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

TL;DR

This paper provides numerical evidence that the fractional quantum Hall states at filling factors 13/5 and 12/5 are described by the non-Abelian Read-Rezayi RR3 state, with implications for topological quantum computation.

Contribution

The study demonstrates, through large-scale DMRG calculations, that the 13/5 and 12/5 FQH states are best described by the non-Abelian RR3 state, revealing their topological order and non-Abelian quasiparticle sectors.

Findings

The 13/5 state is an incompressible FQH state with a finite excitation gap.

The RR3 state has the lowest energy among competing states at 12/5 in the thermodynamic limit.

Entanglement spectrum and topological entanglement entropy support RR3 identification.

Abstract

We investigate the nature of the fractional quantum Hall (FQH) state at filling factor $\nu=13/5$, 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 physics of the Coulomb ground state (GS) at $\nu=13/5$ and $12/5$ is captured by the $k=3$ parafermion Read-Rezayi RR state, $\text{RR}_3$. We first establish that the state at $\nu=13/5$ is an incompressible FQH state, with a GS protected by a finite excitation gap, with the shift in accordance with the RR state. Then, by performing a finite-size scaling analysis of the GS energies for $\nu=12/5$ with different shifts, we find that the $\text{RR}_3$ state has the lowest energy among different competing states in the thermodynamic limit. We find the fingerprint of $\text{RR}_3$ topological order in the FQH $13/5$ and $12/5$ states, based on their entanglement spectrum and topological entanglement entropy, both of which strongly support their identification with the $\text{RR}_3$ state. Furthermore, by considering the shift-free infinite-cylinder geometry, we expose two topologically-distinct GS sectors, one identity sector and a second one matching the non-Abelian sector of the Fibonacci anyonic quasiparticle, which serves as additional evidence for the $\text{RR}_3$ state at $13/5$ and $12/5$.