arXiv:0804.1960 · Physical Review B

Quantum Hall states at ν=2k+2 : Analysis of the particle-hole conjugates of the general level- k Read-Rezayi states

Waheb Bishara, Gregory A. Fiete, Chetan Nayak

Published Jun 19, 2008• 39 citations

Abstract

We study the $\ensuremath{\nu}=\frac{2}{k+2}$ quantum Hall states which are particle-hole conjugates of the $\ensuremath{\nu}=\frac{2}{k+2}$ Read-Rezayi states. We find that equilibration between the different modes at the edge of such a state leads to an emergent $\mathrm{SU}{(2)}_{k}$ algebra in the counter-propagating neutral sector. Heat flow along the edges of these states will be in the opposite direction of charge flow. In the $k=3$ case, which may be relevant to $\ensuremath{\nu}=2+\frac{2}{5}$, the thermal Hall conductance and the exponents associated with quasiparticle and electron tunneling distinguish this state from competing states such as the hierarchy and/or Jain state.

DOI: 10.1103/physrevb.77.241306

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Quasiparticle agglomerates in the Read–Rezayi and anti-Read–Rezayi states

Alessandro Braggio, Dario Ferraro, Nicodemo Magnoli

arXiv:1207.4604Physica ScriptaNov 1, 20123 citations

Parton construction of particle-hole-conjugate Read-Rezayi parafermion fractional quantum Hall states and beyond

Ajit Coimbatore Balram, Maissam Barkeshli, Mark Spencer Rudner

arXiv:1902.04173Physical review. B./Physical review. BJun 19, 201946 citations

Tripartite composite fermion states

G. J. Sreejith, Ying-Hai Wu, Arkadiusz Wójs +1 more

arXiv:1301.6565Physical Review BJun 24, 201331 citations

Parton construction of particle-hole-conjugate Read-Rezayi parafermion fractional quantum Hall states

Ajit Coimbatore Balram, Maissam Barkeshli, Mark Spencer Rudner

Bulletin of the American Physical SocietyMar 4, 20200 citations

‘One-dimensional’ theory of the quantum Hall system

Emil J. Bergholtz, A. Karlhede

arXiv:cond-mat/0509434Journal of Statistical Mechanics Theory and ExperimentApr 12, 200688 citations

Zero Mode Operators and Charge-Conjugation Defects in Non-Abelian Fractional Quantum Hall-Superconductor Heterostructures

Junyi Cao, Eduardo Fradkin

cond-mat.str-elCross-listedarXiv:2610.08907Oct 8, 20260 citations

arXiv:0804.1960 · Physical Review B

Quantum Hall states at ν=2k+2 : Analysis of the particle-hole conjugates of the general level- k Read-Rezayi states

Waheb Bishara, Gregory A. Fiete, Chetan Nayak

Published Jun 19, 2008• 39 citations

arXiv HTML PDF

Abstract

We study the

\ensuremath{\nu}=\frac{2}{k+2}

quantum Hall states which are particle-hole conjugates of the

\ensuremath{\nu}=\frac{2}{k+2}

Read-Rezayi states. We find that equilibration between the different modes at the edge of such a state leads to an emergent

\mathrm{SU}{(2)}_{k}

algebra in the counter-propagating neutral sector. Heat flow along the edges of these states will be in the opposite direction of charge flow. In the

k=3

case, which may be relevant to

\ensuremath{\nu}=2+\frac{2}{5}

, the thermal Hall conductance and the exponents associated with quasiparticle and electron tunneling distinguish this state from competing states such as the hierarchy and/or Jain state.

