Bardeen-Cooper-Schrieffer pairing of composite fermions

The pairing of composite fermions is thought to lead to remarkable physics, such as the topological superconductivity and non-Abelian Majorana modes [1,2]. We investigate the problem by constructing a p-wave paired BCS wave function of composite fermions [3] on torus geometry and minimizing the energy as a function of the gap parameter. Our results show clear evidence of pairing at $\nu=5/2$, but no evidence of pairing at $\nu=1/2$. We find that the lowest energy BCS function has the highest overlap with the Pfaffian wavefunction. We search for but do not find the spin-singlet pairing of composite fermions at $\nu=1/2$. We also consider other pairing symmetries and investigate the Hall viscosity for the BCS wavefunction in different phases. \\ $\left[1\right]$ G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991) \\ $\left[2\right]$ N. Read and D. Green, Phys. Rev. B 61, 10267 (2000) \\ $\left[3\right]$ G. M\”{o}ller and S. H. Simon, Phys. Rev. B 77,075319 (2008)