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

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 $12/5$.

12 pages, 8 figures

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