DOI: 10.1103/physrevb.77.241306

Related papers

Quasiparticle agglomerates in the Read–Rezayi and anti-Read–Rezayi states

Alessandro Braggio, Dario Ferraro, Nicodemo Magnoli

We calculate the dominant excitations for the $k$ -level ( $k\in\mathbb{N}$ ) Read-Rezayi (RR) states and their particle-hole conjugates, the anti Read-Rezayi ( $\bar{\textrm{RR}}$ ), proposed for quantum Hall states. These states are supposed to be build over the second Landau level with total filling factor $ν=2+ν^*$ with $ν^*=k/(k+2)$ for RR and $ν^*=2/(k+2)$ for $\bar{\textrm{RR}}$ . In the $k$ -level RR states, based on $\mathbb{Z}_k$ parafermions, the dominant excitations are the fundamental quasiparticles with fractional charge $e^*_k= e/(k + 2)$ , with $e$ the electron charge, if $k=2,3$ . For k=4 the single-qp and the 2-agglomerate, with charge $2e^*_k$ , have the same scaling and both dominate, while for $k>4$ the 2-agglomerates are dominant. Anyway the dominance of the 2-agglomerates can be affected by the presence of environmental renormalizations. For all the $k$ -level $\bar{\textrm{RR}}$ states the single-qp and the 2-agglomerate have the same scaling and both dominate. In this case only the presence of environmental renormalizations can make one dominant over the other. We determine the conditions where the environmental renormalizations of the charged and neutral modes make dominant the Abelian 2-agglomerates over the non-Abelian single-quasiparticles in the two models and for any value of $k$ . We conclude observing that, according these predictions, the dominance of 2-agglomerates, at very low energies for the $ν=5/2$ , can be an interesting indication supporting the validity of the anti-Pfaffian model in comparison to the Pfaffian.We calculate the dominant excitations for the $k$-level ($k\in\mathbb{N}$) Read-Rezayi (RR) states and their particle-hole conjugates, the anti Read-Rezayi ($\bar{\textrm{RR}}$), proposed for quantum Hall states. These states are supposed to be build over the second Landau level with total filling factor $ν=2+ν^*$ with $ν^*=k/(k+2)$ for RR and $ν^*=2/(k+2)$ for $\bar{\textrm{RR}}$. In the $k$-level RR states, based on $\mathbb{Z}_k$ parafermions, the dominant excitations are the fundamental quasiparticles with fractional charge $e^*_k= e/(k + 2)$, with $e$ the electron charge, if $k=2,3$. For k=4 the single-qp and the 2-agglomerate, with charge $2e^*_k$, have the same scaling and both dominate, while for $k>4$ the 2-agglomerates are dominant. Anyway the dominance of the 2-agglomerates can be affected by the presence of environmental renormalizations. For all the $k$-level $\bar{\textrm{RR}}$ states the single-qp and the 2-agglomerate have the same scaling and both dominate. In this case only the presence of environmental renormalizations can make one dominant over the other. We determine the conditions where the environmental renormalizations of the charged and neutral modes make dominant the Abelian 2-agglomerates over the non-Abelian single-quasiparticles in the two models and for any value of $k$. We conclude observing that, according these predictions, the dominance of 2-agglomerates, at very low energies for the $ν=5/2$, can be an interesting indication supporting the validity of the anti-Pfaffian model in comparison to the Pfaffian.

arXiv:1207.4604Physica ScriptaNov 1, 20123 citations

Parton construction of particle-hole-conjugate Read-Rezayi parafermion fractional quantum Hall states and beyond

