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arXiv:1502.05391 · Physical Review B

Non-Abelian phases in two-component ν=2/3 fractional quantum Hall states: Emergence of Fibonacci anyons

Zhao Liu, Abolhassan Vaezi, Kyungmin Lee, Eun-Ah Kim

Published Aug 5, 2015• 24 citations
arXivHTMLPDF

Abstract

Recent theoretical insights into the possibility of non-Abelian phases in $\ensuremath{\nu}=2/3$ fractional quantum Hall states revived the interest in the numerical phase diagram of the problem. We investigate the effect of various kinds of two-body interlayer couplings on the (330) bilayer state and exactly solve the Hamiltonian for up to 14 electrons on sphere and torus geometries. We consider interlayer tunneling, short-ranged repulsive/attractive pseudopotential interactions, and Coulomb repulsion. We find a 6-fold ground-state degeneracy on the torus when the interlayer hollow-core interaction is dominant. To identify the topological nature of this phase we measure the orbital-cut entanglement spectrum, quasihole counting, topological entanglement entropy, and wave-function overlap. Comparing the numerical results to the theoretical predictions, we interpret this 6-fold ground-state degeneracy phase to be the non-Abelian bilayer Fibonacci state.

DOI
10.1103/physrevb.92.081102

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Recent theoretical insights into the possibility of non-Abelian phases in \ensuremathν=2/3\ensuremath{\nu}=2/3\ensuremathν=2/3 fractional quantum Hall states revived the interest in the numerical phase diagram of the problem. We investigate the effect of various kinds of two-body interlayer couplings on the (330) bilayer state and exactly solve the Hamiltonian for up to 14 electrons on sphere and torus geometries. We consider interlayer tunneling, short-ranged repulsive/attractive pseudopotential interactions, and Coulomb repulsion. We find a 6-fold ground-state degeneracy on the torus when the interlayer hollow-core interaction is dominant. To identify the topological nature of this phase we measure the orbital-cut entanglement spectrum, quasihole counting, topological entanglement entropy, and wave-function overlap. Comparing the numerical results to the theoretical predictions, we interpret this 6-fold ground-state degeneracy phase to be the non-Abelian bilayer Fibonacci state.
There are several possible theoretically allowed non-Abelian fractional quantum Hall (FQH) states that could potentially be realized in one- and two-component FQH systems at total filling fraction $\ensuremath{\nu}=n+2/3$, for integer nnn. Some of these states even possess quasiparticles with non-Abelian statistics that are powerful enough for universal topological quantum computation, and are thus of particular interest. Here we initiate a systematic numerical study, using both exact diagonalization and variational Monte Carlo, to investigate the phase diagram of FQH systems at total filling fraction $\ensuremath{\nu}=n+2/3$, including in particular the possibility of the non-Abelian Z4{Z}_{4}Z4​ parafermion state. In $\ensuremath{\nu}=2/3$ bilayers we determine the phase diagram as a function of interlayer tunneling and repulsion, finding only three competing Abelian states, without the Z4{Z}_{4}Z4​ state. On the other hand, in single-component systems at $\ensuremath{\nu}=8/3$, we find that the Z4{Z}_{4}Z4​ parafermion state has significantly higher overlap with the exact ground state than the Laughlin state, together with a larger gap, suggesting that the experimentally observed $\ensuremath{\nu}=8/3$ state may be non-Abelian. Our results from the two complementary numerical techniques agree well with each other qualitatively.
Identifying and understanding interacting systems that can host non-Abelian topological phases with fractionalized quasiparticles have attracted intense attentions in the past twenty years. Theoretically, it is possible to realize a rich variety of such states by coupling two Abelian fractional quantum Hall (FQH) states together through gapping out part of the low energy degrees of freedom. So far, there are some indications, but no robust example has been established in bilayer systems for realizing the non-Abelian state in the past. Here, we present a phase diagram of a double-layer bosonic FQH system based on the exact diagonalization and density-matrix renormalization group (DMRG) calculations, which demonstrate a potential regime with the emergence of the non-Abelian bosonic Moore-Read state. We start from the Abelian phase with fourfold topological degeneracies on torus geometry when the two layers are weakly coupled. With the increase of interlayer tunneling, we find an intermediate regime with a threefold groundstate degeneracy and a finite fractional drag Hall conductance. We find the different topological sectors in consistent with Moore-Read state by inserting different fluxes in adiabatic DMRG study. We also extract the modular S−\mathcal{S}-S− matrix, which supports the emergence of the non-Abelian Ising anyon quasiparticle in this system.
