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

Physical review. B, Condensed matter

Hierarchical models with inverse-square interaction in harmonic confinement

Norio Kawakami, Yoshio Kuramoto

Published Aug 15, 1994• 6 citations
Publisher

Abstract

A hierarchy of quantum models with inverse-square pairwise interaction is studied for the system with a confining harmonic potential. An approach based on renormalized harmonic oscillators is used to construct the energy spectrum, which naturally results in the renormalized Hartree-Fock description of the interaction. In the limit of weak confinement, the model may serve as a canonical model for multiple chiral Tomonaga-Luttinger liquids for the edge states of the fractional quantum Hall effect. For the SU(\ensuremath{\nu}) symmetric case, we find that the excitation spectrum is classified in terms of the simple roots and the weights of SU(\ensuremath{\nu}) Lie algebra.

DOI
10.1103/physrevb.50.4664

Related papers

Renormalized Harmonic-Oscillator Description of Confined Electron Systems with Inverse-Square Interaction

Norio Kawakami

An integrable model for SU(ν) electrons with inverse-square interaction is studied for the system with confining harmonic potential. We develop a new description of the spectrum based on the renormalized harmonic-oscillators which incorporate interaction effects via the repulsion of energy levels. This approach enables a systematic treatment of the excitation spectrum as well as the ground-state quantities. 1 Quantum integrable systems with inverse-square (1/x2) interaction have been studied extensively with renewed interest [1–11]. Though various models including multicomponent cases have been studied in detail for periodic boundary conditions, the energy spectrum for confined systems has not been investigated systematically so far. In fact various investigations concerning confined systems have been mainly devoted to the single-component case [1,2]. Besides interest in the integrability there have been some attempts to apply the confined models to conductance-oscillation phenomena in mesoscopic systems [12–14], in which the exact ground state has been found for the electron model [14]. It is desirable to develop systematic treatments of the energy spectrum for the class of the integrable multicomponent models with confining potential. In this letter we wish to propose a systematic description of the energy spectrum for the class of one-dimensional multicomponent inverse-square systems with harmonic confinement. The essence of the idea is that the energy spectrum is expressed in terms of the renormalized harmonic oscillators (RHO) for which the interaction effects are taken into account by the renormalized quantum numbers for oscillators. To illustrate the idea, let us start with the integrable spinless-fermion model with confining harmonic potential [1,2],

•Jan 1, 1993•3 citations

Renormalized Harmonic-Oscillator Description of Confined Electron Systems with Inverse-Square Interaction

Norio Kawakami

An integrable model for SU($ν$) electrons with inverse-square interaction is studied for the system with confining harmonic potential. We develop a new description of the spectrum based on the {\it renormalized harmonic-oscillators} which incorporate interaction effects via the repulsion of energy levels. This approach enables a systematic treatment of the excitation spectrum as well as the ground-state quantities.

arXiv:cond-mat/9309038•Journal of the Physical Society of Japan•Dec 15, 1993•15 citations

Critical Properties of Quantum Many-Body Systems with 1/r2 Interaction

N. Kawakami

We review recent results obtained for a class of one-dimensional quantum models with 1/r2 long-range interaction. Based on the asymptotic Bethe-ansatz solution and conformal field theory, we study critical properties of the continuum boson model, the SU(ν) spin chain, the OSp(ν, 1) supersymmetric t-J model and a new hierarchy of models related to the fractional quantum Hall effect. We further investigate the class of 1/r2 models with harmonic confinement by means of a newly proposed method of the renormalized-harmonic oscillator solution.

arXiv:cond-mat/9402011•Progress of Theoretical Physics•Feb 1, 1994•21 citations

Critical Properties of Quantum Many-Body Systems with 1/r2 Interaction

Norio Kawakami

We review recent results obtained for a class of one-dimensional quantum models with 1/r2 long-range interaction. Based on the asymptotic Bethe-ansatz solution and conformal field theory, we study critical properties of the continuum boson model, the SU(ν) spin chain, the OSp(ν, 1) supersymmetric t-J model and a new hierarchy of models related to the fractional quantum Hall effect. We further investigate the class of 1/r2 models with harmonic confinement by means of a newly proposed method of the renormalized-harmonic oscillator solution.

Progress of Theoretical Physics•Feb 1, 1994•4 citations

Novel hierarchy of the SU(N) electron models and edge states of fractional quantum Hall effect

Norio Kawakami

A novel hierarchy of the one-dimensional SU(N) electron models with 1/${\mathit{r}}^{2}$ interaction is proposed and solved by the asymptotic Bethe ansatz both for the continuum and lattice cases. The construction of the hierarchy is closely related to that for the fractional quantum Hall effect (FQHE) of the filling factor ${\ensuremath{\nu}}_{\mathit{c}}$=1/[${\mathit{p}}_{1}$-1/(${\mathit{p}}_{2}$-...-1/${\mathit{p}}_{\mathit{N}}$)...]. Under the chiral constraint the model describes the essential properties of the edge states for the FQHE with the above filling fraction. Furthermore the matrix deduced from the excitation spectrum characterizes the topological order of the FQHE state.