Ajit Coimbatore Balram, Maissam Barkeshli, Mark Spencer Rudner

The Read-Rezayi (RR) parafermion states form a series of exotic non-Abelian fractional quantum Hall (FQH) states at filling $\ensuremath{\nu}=k/(k+2)$. Computationally, the wave functions of these states are prohibitively expensive to generate for large systems. We introduce a series of parton states, denoted ${\overline{2}}^{k}{1}^{k+1}$ , and show that they lie in the same universality classes as the particle-hole-conjugate RR (``anti-RR'') states. Our analytical results imply that a $[U{(1)}_{k+1}\ifmmode\times\else\texttimes\fi{}U{(2k)}_{\ensuremath{-}1}]/[SU{(k)}_{\ensuremath{-}2}\ifmmode\times\else\texttimes\fi{}U{(1)}_{\ensuremath{-}1}]$ coset conformal field theory describes the edge excitations of the ${\overline{2}}^{k}{1}^{k+1}$ state, suggesting nontrivial dualities with respect to previously known descriptions. The parton construction allows wave functions in anti-RR phases to be generated for hundreds of particles. We further propose the parton sequence $\overline{n}{\overline{2}}^{2}{1}^{4}$ , with $n=1,2,3$ , to describe the FQH states observed at $\ensuremath{\nu}=2+1/2$, $2+2/5$ , and $2+3/8$ .The Read-Rezayi (RR) parafermion states form a series of exotic non-Abelian fractional quantum Hall (FQH) states at filling $\ensuremath{\nu}=k/(k+2)$. Computationally, the wave functions of these states are prohibitively expensive to generate for large systems. We introduce a series of parton states, denoted ${\overline{2}}^{k}{1}^{k+1}$, and show that they lie in the same universality classes as the particle-hole-conjugate RR (``anti-RR'') states. Our analytical results imply that a $[U{(1)}_{k+1}\ifmmode\times\else\texttimes\fi{}U{(2k)}_{\ensuremath{-}1}]/[SU{(k)}_{\ensuremath{-}2}\ifmmode\times\else\texttimes\fi{}U{(1)}_{\ensuremath{-}1}]$ coset conformal field theory describes the edge excitations of the ${\overline{2}}^{k}{1}^{k+1}$ state, suggesting nontrivial dualities with respect to previously known descriptions. The parton construction allows wave functions in anti-RR phases to be generated for hundreds of particles. We further propose the parton sequence $\overline{n}{\overline{2}}^{2}{1}^{4}$, with $n=1,2,3$, to describe the FQH states observed at $\ensuremath{\nu}=2+1/2$, $2+2/5$, and $2+3/8$.

arXiv:1902.04173Physical review. B./Physical review. BJun 19, 201946 citations

Tripartite composite fermion states

G. J. Sreejith, Ying-Hai Wu, Arkadiusz Wójs +1 more

The Read-Rezayi wave function is one of the candidates for the fractional quantum Hall effect at filling fraction $\ensuremath{\nu}=2+⅗$, and thereby also its hole conjugate at $2+⅖$ . We study a general class of tripartite composite fermion wave functions, which reduce to the Rezayi-Read ground state and quasiholes for appropriate quantum numbers, but also allow a construction of wave functions for quasiparticles and neutral excitations by analogy to the standard composite fermion theory. We present numerical evidence in finite systems that these trial wave functions capture well the low energy physics of a four-body model interaction. We also compare the tripartite composite fermion wave functions with the exact Coulomb eigenstates at $2+⅗$ , and find reasonably good agreement. The ground state as well as several excited states of the four-body interaction are seen to evolve adiabatically into the corresponding Coulomb states for $N=15$ particles. These results support the plausibility of the Read-Rezayi proposal for the $2+⅖$ and $2+⅗$ fractional quantum Hall effect. However, certain other proposals also remain viable, and further study of excitations and edge states will be necessary for a decisive establishment of the physical mechanism of these fractional quantum Hall states.The Read-Rezayi wave function is one of the candidates for the fractional quantum Hall effect at filling fraction $\ensuremath{\nu}=2+⅗$, and thereby also its hole conjugate at $2+⅖$. We study a general class of tripartite composite fermion wave functions, which reduce to the Rezayi-Read ground state and quasiholes for appropriate quantum numbers, but also allow a construction of wave functions for quasiparticles and neutral excitations by analogy to the standard composite fermion theory. We present numerical evidence in finite systems that these trial wave functions capture well the low energy physics of a four-body model interaction. We also compare the tripartite composite fermion wave functions with the exact Coulomb eigenstates at $2+⅗$, and find reasonably good agreement. The ground state as well as several excited states of the four-body interaction are seen to evolve adiabatically into the corresponding Coulomb states for $N=15$ particles. These results support the plausibility of the Read-Rezayi proposal for the $2+⅖$ and $2+⅗$ fractional quantum Hall effect. However, certain other proposals also remain viable, and further study of excitations and edge states will be necessary for a decisive establishment of the physical mechanism of these fractional quantum Hall states.

