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arXiv · 1505.03050

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

W. Zhu·S. S. Gong·F. D. M. Haldane·D. N. Sheng
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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 $12/5$.

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W. Zhu, S. S. Gong, F. D. M. Haldane, D. N. Sheng. 2015-05-12. The Fractional Quantum Hall States at $\nu=13/5$ and $12/5$ and their Non-Abelian Nature. https://doi.org/10.1103/physrevlett.115.126805

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