Bardeen-Cooper-Schrieffer pairing of composite fermions

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 $\nu=1/2$ and $\nu=5/2$ in GaAs quantum wells, by optimizing two variational parameters that are analogous to the gap and the Debye cut-off energy of the BCS theory. Our results show no evidence for pairing at $\nu=1/2$ but a clear evidence for pairing at $\nu=5/2$. To a good approximation, the highest overlap between the exact Coulomb ground state at $\nu=5/2$ 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.