arXiv:0906.0337 · Physical Review B

Classification of Abelian and non-Abelian multilayer fractional quantum Hall states through the pattern of zeros

Maissam Barkeshli, Xiao-Gang Wen

Published Dec 2, 2010• 25 citations

Abstract

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.

DOI: 10.1103/physrevb.82.245301

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arXiv:0906.0337 · Physical Review B

Classification of Abelian and non-Abelian multilayer fractional quantum Hall states through the pattern of zeros

Maissam Barkeshli, Xiao-Gang Wen

Published Dec 2, 2010• 25 citations

arXiv HTML PDF

Abstract

{\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.

DOI: 10.1103/physrevb.82.245301

Related papers

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Maissam Barkeshli, Xiao-Gang Wen

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.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.

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Abelian and non-Abelian states in ν=2/3 bilayer fractional quantum Hall systems

Michael R. Peterson, Yang-Le Wu, Meng Cheng +3 more

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 $n$ . 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 ${Z}_{4}$ 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 ${Z}_{4}$ state. On the other hand, in single-component systems at $\ensuremath{\nu}=8/3$, we find that the ${Z}_{4}$ 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.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 $n$. 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 ${Z}_{4}$ 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 ${Z}_{4}$ state. On the other hand, in single-component systems at $\ensuremath{\nu}=8/3$, we find that the ${Z}_{4}$ 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.

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Xiao-Gang Wen, Zhenghan Wang

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 ${{S}_{a}}$ . In this paper, we will show how to use the data ${{S}_{a}}$ 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.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 ${{S}_{a}}$. In this paper, we will show how to use the data ${{S}_{a}}$ 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.

arXiv:0803.1016Physical Review BOct 9, 200865 citations

{\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.

(ppq)

Abelian bilayer states to a set of non-Abelian FQH states, which we dub orbifold FQH states, of which the

{Z}_{4}

parafermion (Read-Rezayi) state is a special case. This provides an example in which

{Z}_{2}

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)

bilayer states, they may be experimentally relevant, both as candidates for describing the plateaus in single-layer systems at filling fractions

8/3

and

12/5

and as a way to tune to non-Abelian states in double-layer or wide quantum wells.

n

. 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

{Z}_{4}

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

{Z}_{4}

state. On the other hand, in single-component systems at $\ensuremath{\nu}=8/3$, we find that the

{Z}_{4}

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.

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

{{S}_{a}}

. In this paper, we will show how to use the data

{{S}_{a}}

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.