Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physica Scripta

Generalized Cooperative Ring Exchange Theory of the Fractional Quantum Hall Effect

Steven Allan Kivelson

Published Jan 1, 1987• 0 citations
Publisher

Abstract

Electrons in two dimensions in a high magnetic field can condense into an incompressible quantum fluid state when the electron density is equal to certain rational fractions of the flux density. Much progress has been made in understanding the nature of this state based on the ansatz wavefunction of Laughlin. Here, we show that the incompressible nature of the quantum fluid can be derived from a semiclassical evaluation of the imaginary time propagator. The dominant classical paths consist of cooperative ring exchange events. When these condense, they mediate a logarithmic interaction between density fluctuations, and produce a precise fractional quantization condition for the quasi-particle charges. We further show that the essential long wavelength properties of Laughlin's ground state and quasi-particle wavefunctions can be derived constructively from this approach. We conclude with some speculations concerning the nature of the off-diagonal order in the quantum Hall liquid.

DOI
10.1088/0031-8949/1987/t19a/011

Related papers

Cooperative ring exchange and the fractional quantum Hall effect

Steven Allan Kivelson, Catherine Kallin, Daniel P. Arovas +1 more

We study the transition of the interacting two-dimensional electron gas at high magnetic field from a low-density Wigner crystal to a higher-density correlated state which exhibits the fractionally quantized Hall effect. The phase transition is precipitated by a condensation of ring-exchange processes, which by nature of their sensitivity to enclosed magnetic flux allow for especially low-energy exchange condensates at certain values of the Landau-level filling factor, \ensuremath{\nu}=${\ensuremath{\nu}}_{i}$. In the condensate, the exchanges mediate a logarithmic potential between local-density fluctuations, leading to a cusp in the ground-state energy as a function of \ensuremath{\Vert}\ensuremath{\nu}-${\ensuremath{\nu}}_{i}$\ensuremath{\Vert} in the vicinity of a preferred state. In addition, we derive quasiparticle excitations of sharp fractional charge which are analogous to the excitations derived by Laughlin.

Physical review. B, Condensed matter•Jul 15, 1987•92 citations

Generalized Cooperative-Ring-Exchange Theory of the Fractional Quantum Hall Effect

D. H. Lee, G. Baskaran, Steven Allan Kivelson

Through the calculations of the diagonal and off-diagonal elements of the full $N$-particle density matrix we show that all essential features of the Laughlin variational wave functions can be derived in a generalized cooperative-ring-exchange theory of the fractional quantum Hall fluid.

Physical Review Letters•Nov 23, 1987•25 citations

Toward a New Theory of the Fractional Quantum Hall Effect

Sergey A. Mikhailov

The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx=e2/3h, in the vicinity of the Landau level filling factor ν=1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N≤7 electrons continuously as a function of magnetic field in the range 1/4≲ν<1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν=1/3 Laughlin state for N≤8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.

arXiv:2206.05152•Nanomaterials•Jan 31, 2024•1 citation

Incompressible quantum Hall states

Jainendra K. Jain

Considering each electron as a superposition of fictitious fractionally charged particles allows, with the help of a natural ansatz, a systematic identification and characterization of the incompressible quantum Hall states at noninteger filling factors. Explicit Laughlin-type wave functions are obtained for all the incompressible states and their quasiparticles. The order of stability of the various states predicted on the basis of physically plausible rules is in agreement with experiments. Although in principle all rational fractions are observable, these rules imply that the even-denominator fractions are in general much less stable than the odd-denominator ones.

