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:cond-mat/9808221

Jastrow-Luttinger Fractional Liquids

K.-V. Pham, M. Gabay, Pascal Lederer

Published Aug 20, 1998• 1 citation
arXivHTMLPDF

Abstract

In this paper, we present a description of Haldane's Luttinger liquid which parallels Laughlin's theory of the Fractional Quantum Hall (FQH) incompressible fluid, both exhibiting similar ground states as well as fractional excitations. These two non-Fermi liquids are instances of a generic structure for low-dimensional quantum liquids which we propose to dub Jastrow-Luttinger Fractional Liquids. An important feature of such liquids is the complete fractionalization of the parent particle. In particular, in both one and two dimensions spin-charge separation can be achieved and is indeed suggested to occur for unpolarized quantum Hall systems both at the edge and in the bulk.

DOI
10.48550/arxiv.cond-mat/9808221

Related papers

Fractional excitations in the Luttinger liquid

K.-V. Pham, M. Gabay, Pascal Lederer

We reconsider the spectrum of the Luttinger liquid (LL) usually understood in terms of phonons (density fluctuations), and within the context of bosonization we give an alternative representation in terms of fractional states. This allows us to make contact with the Bethe Ansatz which predicts similar fractional states. As an example we study the spinon operator in the absence of spin rotational invariance and derive it from first principles: we find that it is not a semion in general; a trial Jastrow wave function is also given for that spinon state. Our construction of the spectroscopy based on fractional states leads to several unique physical insights: in the low-energy limit, we find that the ${S}_{z}=0$ continuum of gapless spin chains is due to pairs of fractional quasiparticle-quasihole states which are the one-dimensional counterpart of the Laughlin fractional Quantum Hall effect quasiparticles. The holon operator for the Luttinger liquid with spin is also derived. In the presence of a magnetic field, spin-charge separation is not realized any longer in a LL: the holon and the spinon are then replaced by new fractional states that we are able to describe.

arXiv:cond-mat/9909295•Physical review. B, Condensed matter•Jun 15, 2000•135 citations

One-dimensional Fermi liquids

Johannes Voit

We review the progress in the theory of one-dimensional (ID) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory, based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle-hole bubble, producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless 1D quantum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a "Luttinger liquid" implies that their low-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, and variants such as the t-J- or the extended Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.

arXiv:cond-mat/9510014•Reports on Progress in Physics•Sep 1, 1995•1,282 citations

‘Luttinger liquid theory’ of one-dimensional quantum fluids I: Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas

F. D. M. Haldane

The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.

Advanced series in mathematical physics•May 1, 1994•1 citation

'Luttinger liquid theory' of one-dimensional quantum fluids. I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas

F. D. M. Haldane

The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.

Journal of Physics C Solid State Physics•Jul 10, 1981•2,495 citations

Entanglement spectrum edge reconstruction and correlation hole of fractional quantum Hall liquids

Li-Xia Wei, Na Jiang, Qi Li +1 more

The edge of the electronic fractional quantum Hall (FQH) system obeys the law of the chiral Luttinger liquid theory due to its intrinsic topological properties and the relation of bulk-edge correspondence. However, in a realistic experimental system, such as the usual Hall bar setup, the softening of the background confinement potential can induce the reconstruction of the edge spectrum, which breaks the chirality and universality of the FQH edge. The entanglement spectrum (ES) of the FQH ground state has the same counting structure as that of the energy spectrum indicating the topological characters of the quantum state. In this work, we report that the ES can also have an edge reconstruction while sweeping the area of the subsystem in real space cut. Moreover, we found that the critical area of the subsystem matches accurately with the intrinsic building block of the FQH liquids, namely the correlation hole of the FQH liquids. The above results seem to be universal after our study of a series of typical FQH states, such as two Laughlin states at $\ensuremath{\nu}=1/3$ and $1/5$, and the Moore-Read state for $\ensuremath{\nu}=5/2$.

arXiv:1910.11975•Physical review. B./Physical review. B•Feb 27, 2020•4 citations

Fermi Liquids and Fractional Statistics in One Dimension

Jon Magne Leinaas

Interacting fermion systems in one dimension, which in the low energy approximation are described by Luttinger liquid theory, can be reformulated as systems of weakly interacting particles with fractional exchange statistics. This is shown by use of Landau's Fermi liquid theory, with quasiparticles interpreted as adiabatically dressed fermions. An application of this method is included, where boundary excitations of a two-dimensional quantum Hall electron system are studied.

arXiv:2104.07775•WORLD SCIENTIFIC eBooks•Mar 31, 2022•0 citations
In this paper, we present a description of Haldane's Luttinger liquid which parallels Laughlin's theory of the Fractional Quantum Hall (FQH) incompressible fluid, both exhibiting similar ground states as well as fractional excitations. These two non-Fermi liquids are instances of a generic structure for low-dimensional quantum liquids which we propose to dub Jastrow-Luttinger Fractional Liquids. An important feature of such liquids is the complete fractionalization of the parent particle. In particular, in both one and two dimensions spin-charge separation can be achieved and is indeed suggested to occur for unpolarized quantum Hall systems both at the edge and in the bulk.
We reconsider the spectrum of the Luttinger liquid (LL) usually understood in terms of phonons (density fluctuations), and within the context of bosonization we give an alternative representation in terms of fractional states. This allows us to make contact with the Bethe Ansatz which predicts similar fractional states. As an example we study the spinon operator in the absence of spin rotational invariance and derive it from first principles: we find that it is not a semion in general; a trial Jastrow wave function is also given for that spinon state. Our construction of the spectroscopy based on fractional states leads to several unique physical insights: in the low-energy limit, we find that the Sz=0{S}_{z}=0Sz​=0 continuum of gapless spin chains is due to pairs of fractional quasiparticle-quasihole states which are the one-dimensional counterpart of the Laughlin fractional Quantum Hall effect quasiparticles. The holon operator for the Luttinger liquid with spin is also derived. In the presence of a magnetic field, spin-charge separation is not realized any longer in a LL: the holon and the spinon are then replaced by new fractional states that we are able to describe.
We review the progress in the theory of one-dimensional (ID) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory, based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle-hole bubble, producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless 1D quantum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a "Luttinger liquid" implies that their low-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, and variants such as the t-J- or the extended Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.
The edge of the electronic fractional quantum Hall (FQH) system obeys the law of the chiral Luttinger liquid theory due to its intrinsic topological properties and the relation of bulk-edge correspondence. However, in a realistic experimental system, such as the usual Hall bar setup, the softening of the background confinement potential can induce the reconstruction of the edge spectrum, which breaks the chirality and universality of the FQH edge. The entanglement spectrum (ES) of the FQH ground state has the same counting structure as that of the energy spectrum indicating the topological characters of the quantum state. In this work, we report that the ES can also have an edge reconstruction while sweeping the area of the subsystem in real space cut. Moreover, we found that the critical area of the subsystem matches accurately with the intrinsic building block of the FQH liquids, namely the correlation hole of the FQH liquids. The above results seem to be universal after our study of a series of typical FQH states, such as two Laughlin states at $\ensuremath{\nu}=1/3$ and 1/51/51/5, and the Moore-Read state for $\ensuremath{\nu}=5/2$.