5/2$ fractional quantum Hall effect. We discuss the issue of modular covariance of the composite-fermion BCS wave function, and calculate its Hall viscosity and pair correlation function. By similar methods, we look for but do not find an instability to $s$-wave pairing for a spin-singlet composite-fermion Fermi sea at half-filled lowest Landau level in a system where the Zeeman splitting has been set to zero.