Physical Review Letters•Jul 12, 1993•50 citations

A Microscopic Model of Edge States of Fractional Quantum Hall Liquid: From Composite Fermions to Calogero-Sutherland Model

Yue Yu

Based on the composite fermion approach, we derive a microscopic theory describing the low-lying edge excitations in the fractional quantum Hall liquid with $ν=\frac{ν^*}{\tildeϕν^*+1}$. For $ν^*>0$, it is found that the composite fermion model reduces to an SU$(ν^*)$ Calogero-Sutherland model in the one-dimensional limit, whereas it is not exact soluble for $ν^*<0$. However, the ground states in both cases can be found and the low-lying excitations can be shown the chiral Luttinger liquid behaviors since a gap exists between the right- and left-moving sectors in each branch of the azimuthal excitations.

arXiv:cond-mat/9906123•Jun 9, 1999•0 citations
A hierarchy of quantum models with inverse-square pairwise interaction is studied for the system with a confining harmonic potential. An approach based on renormalized harmonic oscillators is used to construct the energy spectrum, which naturally results in the renormalized Hartree-Fock description of the interaction. In the limit of weak confinement, the model may serve as a canonical model for multiple chiral Tomonaga-Luttinger liquids for the edge states of the fractional quantum Hall effect. For the SU(\ensuremath{\nu}) symmetric case, we find that the excitation spectrum is classified in terms of the simple roots and the weights of SU(\ensuremath{\nu}) Lie algebra.
An integrable model for SU(ν) electrons with inverse-square interaction is studied for the system with confining harmonic potential. We develop a new description of the spectrum based on the renormalized harmonic-oscillators which incorporate interaction effects via the repulsion of energy levels. This approach enables a systematic treatment of the excitation spectrum as well as the ground-state quantities. 1 Quantum integrable systems with inverse-square (1/x2) interaction have been studied extensively with renewed interest [1–11]. Though various models including multicomponent cases have been studied in detail for periodic boundary conditions, the energy spectrum for confined systems has not been investigated systematically so far. In fact various investigations concerning confined systems have been mainly devoted to the single-component case [1,2]. Besides interest in the integrability there have been some attempts to apply the confined models to conductance-oscillation phenomena in mesoscopic systems [12–14], in which the exact ground state has been found for the electron model [14]. It is desirable to develop systematic treatments of the energy spectrum for the class of the integrable multicomponent models with confining potential. In this letter we wish to propose a systematic description of the energy spectrum for the class of one-dimensional multicomponent inverse-square systems with harmonic confinement. The essence of the idea is that the energy spectrum is expressed in terms of the renormalized harmonic oscillators (RHO) for which the interaction effects are taken into account by the renormalized quantum numbers for oscillators. To illustrate the idea, let us start with the integrable spinless-fermion model with confining harmonic potential [1,2],
An integrable model for SU(ννν) electrons with inverse-square interaction is studied for the system with confining harmonic potential. We develop a new description of the spectrum based on the {\it renormalized harmonic-oscillators} which incorporate interaction effects via the repulsion of energy levels. This approach enables a systematic treatment of the excitation spectrum as well as the ground-state quantities.
We review recent results obtained for a class of one-dimensional quantum models with 1/r2 long-range interaction. Based on the asymptotic Bethe-ansatz solution and conformal field theory, we study critical properties of the continuum boson model, the SU(ν) spin chain, the OSp(ν, 1) supersymmetric t-J model and a new hierarchy of models related to the fractional quantum Hall effect. We further investigate the class of 1/r2 models with harmonic confinement by means of a newly proposed method of the renormalized-harmonic oscillator solution.
We review recent results obtained for a class of one-dimensional quantum models with 1/r2 long-range interaction. Based on the asymptotic Bethe-ansatz solution and conformal field theory, we study critical properties of the continuum boson model, the SU(ν) spin chain, the OSp(ν, 1) supersymmetric t-J model and a new hierarchy of models related to the fractional quantum Hall effect. We further investigate the class of 1/r2 models with harmonic confinement by means of a newly proposed method of the renormalized-harmonic oscillator solution.
A novel hierarchy of the one-dimensional SU(N) electron models with 1/r2{\mathit{r}}^{2}r2 interaction is proposed and solved by the asymptotic Bethe ansatz both for the continuum and lattice cases. The construction of the hierarchy is closely related to that for the fractional quantum Hall effect (FQHE) of the filling factor ${\ensuremath{\nu}}_{\mathit{c}}$=1/[p1{\mathit{p}}_{1}p1​-1/(p2{\mathit{p}}_{2}p2​-...-1/pN{\mathit{p}}_{\mathit{N}}pN​)...]. Under the chiral constraint the model describes the essential properties of the edge states for the FQHE with the above filling fraction. Furthermore the matrix deduced from the excitation spectrum characterizes the topological order of the FQHE state.
Based on the composite fermion approach, we derive a microscopic theory describing the low-lying edge excitations in the fractional quantum Hall liquid with ν=ν∗ϕ~ν∗+1ν=\frac{ν^*}{\tildeϕν^*+1}ν=ϕ~​ν∗+1ν∗​. For ν∗>0ν^*>0ν∗>0, it is found that the composite fermion model reduces to an SU(ν∗)(ν^*)(ν∗) Calogero-Sutherland model in the one-dimensional limit, whereas it is not exact soluble for ν∗<0ν^*<0ν∗<0. However, the ground states in both cases can be found and the low-lying excitations can be shown the chiral Luttinger liquid behaviors since a gap exists between the right- and left-moving sectors in each branch of the azimuthal excitations.