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

arXiv:1111.2519 · Physical Review B

Exact wave functions for excitations of the ν=13 fractional quantum Hall state from a model Hamiltonian

Paul Soulé, Th. Jolicœur

Published Apr 9, 2012• 13 citations
arXivHTMLPDF

Abstract

We study fractional quantum Hall states in the cylinder geometry with open boundaries. We introduce solvable Hamiltonians for which we are able to obtain exact results. We give a simple construction of the ground state, quasiholes, quasielectrons, and the magnetoroton branch of excited states for spin-polarized electrons at filling factor $\ensuremath{\nu}=1/3$ and spinless bosons at filling $\ensuremath{\nu}=1/2$. The wave functions are simple in the second-quantized language. These model Hamiltonians have all the features we expect from composite fermion theory.

DOI
10.1103/physrevb.85.155116

Related papers

Exact wavefunctions for excitations of the nu=1/3 fractional quantum Hall state from a model Hamiltonian

Paul Soulé, Th. Jolicœur

We study fractional quantum Hall states in the cylinder geometry with open boundaries. By truncating the Coulomb interactions between electrons we show that it is possible to construct infinitely many exact eigenstates including the ground state, quasiholes, quasielectrons and the magnetoroton branch of excited states.

•Jan 1, 2012•0 citations

Quantum Hall hierarchy wave functions: From conformal correlators to Tao-Thouless states

Emil J. Bergholtz, T. H. Hansson, Maria Hermanns +2 more

Laughlin's wave functions, which describe the fractional quantum Hall effect at filling factors $\ensuremath{\nu}=1/(2k+1)$, can be obtained as correlation functions in a conformal field theory, and recently, this construction was extended to Jain's composite fermion wave functions at filling factors $\ensuremath{\nu}=n/(2kn+1)$. Here, we generalize this latter construction and present ground state wave functions for all quantum Hall hierarchy states that are obtained by successive condensation of quasielectrons (as opposed to quasiholes) in the original hierarchy construction. By considering these wave functions on a cylinder, we show that they approach the exact ground states, which are the Tao-Thouless states, when the cylinder becomes thin. We also present wave functions for the multihole states, make the connection to Wen's general classification of Abelian quantum Hall fluids, and discuss whether the fractional statistics of the quasiparticles can be analytically determined. Finally, we discuss to what extent our wave functions can be described in the language of composite fermions.

arXiv:0712.3848•Physical Review B•Apr 17, 2008•31 citations

New Trial Wave Functions for Quantum Hall States at Half Filling

Jian Cheng Yang

New trial wave functions corresponding to half filling quantum Hall states are proposed. These wave functions are constructed by first pairing up the quasielectrons of the 1/3 Laughlin quantum Hall state, with the same relative angular momentum for each pair, and then making the paired quasielectrons condense into a 1/4 Laughlin state. The quasiparticle excitations of the proposed wave functions carry $\pm1/4$ of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta $N_ϕ$ is shown to be related to the number of electrons $N$ by $N_ϕ = 2N-(5-q)$ with $q$ being the relative angular momentum between the quasielectrons in each pair which takes values of non-negative even integers. The overlaps are calculated between the proposed trial wave functions, including the ground state, quasiexciton states, and quasihole states, and the exact states of the finite size systems at $N_ϕ = 2N-3$. The near unity overlaps are obtained in the lowest Landau level, while the moderate overlaps are obtained in the second Landau level. The relevance of the wave functions to the yet to be discovered fractional quantum Hall effect in the lowest Landau level, as well as the 5/2 quantum Hall effect is also discussed.

arXiv:1112.4650•Dec 20, 2011•0 citations

Quantum Hall hierarchy wave functions from conformal field theory

S. Viefers, Emil J. Bergholtz, T. H. Hansson +2 more

It has long been known that Laughlin's wave functions, describing the fractional quantum Hall effect at filling fractions ν=1/(2k+1), can be obtained as correlation functions in conformal field theory. We show how to generalize this approach to construct explicit trial wave functions for all states in the quantum Hall hierarchy corresponding to quasiparticle (as opposed to quasihole) condensates, including the recently observed state at ν= 4/11. At the filling fractions ν= n/(2np+1) this construction exactly reproduces Jain's composite fermion wave functions. An explicit connection is made to Wen's topological classification of FQH states.

