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/9607041 · Physical review. B, Condensed matter

Exact results of a hard-core interacting system with a single impurity

You‐Quan Li, Zhongshui Ma

Published Nov 1, 1995• 8 citations
arXivHTMLPDF

Abstract

The quantum-mechanical problem of a many-particle system with a single impurity in one dimension, interacting by a \ensuremath{\delta} function, is solved. The wave function for a bosonic system and the related secular equation for the spectrum are obtained. The energy per length is calculated in the thermodynamic limit.
DOI
10.1103/physrevb.52.r13071

Related papers

δ-function bosons in a one-dimensional potential well

You‐Quan Li

The quantum-mechanical problem of bosons, in a square well in one dimension and interacting with a \ensuremath{\delta} function, is analyzed and solved. The resulting Bethe-ansatz wave function is a sum, over the Weyl group of Lie algebra ${\mathit{B}}_{\mathit{N}}$, of nondiffractive waves. The secular equation of the spectrum is obtained and its thermodynamic limit is discussed.

Physical Review A•Jul 1, 1995•7 citations

Motion of an impurity particle in an ultracold quasi-one-dimensional gas of hard-core bosons

Marvin D. Girardeau, Anna Minguzzi

The low-lying energy eigenstates of a one-dimensional (1D) system of many impenetrable point bosons and one moving impurity particle with repulsive zero-range impurity-boson interaction are found for all values of the impurity-boson mass ratio and coupling constant. The moving entity is a polaronlike composite object consisting of the impurity clothed by a comoving gray soliton. The special case with impurity-boson interaction of point hard-core form and impurity-boson mass ratio ${m}_{i}/m$ unity is first solved exactly as a special case of a previous Fermi-Bose (FB) mapping treatment of soluble 1D Bose-Fermi mixture problems. Then a more general treatment is given using second quantization for the bosons and the second-quantized form of the FB mapping, eliminating the impurity degrees of freedom by a Lee-Low-Pines canonical transformation. This yields the exact ground state (total linear momentum $q=0$) and exact boson-impurity distribution function in the thermodynamic limit for arbitrary ${m}_{i}/m$ and arbitrary impurity-boson interaction strength. These results are then extended to states with $q>0$.

arXiv:0807.3366•Physical Review A•Mar 12, 2009•23 citations

Solution of the Integral Equations for the Many-Body Problem in One Dimension

A. D. Jannussis

The integral equations for the calculation of the ground-state energy of the repulsive delta-function interaction problem in one dimension for $N$ particles are reduced to a system of algebraic equations by using the method of moments. We obtain explicit results in a limiting case.

Physical Review Letters•Aug 19, 1968•2 citations

Ground-state energy of one-dimensional many-boson systems with attractive delta interactions

Steven A. Moszkowski, George L. Strobel

We have calculated the ground-state energy of a system of bosons in one dimension interacting via attractive δ-function interactions. In this example, the K harmonic method in its simplest form gives slightly larger binding than the independent particle method and slightly less than the exact result.

American Journal of Physics•Nov 1, 1979•2 citations

Thermodynamic properties of a finite one-dimensional system of bosons with repulsive delta-function interaction

Ka-Wo Poon, K. Young, David Kiang

The thermodynamics of N bosons in a length L in one dimension, with repulsive delta-function interaction, is studied numerically for finite N, L. The results show the nature of finite-size corrections and how the thermodynamic limit is approached, and hopefully will be of some guidance in seeking the solution of a more general model.

Canadian Journal of Physics•Mar 1, 1995•0 citations

Analysis of the Many-Body Problem in One Dimension with Repulsive Delta-Function Interaction

M. Albertsson

The repulsive delta-function interaction model in one dimension is reviewed for spinless particles and for spin-1/2 fermions. The problem of solving the differential equation related to the Schrödinger equation is reduced by the Bethe ansatz to a system of algebraic equations. The delta-function interaction is shown to have no effect on spinless fermions which therefore behave like free fermions, in agreement with Pauli's exclusion principle. The ground-state problem of spinless bosons is reduced to an inhomogeneous Fredholm equation of the second kind. In the limit of impenetrable interactions, the spinless bosons are shown to have the energy spectrum of free fermions. The model for spin-1/2 fermions is reduced by the Bethe ansatz to an eigenvalue problem of matrices of the same sizes as the irreducible representations R of the permutation group of N elements. For some R's this eigenvalue problem itself is solved by a generalized Bethe ansatz. The ground-state problem of spin-1/2 fermions is reduced to a generalized Fredholm equation.

•Jan 1, 2014•1 citation
The quantum-mechanical problem of bosons, in a square well in one dimension and interacting with a \ensuremath{\delta} function, is analyzed and solved. The resulting Bethe-ansatz wave function is a sum, over the Weyl group of Lie algebra BN{\mathit{B}}_{\mathit{N}}BN​, of nondiffractive waves. The secular equation of the spectrum is obtained and its thermodynamic limit is discussed.
The low-lying energy eigenstates of a one-dimensional (1D) system of many impenetrable point bosons and one moving impurity particle with repulsive zero-range impurity-boson interaction are found for all values of the impurity-boson mass ratio and coupling constant. The moving entity is a polaronlike composite object consisting of the impurity clothed by a comoving gray soliton. The special case with impurity-boson interaction of point hard-core form and impurity-boson mass ratio mi/m{m}_{i}/mmi​/m unity is first solved exactly as a special case of a previous Fermi-Bose (FB) mapping treatment of soluble 1D Bose-Fermi mixture problems. Then a more general treatment is given using second quantization for the bosons and the second-quantized form of the FB mapping, eliminating the impurity degrees of freedom by a Lee-Low-Pines canonical transformation. This yields the exact ground state (total linear momentum q=0q=0q=0) and exact boson-impurity distribution function in the thermodynamic limit for arbitrary mi/m{m}_{i}/mmi​/m and arbitrary impurity-boson interaction strength. These results are then extended to states with q>0q>0q>0.
The integral equations for the calculation of the ground-state energy of the repulsive delta-function interaction problem in one dimension for NNN particles are reduced to a system of algebraic equations by using the method of moments. We obtain explicit results in a limiting case.
The repulsive delta-function interaction model in one dimension is reviewed for spinless particles and for spin-1/2 fermions. The problem of solving the differential equation related to the Schrödinger equation is reduced by the Bethe ansatz to a system of algebraic equations. The delta-function interaction is shown to have no effect on spinless fermions which therefore behave like free fermions, in agreement with Pauli's exclusion principle. The ground-state problem of spinless bosons is reduced to an inhomogeneous Fredholm equation of the second kind. In the limit of impenetrable interactions, the spinless bosons are shown to have the energy spectrum of free fermions. The model for spin-1/2 fermions is reduced by the Bethe ansatz to an eigenvalue problem of matrices of the same sizes as the irreducible representations R of the permutation group of N elements. For some R's this eigenvalue problem itself is solved by a generalized Bethe ansatz. The ground-state problem of spin-1/2 fermions is reduced to a generalized Fredholm equation.