arXiv:1301.6565Physical Review BJun 24, 201331 citations

Parton construction of particle-hole-conjugate Read-Rezayi parafermion fractional quantum Hall states

Ajit Coimbatore Balram, Maissam Barkeshli, Mark Spencer Rudner

Bulletin of the American Physical SocietyMar 4, 20200 citations

‘One-dimensional’ theory of the quantum Hall system

Emil J. Bergholtz, A. Karlhede

We consider the lowest Landau level on a torus as a function of its circumference L 1 . When L 1 → 0, the ground state at general rational filling fraction is a crystal with a gap—a Tao–Thouless state. For filling fractions ν = p /(2 pm + 1), these states are the limits of Laughlin's or Jain's wavefunctions describing the gapped quantum Hall states when L 1 → ∞ . For the half-filled Landau level, there is a transition to a Fermi sea of non-interacting neutral fermions (dipoles), or rather to a Luttinger liquid modification thereof, at L 1 ∼ 5 magnetic lengths. Using exact diagonalization we identify this state as a version of the Rezayi–Read state, and find that it develops continuously into the state that is believed to describe the observed metallic phase as L 1 → ∞ . Furthermore, the effective Landau level structure that emerges within the lowest Landau level is found to be a consequence of the magnetic symmetries.We consider the lowest Landau level on a torus as a function of its circumference L 1 . When L 1 → 0, the ground state at general rational filling fraction is a crystal with a gap—a Tao–Thouless state. For filling fractions ν = p /(2 pm + 1), these states are the limits of Laughlin's or Jain's wavefunctions describing the gapped quantum Hall states when L 1 → ∞ . For the half-filled Landau level, there is a transition to a Fermi sea of non-interacting neutral fermions (dipoles), or rather to a Luttinger liquid modification thereof, at L 1 ∼ 5 magnetic lengths. Using exact diagonalization we identify this state as a version of the Rezayi–Read state, and find that it develops continuously into the state that is believed to describe the observed metallic phase as L 1 → ∞ . Furthermore, the effective Landau level structure that emerges within the lowest Landau level is found to be a consequence of the magnetic symmetries.

arXiv:cond-mat/0509434Journal of Statistical Mechanics Theory and ExperimentApr 12, 200688 citations

Zero Mode Operators and Charge-Conjugation Defects in Non-Abelian Fractional Quantum Hall-Superconductor Heterostructures