Bulletin of the American Physical Society•Mar 13, 2008•0 citations

Fractional quantum Hall states of clustered composite fermions

Arkadiusz Wójs, Kyung-Soo Yi, John J. Quinn

The energy spectra and wave functions of up to 14 interacting quasielectrons (QE's) in the Laughlin $\ensuremath{\nu}=\frac{1}{3}$ fractional quantum Hall (FQH) state are investigated using exact numerical diagonalization. It is shown that at sufficiently high density the QE's form pairs or larger clusters. This behavior, opposite to Laughlin correlations, invalidates the (sometimes invoked) reapplication of the composite fermion picture to the individual QE's. The series of finite-size incompressible ground states are identified at the QE filling factors ${\ensuremath{\nu}}_{\mathrm{QE}}=\frac{1}{2},$ $\frac{1}{3},$ $\frac{2}{3},$ corresponding to the electron fillings $\ensuremath{\nu}=\frac{3}{8},$ $\frac{4}{11},$ $\frac{5}{13}.$ The equivalent quasihole (QH) states occur at ${\ensuremath{\nu}}_{\mathrm{QH}}=\frac{1}{4},$ $\frac{1}{5},$ $\frac{2}{7},$ corresponding to $\ensuremath{\nu}=\frac{3}{10},$ $\frac{4}{13},$ $\frac{5}{17}.$ All these six novel FQH states were recently discovered experimentally. Detailed analysis indicates that QE or QH correlations in these states are different from those of well-known FQH electron states (e.g., Laughlin or Moore--Read states), leaving the origin of their incompressibility uncertain. Halperin's idea of Laughlin states of QP pairs is also explored, but is does not seem adequate.

arXiv:cond-mat/0312290•Physical Review B•May 28, 2004•73 citations

Exactly Solvable Fermion Chain Describing a ν=1/3 Fractional Quantum Hall State

Masaaki Nakamura, Zheng-Yuan Wang, Emil J. Bergholtz

We introduce an exactly solvable fermion chain that describes a ν=1/3 fractional quantum Hall (FQH) state beyond the thin-torus limit. The ground state of our model is shown to be unique for each center-of-mass sector, and it has a matrix product representation that enables us to exactly calculate order parameters, correlation functions, and entanglement spectra. The ground state of our model shows striking similarities with the BCS wave functions and quantum spin-1 chains. Using the variational method with matrix product ansatz, we analytically calculate excitation gaps and vanishing of the compressibility expected in the FQH state. We also show that the above results can be related to a ν=1/2 bosonic FQH state.

