Competing Topological Orders in the ?=12/5 Quantum Hall State

We provide numerical evidence that a p_x-i p_y paired Bonderson-Slingerland (BS) non-Abelian hierarchy state is a strong candidate for the observed ? =12/5 quantum Hall plateau. We confirm the existence of a gapped incompressible ?=12/5 quantum Hall state with shift S = 2 on the sphere, matching that of the BS state. The exact ground state of the Coulomb interaction at S = 2 is shown to have a large overlap with the BS trial wave function. Larger overlaps are obtained with BS-type wave functions that are hierarchical descendants of general p_x - i p_y weakly paired states at ?=12/5. We perform a finite-size scaling analysis of the ground-state energies for ?=12/5 states at shifts corresponding to the BS (S = 2) and 3-clustered Read-Rezayi (S = -2) universality classes. This analysis reveals very tight competition between these two non-Abelian topological orders.