Physical review. B, Condensed matter•Oct 1, 1989•403 citations

Graviton modes in fractional quantum hall liquids

Yuzhu Wang

The fractional quantum Hall fluid is a two-dimensional quantum fluid of electrons subject to a strong magnetic field at low temperatures.Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase.Among these states, there are some specific modes in the long-wavelength limit that can be understood as "spin-2 gravitons".In this thesis, we will introduce a set of analytical results for the energy gap of the graviton modes for model Hamiltonians in the thermodynamic limit, which is governed by a well-defined and universal characteristic tensor.These results can help to construct model Hamiltonians for the graviton modes of different FQH phases and elucidate a hierarchical structure of conformal Hilbert spaces (null spaces of model Hamiltonians) containing the graviton modes and their corresponding ground states.An isomorphism can be defined for these conformal Hilbert spaces, and the mapping between them can be regarded as a more rigorous and general reinterpretation of the composite fermionization of FQH states, with naturally emergent composite fermions (each consisting of one electron and an even number of fluxes).The results from exact diagonalization will be shown, which confirm that for gapped phases, low-lying neutral excitations can undergo a phase transition even when the ground state remains in the same phase.Furthermore, the gaplessness of the Gaffnian state could be testified based on this formalism with further numerical experiments.Recently there has been numerical evidence implying the signature of multiple graviton modes.We will introduce the microscopic theory for the emergence of multiple gravitons in fractional quantum Hall droplets based on composite fermionization and the well-defined particle-hole conjugate within a specific conformal Hilbert space.This reveals the dynamical nature of this phenomenon and provides theoretical insights into the chirality and the "merging and splitting" behaviors of the graviton modes.The experimental relevance of multiple graviton modes will be discussed.The microscopic theory of gravitons can also provide valuable insights into the field-theoretical approaches.

•Jan 1, 2023•1 citation

Mechanism for particle fractionalization and universal edge physics in quantum Hall fluids

Arkadiusz Bochniak, Zohar Nussinov, Alexander Seidel +1 more

Abstract Advancing a microscopic framework that rigorously unveils the underlying topological hallmarks of fractional quantum Hall (FQH) fluids is a prerequisite for making progress in the classification of strongly-coupled topological matter. We present a second-quantization framework that reveals an exact fusion mechanism for particle fractionalization in FQH fluids, and uncovers the fundamental structure behind the condensation of non-local operators characterizing topological order in the lowest-Landau-level. We show the first exact analytic computation of the quasielectron Berry connections leading to its fractional charge and exchange statistics, and perform Monte Carlo simulations that numerically confirm the fusion mechanism for quasiparticles. We express the sequence of (bosonic and fermionic) Laughlin second-quantized states, highlighting the lack of local condensation, and present a rigorous constructive subspace bosonization dictionary for the bulk fluid. Finally, we establish universal long-distance behavior of edge excitations by formulating a conjecture based on the DNA, or root state, of the FQH fluid.