Junyi Cao, Eduardo Fradkin

We study zero modes bound to charge-conjugation defects in fractional quantum Hall-superconductor (FQH-SC) heterostructures built from Abelian and non-Abelian FQH states. Although the associated defects are well described by $G$ -crossed braided tensor categories, a local operator description of their zero modes is not generally known. We generalize the bulk-to-boundary construction of localized zero mode operators to FQH-SCs with non-Abelian parent topological orders, mapping bulk Wilson operators in the folded theory to localized zero mode operators in the one-dimensional FQH-SC theory. This construction distinguishes zero mode anyons from the local zero mode operators that act on the defect Hilbert space and yields a generalized zero mode algebra. It also determines the fusion-channel dependence of tunneling: tunneling of zero mode operators probes bulk monodromy, while direct zero mode anyon transfer probes $G$ -crossed associativity data. Applying this framework, we recover the $\mathbb{Z}_{2m}$ parafermion zero mode structure of fermionic Laughlin FQH-SCs at filling $ν=1/m$ , obtain a $\mathbb{Z}_2^{(n)}\times\mathbb{Z}_N$ structure for generalized Moore-Read states at filling $ν=1/m$ , with $N=2m$ for fermionic states and $N=m$ for bosonic states, and find a $\mathrm{Fib}\times\mathbb{Z}_N$ structure for the $k=3$ Read-Rezayi state at filling $ν=3/(3M+2)$ , with $N=2(3M+2)$ for fermionic states and $N=(3M+2)/2$ for bosonic states. In the Read-Rezayi case, zero mode operator tunneling and direct zero mode anyon transfer yield distinct universal amplitude ratios between the neutral fusion channels, providing a characteristic signature of the underlying Fibonacci defect structure.We study zero modes bound to charge-conjugation defects in fractional quantum Hall-superconductor (FQH-SC) heterostructures built from Abelian and non-Abelian FQH states. Although the associated defects are well described by $G$-crossed braided tensor categories, a local operator description of their zero modes is not generally known. We generalize the bulk-to-boundary construction of localized zero mode operators to FQH-SCs with non-Abelian parent topological orders, mapping bulk Wilson operators in the folded theory to localized zero mode operators in the one-dimensional FQH-SC theory. This construction distinguishes zero mode anyons from the local zero mode operators that act on the defect Hilbert space and yields a generalized zero mode algebra. It also determines the fusion-channel dependence of tunneling: tunneling of zero mode operators probes bulk monodromy, while direct zero mode anyon transfer probes $G$-crossed associativity data. Applying this framework, we recover the $\mathbb{Z}_{2m}$ parafermion zero mode structure of fermionic Laughlin FQH-SCs at filling $ν=1/m$, obtain a $\mathbb{Z}_2^{(n)}\times\mathbb{Z}_N$ structure for generalized Moore-Read states at filling $ν=1/m$, with $N=2m$ for fermionic states and $N=m$ for bosonic states, and find a $\mathrm{Fib}\times\mathbb{Z}_N$ structure for the $k=3$ Read-Rezayi state at filling $ν=3/(3M+2)$, with $N=2(3M+2)$ for fermionic states and $N=(3M+2)/2$ for bosonic states. In the Read-Rezayi case, zero mode operator tunneling and direct zero mode anyon transfer yield distinct universal amplitude ratios between the neutral fusion channels, providing a characteristic signature of the underlying Fibonacci defect structure.

cond-mat.str-elCross-listedarXiv:2610.08907Oct 8, 20260 citations

We study the

\ensuremath{\nu}=\frac{2}{k+2}

quantum Hall states which are particle-hole conjugates of the

\ensuremath{\nu}=\frac{2}{k+2}

Read-Rezayi states. We find that equilibration between the different modes at the edge of such a state leads to an emergent

\mathrm{SU}{(2)}_{k}

algebra in the counter-propagating neutral sector. Heat flow along the edges of these states will be in the opposite direction of charge flow. In the

k=3

case, which may be relevant to

\ensuremath{\nu}=2+\frac{2}{5}

, the thermal Hall conductance and the exponents associated with quasiparticle and electron tunneling distinguish this state from competing states such as the hierarchy and/or Jain state.

We calculate the dominant excitations for the

k

-level (

k\in\mathbb{N}

) Read-Rezayi (RR) states and their particle-hole conjugates, the anti Read-Rezayi (

\bar{\textrm{RR}}

), proposed for quantum Hall states. These states are supposed to be build over the second Landau level with total filling factor

ν=2+ν^*

with

ν^*=k/(k+2)

for RR and

ν^*=2/(k+2)

for

\bar{\textrm{RR}}

. In the

k

-level RR states, based on

\mathbb{Z}_k

parafermions, the dominant excitations are the fundamental quasiparticles with fractional charge

e^*_k= e/(k + 2)

, with

e

the electron charge, if

k=2,3

. For k=4 the single-qp and the 2-agglomerate, with charge

2e^*_k

, have the same scaling and both dominate, while for

k>4

the 2-agglomerates are dominant. Anyway the dominance of the 2-agglomerates can be affected by the presence of environmental renormalizations. For all the

k

-level

\bar{\textrm{RR}}

states the single-qp and the 2-agglomerate have the same scaling and both dominate. In this case only the presence of environmental renormalizations can make one dominant over the other. We determine the conditions where the environmental renormalizations of the charged and neutral modes make dominant the Abelian 2-agglomerates over the non-Abelian single-quasiparticles in the two models and for any value of

k

. We conclude observing that, according these predictions, the dominance of 2-agglomerates, at very low energies for the

ν=5/2

, can be an interesting indication supporting the validity of the anti-Pfaffian model in comparison to the Pfaffian.

