The Fractional Quantum Hall States at $\nu=13/5$ and $12/5$ and their Non-Abelian Nature

Topological quantum states with non-Abelian Fibonacci anyonic excitations are widely sought after for their exotic fundamental physics and potential applications in universal quantum computing. The fractional quantum Hall (FQH) state at filling factor $\nu=12/5$ is such a promising candidate, however, its precise nature is still under debate and no consensus has been achieved so far. Here, we investigate the nature of the FQH $\nu=13/5$ state and its particle-hole conjugate state at $12/5$ with the Coulomb interaction, and address the issue of possible competing states. Based on a large-scale density-matrix renormalization group (DMRG) calculation in spherical geometry, we present evidence that the essential physics of the Coulomb ground state (GS) at $\nu=13/5$ and $12/5$ is captured by the $k=3$ parafermion Read-Rezayi state ($\text{RR}_3$), including a robust excitation gap and the topological fingerprint from entanglement spectrum and topological entanglement entropy. Furthermore, by considering the infinite-cylinder geometry (topologically equivalent to torus geometry), we expose the non-Abelian GS sector corresponding to a Fibonacci anyonic quasiparticle, which serves as a signature of the $\text{RR}_3$ state at $13/5$ and $12/5$ filling numbers.