arXiv:1110.5033•Physical Review Letters•Jul 2, 2012•55 citations
We study fractional quantum Hall states in the cylinder geometry with open boundaries. We introduce solvable Hamiltonians for which we are able to obtain exact results. We give a simple construction of the ground state, quasiholes, quasielectrons, and the magnetoroton branch of excited states for spin-polarized electrons at filling factor \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3 and spinless bosons at filling \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=1/2. The wave functions are simple in the second-quantized language. These model Hamiltonians have all the features we expect from composite fermion theory.
Laughlin's wave functions, which describe the fractional quantum Hall effect at filling factors $\ensuremath{\nu}=1/(2k+1)$, can be obtained as correlation functions in a conformal field theory, and recently, this construction was extended to Jain's composite fermion wave functions at filling factors $\ensuremath{\nu}=n/(2kn+1)$. Here, we generalize this latter construction and present ground state wave functions for all quantum Hall hierarchy states that are obtained by successive condensation of quasielectrons (as opposed to quasiholes) in the original hierarchy construction. By considering these wave functions on a cylinder, we show that they approach the exact ground states, which are the Tao-Thouless states, when the cylinder becomes thin. We also present wave functions for the multihole states, make the connection to Wen's general classification of Abelian quantum Hall fluids, and discuss whether the fractional statistics of the quasiparticles can be analytically determined. Finally, we discuss to what extent our wave functions can be described in the language of composite fermions.
New trial wave functions corresponding to half filling quantum Hall states are proposed. These wave functions are constructed by first pairing up the quasielectrons of the 1/3 Laughlin quantum Hall state, with the same relative angular momentum for each pair, and then making the paired quasielectrons condense into a 1/4 Laughlin state. The quasiparticle excitations of the proposed wave functions carry ±1/4\pm1/4±1/4 of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta NϕN_ϕNϕ​ is shown to be related to the number of electrons NNN by Nϕ=2N−(5−q)N_ϕ = 2N-(5-q)Nϕ​=2N−(5−q) with qqq being the relative angular momentum between the quasielectrons in each pair which takes values of non-negative even integers. The overlaps are calculated between the proposed trial wave functions, including the ground state, quasiexciton states, and quasihole states, and the exact states of the finite size systems at Nϕ=2N−3N_ϕ = 2N-3Nϕ​=2N−3. The near unity overlaps are obtained in the lowest Landau level, while the moderate overlaps are obtained in the second Landau level. The relevance of the wave functions to the yet to be discovered fractional quantum Hall effect in the lowest Landau level, as well as the 5/2 quantum Hall effect is also discussed.
It has long been known that Laughlin's wave functions, describing the fractional quantum Hall effect at filling fractions ν=1/(2k+1), can be obtained as correlation functions in conformal field theory. We show how to generalize this approach to construct explicit trial wave functions for all states in the quantum Hall hierarchy corresponding to quasiparticle (as opposed to quasihole) condensates, including the recently observed state at ν= 4/11. At the filling fractions ν= n/(2np+1) this construction exactly reproduces Jain's composite fermion wave functions. An explicit connection is made to Wen's topological classification of FQH states.
The energy spectra and wave functions of up to 14 interacting quasielectrons (QE's) in the Laughlin $\ensuremath{\nu}=\frac{1}{3}$ fractional quantum Hall (FQH) state are investigated using exact numerical diagonalization. It is shown that at sufficiently high density the QE's form pairs or larger clusters. This behavior, opposite to Laughlin correlations, invalidates the (sometimes invoked) reapplication of the composite fermion picture to the individual QE's. The series of finite-size incompressible ground states are identified at the QE filling factors ${\ensuremath{\nu}}_{\mathrm{QE}}=\frac{1}{2},$ 13,\frac{1}{3},31​, 23,\frac{2}{3},32​, corresponding to the electron fillings $\ensuremath{\nu}=\frac{3}{8},$ 411,\frac{4}{11},114​, 513.\frac{5}{13}.135​. The equivalent quasihole (QH) states occur at ${\ensuremath{\nu}}_{\mathrm{QH}}=\frac{1}{4},$ 15,\frac{1}{5},51​, 27,\frac{2}{7},72​, corresponding to $\ensuremath{\nu}=\frac{3}{10},$ 413,\frac{4}{13},134​, 517.\frac{5}{17}.175​. All these six novel FQH states were recently discovered experimentally. Detailed analysis indicates that QE or QH correlations in these states are different from those of well-known FQH electron states (e.g., Laughlin or Moore--Read states), leaving the origin of their incompressibility uncertain. Halperin's idea of Laughlin states of QP pairs is also explored, but is does not seem adequate.
We introduce an exactly solvable fermion chain that describes a ν=1/3 fractional quantum Hall (FQH) state beyond the thin-torus limit. The ground state of our model is shown to be unique for each center-of-mass sector, and it has a matrix product representation that enables us to exactly calculate order parameters, correlation functions, and entanglement spectra. The ground state of our model shows striking similarities with the BCS wave functions and quantum spin-1 chains. Using the variational method with matrix product ansatz, we analytically calculate excitation gaps and vanishing of the compressibility expected in the FQH state. We also show that the above results can be related to a ν=1/2 bosonic FQH state.