Multicomponent quantum Hall systems with internal degrees of freedom provide a fertile ground for the emergence of exotic quantum liquids. Here, we investigate the possibility of non-Abelian topological order in the half-filled fractional quantum Hall (FQH) bilayer system driven by the tunneling effect between two layers. By means of the state-of-the-art density-matrix renormalization group, we unveil ``fingerprint'' evidence of the non-Abelian Moore-Read Pfaffian state emerging in the intermediate-tunneling regime, including the ground-state degeneracy on the torus geometry and the topological entanglement spectroscopy (entanglement spectrum and topological entanglement entropy) on the spherical geometry, respectively. Remarkably, the phase transition from the previously identified Abelian (331) Halperin state to the non-Abelian Moore-Read Pfaffian state is determined to be continuous, which is signaled by the continuous evolution of the universal part of the entanglement spectrum, and discontinuities in the excitation gap and the derivative of the ground-state energy. Our results not only provide a ``proof-of-principle'' demonstration of realizing a non-Abelian state through coupling different degrees of freedom, but also open up a possibility in FQH bilayer systems for detecting different chiral $p\text{\ensuremath{-}}\text{wave}$ pairing states.
Identifying and understanding interacting systems that can host non-Abelian topological phases with fractionalized quasiparticles have attracted intense attention in the past 20 years. Theoretically, it is possible to realize a rich variety of such states by coupling together two Abelian fractional quantum Hall (FQH) states through gapping out part of the low-energy degrees of freedom. So far, there are some indications, but no robust examples have been established in bilayer systems for realizing the non-Abelian state in the past. Here, we present a phase diagram of a double-layer bosonic FQH system based on exact diagonalization and density-matrix renormalization group (DMRG) calculations, which demonstrates a potential regime with the emergence of a non-Abelian bosonic Moore-Read state. Starting from the Abelian phase with fourfold topological degeneracy at weak coupling, with an increase of interlayer tunneling, we find an intermediate regime with a threefold ground-state degeneracy and a finite drag Hall conductance. We find different topological sectors consistent with the Moore-Read state by inserting different fluxes in an adiabatic DMRG study. We also extract the modular S\mathcal{S}S matrix, which supports the emergence of a non-Abelian Ising anyon quasiparticle in this system.
The possibility of realizing non-Abelian statistics and utilizing it for topological quantum computation (TQC) has generated widespread interest. However, the non-Abelian statistics that can be realized in most accessible proposals is not powerful enough for universal TQC. In this Letter, we consider a simple bilayer fractional quantum Hall system with the 1/3 Laughlin state in each layer. We show that interlayer tunneling can drive a transition to an exotic non-Abelian state that contains the famous "Fibonacci" anyon, whose non-Abelian statistics is powerful enough for universal TQC. Our analysis rests on startling agreements from a variety of distinct methods, including thin torus limits, effective field theories, and coupled wire constructions. We provide evidence that the transition can be continuous, at which point the charge gap remains open while the neutral gap closes. This raises the question of whether these exotic phases may have already been realized at ν=2/3 in bilayers, as past experiments may not have definitively ruled them out.
Topological quantum states with non-Abelian Fibonacci anyonic excitations are widely sought after for the exotic fundamental physics they would exhibit, and for universal quantum computing applications. The fractional quantum Hall (FQH) state at a filling factor of ν=12/5 is 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 ν=13/5 state and its particle-hole conjugate state at 12/5 with the Coulomb interaction, and we address the issue of possible competing states. Based on a large-scale density-matrix renormalization group calculation in spherical geometry, we present evidence that the essential physics of the Coulomb ground state (GS) at ν=13/5 and 12/5 is captured by the k=3 parafermion Read-Rezayi state (RR_{3}), including a robust excitation gap and the topological fingerprint from the 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 RR_{3} state at 13/5 and 12/5 filling numbers.