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

Norio Kawakami

Published Jan 1, 1993• 3 citations
OpenAlex

Abstract

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],

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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

Hierarchical models with inverse-square interaction in harmonic confinement

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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.

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The integrals of motion are constructed for the Sutherland hyperbolic quantum systems of particles with internal degrees of freedom (SU( n ) spins) interacting with an external field with the Morse potential of an arbitrary strength τ 2 . These systems are confined if some constraint is imposed on τ, the strength λ of the pairwise interaction and the number of particles. The ground state is described by the wave function of the Jastrow form. The known results for the systems with inverse-square interaction within harmonic well are reproduced in the limit τ → ∞ .

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Eigenfunctions for SU( nu ) particles with 1/r2interaction in harmonic confinement

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We find a set of exact eigenfunctions which provide the energy spectrum for the quantum N-body Calogero-Sutherland model for fermions or bosons with SU( nu ) spin degrees of freedom moving in a harmonic confinement potential. The eigenfunctions are explicitly constructed as a simple product of the Jastrow wavefunction for the ground state and the Hermite polynomials introduced to generate the excited states. The corresponding energy spectrum is given by a sum of the correlated ground-state energy and of the excited-state energy for SU( nu ) free particles in a harmonic well.

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Finite size effects and dynamics in one-dimensional integrable systems

Maria Colomé‐Tatché

Many physical systems can be described by one-dimensional (1D) models. It is the case of ultra-cold atoms: under certain circumstances their dynamics occurs only in one dimension. During my PhD I studied some aspects of 1D integrable systems. First, I present a study on the ground state of a system of 2-component repulsive fermions in 1D under harmonic confinement. I use the Bethe ansatz solution to calculate the phase diagram of the system in the homogeneous case. Adding a harmonic confinement I show that the atoms are distributed in a two-shell structure: the partially polarised phase in the inner shell and the fully polarised phase at the edges of the trap. Next I study the finite size effects for the gap of the quasiparticle excitation spectrum in the 1D Hubbard model. Two type of corrections to the result of the thermodynamic limit are obtained: a power law correction inversely proportional to the size of the system L, due to gapless excitations, and an exponential correction on L related to the existence of gapped excitations. In the weakly interacting regime this last correction can become important. Finally I study the response of a highly excited 1D gas to a periodic modulation of the coupling constant. I consider the Lieb-Liniger model and the non-integrable model of a single mobile impurity in a Fermi gas. I show that the non-integrable system is sensitive to excitations with frequencies as low as the mean level spacing, whereas the threshold frequency in the integrable case is much larger. This effect can be used as a probe of integrability for mesoscopic 1D systems, and can be observed experimentally by measuring the heating rate of a parametrically excited gas.

•Dec 17, 2008•0 citations

Models with Inverse-Square Exchange

A one-dimensional quantum N-body system of either fermions or bosons with SU (n) "spins" (or colors in particle physics language) interacting via inverse-square exchange is presented in this article.A class of eigenstates of both the continuum and lattice version of the model Hamiltonians is constructed in terms of the Jastrow-product type wave function.The class of states we construct in this paper corresponds to the ground state and the low energy excitations of the model that can be described by the effective harmonic fluid Hamiltonian.By expanding the energy about the ground state we find the harmonic fluid parameters (i.e. the charge, spin velocities, etc.), explicitly.The correlation exponent and the compressibility are also found.As expected the general harmonic relation(i.e.v S = (v N v J ) 1/2 ) is satisfied among the charge and spin velocities.

WORLD SCIENTIFIC eBooks•Sep 1, 1996•0 citations
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.
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.
The integrals of motion are constructed for the Sutherland hyperbolic quantum systems of particles with internal degrees of freedom (SU( n ) spins) interacting with an external field with the Morse potential of an arbitrary strength τ 2 . These systems are confined if some constraint is imposed on τ, the strength λ of the pairwise interaction and the number of particles. The ground state is described by the wave function of the Jastrow form. The known results for the systems with inverse-square interaction within harmonic well are reproduced in the limit τ → ∞ .
We find a set of exact eigenfunctions which provide the energy spectrum for the quantum N-body Calogero-Sutherland model for fermions or bosons with SU( nu ) spin degrees of freedom moving in a harmonic confinement potential. The eigenfunctions are explicitly constructed as a simple product of the Jastrow wavefunction for the ground state and the Hermite polynomials introduced to generate the excited states. The corresponding energy spectrum is given by a sum of the correlated ground-state energy and of the excited-state energy for SU( nu ) free particles in a harmonic well.
Many physical systems can be described by one-dimensional (1D) models. It is the case of ultra-cold atoms: under certain circumstances their dynamics occurs only in one dimension. During my PhD I studied some aspects of 1D integrable systems. First, I present a study on the ground state of a system of 2-component repulsive fermions in 1D under harmonic confinement. I use the Bethe ansatz solution to calculate the phase diagram of the system in the homogeneous case. Adding a harmonic confinement I show that the atoms are distributed in a two-shell structure: the partially polarised phase in the inner shell and the fully polarised phase at the edges of the trap. Next I study the finite size effects for the gap of the quasiparticle excitation spectrum in the 1D Hubbard model. Two type of corrections to the result of the thermodynamic limit are obtained: a power law correction inversely proportional to the size of the system L, due to gapless excitations, and an exponential correction on L related to the existence of gapped excitations. In the weakly interacting regime this last correction can become important. Finally I study the response of a highly excited 1D gas to a periodic modulation of the coupling constant. I consider the Lieb-Liniger model and the non-integrable model of a single mobile impurity in a Fermi gas. I show that the non-integrable system is sensitive to excitations with frequencies as low as the mean level spacing, whereas the threshold frequency in the integrable case is much larger. This effect can be used as a probe of integrability for mesoscopic 1D systems, and can be observed experimentally by measuring the heating rate of a parametrically excited gas.
A one-dimensional quantum N-body system of either fermions or bosons with SU (n) "spins" (or colors in particle physics language) interacting via inverse-square exchange is presented in this article.A class of eigenstates of both the continuum and lattice version of the model Hamiltonians is constructed in terms of the Jastrow-product type wave function.The class of states we construct in this paper corresponds to the ground state and the low energy excitations of the model that can be described by the effective harmonic fluid Hamiltonian.By expanding the energy about the ground state we find the harmonic fluid parameters (i.e. the charge, spin velocities, etc.), explicitly.The correlation exponent and the compressibility are also found.As expected the general harmonic relation(i.e.v S = (v N v J ) 1/2 ) is satisfied among the charge and spin velocities.