arXiv:2110.06216•Communications Physics•Jul 1, 2022•5 citations
Electrons in two dimensions in a high magnetic field can condense into an incompressible quantum fluid state when the electron density is equal to certain rational fractions of the flux density. Much progress has been made in understanding the nature of this state based on the ansatz wavefunction of Laughlin. Here, we show that the incompressible nature of the quantum fluid can be derived from a semiclassical evaluation of the imaginary time propagator. The dominant classical paths consist of cooperative ring exchange events. When these condense, they mediate a logarithmic interaction between density fluctuations, and produce a precise fractional quantization condition for the quasi-particle charges. We further show that the essential long wavelength properties of Laughlin's ground state and quasi-particle wavefunctions can be derived constructively from this approach. We conclude with some speculations concerning the nature of the off-diagonal order in the quantum Hall liquid.
We study the transition of the interacting two-dimensional electron gas at high magnetic field from a low-density Wigner crystal to a higher-density correlated state which exhibits the fractionally quantized Hall effect. The phase transition is precipitated by a condensation of ring-exchange processes, which by nature of their sensitivity to enclosed magnetic flux allow for especially low-energy exchange condensates at certain values of the Landau-level filling factor, \ensuremath{\nu}=${\ensuremath{\nu}}_{i}$. In the condensate, the exchanges mediate a logarithmic potential between local-density fluctuations, leading to a cusp in the ground-state energy as a function of \ensuremath{\Vert}\ensuremath{\nu}-${\ensuremath{\nu}}_{i}$\ensuremath{\Vert} in the vicinity of a preferred state. In addition, we derive quasiparticle excitations of sharp fractional charge which are analogous to the excitations derived by Laughlin.
Through the calculations of the diagonal and off-diagonal elements of the full NNN-particle density matrix we show that all essential features of the Laughlin variational wave functions can be derived in a generalized cooperative-ring-exchange theory of the fractional quantum Hall fluid.
The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx=e2/3h, in the vicinity of the Landau level filling factor ν=1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N≤7 electrons continuously as a function of magnetic field in the range 1/4≲ν<1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν=1/3 Laughlin state for N≤8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.
Considering each electron as a superposition of fictitious fractionally charged particles allows, with the help of a natural ansatz, a systematic identification and characterization of the incompressible quantum Hall states at noninteger filling factors. Explicit Laughlin-type wave functions are obtained for all the incompressible states and their quasiparticles. The order of stability of the various states predicted on the basis of physically plausible rules is in agreement with experiments. Although in principle all rational fractions are observable, these rules imply that the even-denominator fractions are in general much less stable than the odd-denominator ones.
The fractional quantum Hall fluid is a two-dimensional quantum fluid of electrons subject to a strong magnetic field at low temperatures.Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase.Among these states, there are some specific modes in the long-wavelength limit that can be understood as "spin-2 gravitons".In this thesis, we will introduce a set of analytical results for the energy gap of the graviton modes for model Hamiltonians in the thermodynamic limit, which is governed by a well-defined and universal characteristic tensor.These results can help to construct model Hamiltonians for the graviton modes of different FQH phases and elucidate a hierarchical structure of conformal Hilbert spaces (null spaces of model Hamiltonians) containing the graviton modes and their corresponding ground states.An isomorphism can be defined for these conformal Hilbert spaces, and the mapping between them can be regarded as a more rigorous and general reinterpretation of the composite fermionization of FQH states, with naturally emergent composite fermions (each consisting of one electron and an even number of fluxes).The results from exact diagonalization will be shown, which confirm that for gapped phases, low-lying neutral excitations can undergo a phase transition even when the ground state remains in the same phase.Furthermore, the gaplessness of the Gaffnian state could be testified based on this formalism with further numerical experiments.Recently there has been numerical evidence implying the signature of multiple graviton modes.We will introduce the microscopic theory for the emergence of multiple gravitons in fractional quantum Hall droplets based on composite fermionization and the well-defined particle-hole conjugate within a specific conformal Hilbert space.This reveals the dynamical nature of this phenomenon and provides theoretical insights into the chirality and the "merging and splitting" behaviors of the graviton modes.The experimental relevance of multiple graviton modes will be discussed.The microscopic theory of gravitons can also provide valuable insights into the field-theoretical approaches.
Abstract Advancing a microscopic framework that rigorously unveils the underlying topological hallmarks of fractional quantum Hall (FQH) fluids is a prerequisite for making progress in the classification of strongly-coupled topological matter. We present a second-quantization framework that reveals an exact fusion mechanism for particle fractionalization in FQH fluids, and uncovers the fundamental structure behind the condensation of non-local operators characterizing topological order in the lowest-Landau-level. We show the first exact analytic computation of the quasielectron Berry connections leading to its fractional charge and exchange statistics, and perform Monte Carlo simulations that numerically confirm the fusion mechanism for quasiparticles. We express the sequence of (bosonic and fermionic) Laughlin second-quantized states, highlighting the lack of local condensation, and present a rigorous constructive subspace bosonization dictionary for the bulk fluid. Finally, we establish universal long-distance behavior of edge excitations by formulating a conjecture based on the DNA, or root state, of the FQH fluid.