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arXiv:0906.0356 · Physical Review B

Non-Abelian two-component fractional quantum Hall states

Maissam Barkeshli, Xiao-Gang Wen

Published Dec 2, 2010• 24 citations
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Abstract

A large class of fractional quantum Hall (FQH) states can be classified according to their pattern of zeros, which describes the order of zeros in ground-state wave functions as various clusters of electrons are brought together. The pattern-of-zeros approach can be generalized to systematically classify multilayer/spin-unpolarized FQH states, which has led to the construction of a class of non-Abelian multicomponent FQH states. Here we discuss some of the simplest non-Abelian two-component states that we find and the possibility of their experimental realization in bilayer systems at $\ensuremath{\nu}=2/3$, 4/5, 4/7, 4/9, 1/4, etc.

DOI
10.1103/physrevb.82.233301

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A large class of fractional quantum Hall (FQH) states can be classified according to their pattern of zeros, which describes the order of zeros in ground-state wave functions as various clusters of electrons are brought together. The pattern-of-zeros approach can be generalized to systematically classify multilayer/spin-unpolarized FQH states, which has led to the construction of a class of non-Abelian multicomponent FQH states. Here we discuss some of the simplest non-Abelian two-component states that we find and the possibility of their experimental realization in bilayer systems at \ensuremathν=2/3\ensuremath{\nu}=2/3\ensuremathν=2/3, 4/5, 4/7, 4/9, 1/4, etc.
A large class of fractional quantum Hall (FQH) states can be classified according to their pattern of zeros, which describes the way ideal ground-state wave functions go to zero as various clusters of electrons are brought together. In this paper, we generalize this approach to classify multilayer FQH states. Such a classification leads to the construction of a class of non-Abelian multilayer FQH states that are closely related to ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{g}}_{k}$ parafermion conformal field theories, where ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{g}}_{k}$ is an affine simple Lie algebra. We discuss the possibility of some of the simplest of these non-Abelian states occurring in experiments on bilayer FQH systems at $\ensuremath{\nu}=2/3$, 4/5, 4/7, etc.
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.
We study continuous quantum phase transitions that can occur in bilayer fractional quantum Hall (FQH) systems as the interlayer tunneling and interlayer repulsion are tuned. We introduce a slave-particle gauge theory description of a series of continuous transitions from the (ppq)(ppq)(ppq) Abelian bilayer states to a set of non-Abelian FQH states, which we dub orbifold FQH states, of which the Z4{Z}_{4}Z4​ parafermion (Read-Rezayi) state is a special case. This provides an example in which Z2{Z}_{2}Z2​ electron fractionalization leads to non-Abelian topological phases. The naive ``ideal'' wave functions and ideal Hamiltonians associated with these orbifold states do not in general correspond to incompressible phases but, instead, lie at a nearby critical point. We discuss this unusual situation from the perspective of the pattern-of-zeros/vertex algebra frameworks and discuss implications for the conceptual foundations of these approaches. Due to the proximity in the phase diagram of these non-Abelian states to the (ppq)(ppq)(ppq) bilayer states, they may be experimentally relevant, both as candidates for describing the plateaus in single-layer systems at filling fractions 8/38/38/3 and 12/512/512/5 and as a way to tune to non-Abelian states in double-layer or wide quantum wells.
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.
It has been shown that different Abelian and non-Abelian fractional quantum Hall states can be characterized by patterns of zeros described by sequences of integers Sa{{S}_{a}}Sa​. In this paper, we will show how to use the data Sa{{S}_{a}}Sa​ to calculate various topological properties of the corresponding fraction quantum Hall state, such as the number of possible quasiparticle types and their quantum numbers, as well as the actions of the quasiparticle tunneling and modular transformations on the degenerate ground states on torus.