{\overline{2}}^{k}{1}^{k+1}

, and show that they lie in the same universality classes as the particle-hole-conjugate RR (``anti-RR'') states. Our analytical results imply that a $[U{(1)}_{k+1}\ifmmode\times\else\texttimes\fi{}U{(2k)}_{\ensuremath{-}1}]/[SU{(k)}_{\ensuremath{-}2}\ifmmode\times\else\texttimes\fi{}U{(1)}_{\ensuremath{-}1}]$ coset conformal field theory describes the edge excitations of the

{\overline{2}}^{k}{1}^{k+1}

state, suggesting nontrivial dualities with respect to previously known descriptions. The parton construction allows wave functions in anti-RR phases to be generated for hundreds of particles. We further propose the parton sequence

\overline{n}{\overline{2}}^{2}{1}^{4}

, with

n=1,2,3

, to describe the FQH states observed at $\ensuremath{\nu}=2+1/2$,

2+2/5

, and

2+3/8

The Read-Rezayi wave function is one of the candidates for the fractional quantum Hall effect at filling fraction $\ensuremath{\nu}=2+⅗$, and thereby also its hole conjugate at

2+⅖

. We study a general class of tripartite composite fermion wave functions, which reduce to the Rezayi-Read ground state and quasiholes for appropriate quantum numbers, but also allow a construction of wave functions for quasiparticles and neutral excitations by analogy to the standard composite fermion theory. We present numerical evidence in finite systems that these trial wave functions capture well the low energy physics of a four-body model interaction. We also compare the tripartite composite fermion wave functions with the exact Coulomb eigenstates at

2+⅗

, and find reasonably good agreement. The ground state as well as several excited states of the four-body interaction are seen to evolve adiabatically into the corresponding Coulomb states for

N=15

particles. These results support the plausibility of the Read-Rezayi proposal for the

2+⅖

and

2+⅗

fractional quantum Hall effect. However, certain other proposals also remain viable, and further study of excitations and edge states will be necessary for a decisive establishment of the physical mechanism of these fractional quantum Hall states.

G

-crossed braided tensor categories, a local operator description of their zero modes is not generally known. We generalize the bulk-to-boundary construction of localized zero mode operators to FQH-SCs with non-Abelian parent topological orders, mapping bulk Wilson operators in the folded theory to localized zero mode operators in the one-dimensional FQH-SC theory. This construction distinguishes zero mode anyons from the local zero mode operators that act on the defect Hilbert space and yields a generalized zero mode algebra. It also determines the fusion-channel dependence of tunneling: tunneling of zero mode operators probes bulk monodromy, while direct zero mode anyon transfer probes

G

-crossed associativity data. Applying this framework, we recover the

\mathbb{Z}_{2m}

parafermion zero mode structure of fermionic Laughlin FQH-SCs at filling

ν=1/m

, obtain a

\mathbb{Z}_2^{(n)}\times\mathbb{Z}_N

structure for generalized Moore-Read states at filling

ν=1/m

, with

N=2m

for fermionic states and

N=m

for bosonic states, and find a

\mathrm{Fib}\times\mathbb{Z}_N

structure for the

k=3

Read-Rezayi state at filling

ν=3/(3M+2)

, with

N=2(3M+2)

for fermionic states and

N=(3M+2)/2

for bosonic states. In the Read-Rezayi case, zero mode operator tunneling and direct zero mode anyon transfer yield distinct universal amplitude ratios between the neutral fusion channels, providing a characteristic signature of the underlying Fibonacci defect structure.