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

Journal of Physics A Mathematical and General

Analytical and numerical approximations to the delta interaction

Renzo Alzetta, A M Pinna

Published May 11, 1987• 0 citations
Publisher

Abstract

First-order perturbative corrections to the ground-state energy and Hartree-Fock equations are derived for the system of (N+N) identical fermions attracted by a delta-function interaction on a circle of length L. Numerical results of Monte Carlo calculations for few fermions are presented and discussed.
DOI
10.1088/0305-4470/20/7/023

Related papers

Hartree-Fock states in the thermodynamic limit. III. Low-density clustering effects

L. Döhnert, MANUEL DE LLANO, Angelo Plastino

We present an orthonormal set of single-particle orbital functions explicitly satisfying the Hartree-Fock (HF) equations for occupied states, and which give low-density few-particle clustering effects in a many-fermion system. The initial study given here is in one and three dimensions for attractive $\ensuremath{\delta}$ pair interactions. In one dimension the numerical results compare favorably with the exact (lowest) HF state at zero density. The results turn out to be superior to the classic Overhauser HF orbitals, as well as to some more recent ones.

Physical Review A•Feb 1, 1978•14 citations

Solitons and the delta function fermion gas in Hartree–Fock theory

Élliott H. Lieb, MANUEL DE LLANO

The quantum-mechanical, fermion, attractive delta function gas in one dimension has many interesting, known physical properties, including some aspects of soliton behavior. Motivated by a desire to test the validity of the HF approximation to many-body systems, we have made a rigorous study of the HF theory of this gas. When the particle number is even we can show that although HF theory does not give energies accurately, it does correctly predict the known qualitative properties; in particular, only two-particle bound states exist.

Journal of Mathematical Physics•Apr 1, 1978•29 citations

Ground state energy for the Hartree–Fock equations with Dirichlet boundary conditions

Joel F. Campbell

The Hartree–Fock equations for 2 N fermions in a one-dimensional box are solved in terms of a power series in the coupling constant for various values of N and the quantum numbers. A general formula is found for the total energy and wave function for N=1, which is accurate to sixth order. Another formula for the total energy is conjectured for any N, which is accurate to second order. Using these results, the ground state energy, energy of interaction, and boundary energy are found to second-order accuracy for arbitrary N.

Journal of Mathematical Physics•Apr 1, 1994•3 citations

Interacting Fermions in One Dimension. II. Attractive Potential

James McGuire

The exact energies and wavefunctions for a system of (N − 1) one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large volume, finite density limit, when the particles interact via an attractive delta function potential. It is found that the attractive potential gives rise to a bound state, but, in spite of the presence of this bound state, all of the physical properties which are calculated (ground-state energy, effective mass of a certain class of excitations, etc.) are analytic continuations in the coupling constant of the corresponding results in the repulsive case. In addition, it is possible to have eigenstates which do not have the bound state present. These excited states are also discussed and are found to exhibit a negative effective mass and to modify the particle density at very large distances from the different particle.

Journal of Mathematical Physics•Jan 1, 1966•160 citations

Ground-state energy and density response of a one-dimensional fermion system with repulsive δ -function interaction

Seido Nagano, K. S. Singwi

The ground-state energy of a one-dimensional fermion system interacting via a repulsive $\ensuremath{\delta}$-function potential is calculated rigorously within the "ladder" approximation as a function of the coupling constant and the results compared with those given by the exact solution of Yang. Comparison with other many-body approximate theories has also been made. The density-density response function has also been calculated in an approximation in which the particle-hole interaction in the polarization bubble is taken into account to all orders. Results of this dynamic theory are compared with those of other approximations.

Physical review. B, Condensed matter•Jun 1, 1983•30 citations

Many-Body Problem of Attractive Fermions with Arbitrary Spin in One Dimension

Minoru Takahashi

A set of coupled integral equations which determine the ground state energy of N fermions with arbitrary spin attracting via a delta-function potential are obtained. In the case of ½ spin our equations are equivalent to those of Gaudin.

Progress of Theoretical Physics•Oct 1, 1970•68 citations
We present an orthonormal set of single-particle orbital functions explicitly satisfying the Hartree-Fock (HF) equations for occupied states, and which give low-density few-particle clustering effects in a many-fermion system. The initial study given here is in one and three dimensions for attractive $\ensuremath{\delta}$ pair interactions. In one dimension the numerical results compare favorably with the exact (lowest) HF state at zero density. The results turn out to be superior to the classic Overhauser HF orbitals, as well as to some more recent ones.
The quantum-mechanical, fermion, attractive delta function gas in one dimension has many interesting, known physical properties, including some aspects of soliton behavior. Motivated by a desire to test the validity of the HF approximation to many-body systems, we have made a rigorous study of the HF theory of this gas. When the particle number is even we can show that although HF theory does not give energies accurately, it does correctly predict the known qualitative properties; in particular, only two-particle bound states exist.
The Hartree–Fock equations for 2 N fermions in a one-dimensional box are solved in terms of a power series in the coupling constant for various values of N and the quantum numbers. A general formula is found for the total energy and wave function for N=1, which is accurate to sixth order. Another formula for the total energy is conjectured for any N, which is accurate to second order. Using these results, the ground state energy, energy of interaction, and boundary energy are found to second-order accuracy for arbitrary N.
The exact energies and wavefunctions for a system of (N − 1) one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large volume, finite density limit, when the particles interact via an attractive delta function potential. It is found that the attractive potential gives rise to a bound state, but, in spite of the presence of this bound state, all of the physical properties which are calculated (ground-state energy, effective mass of a certain class of excitations, etc.) are analytic continuations in the coupling constant of the corresponding results in the repulsive case. In addition, it is possible to have eigenstates which do not have the bound state present. These excited states are also discussed and are found to exhibit a negative effective mass and to modify the particle density at very large distances from the different particle.
The ground-state energy of a one-dimensional fermion system interacting via a repulsive $\ensuremath{\delta}$-function potential is calculated rigorously within the "ladder" approximation as a function of the coupling constant and the results compared with those given by the exact solution of Yang. Comparison with other many-body approximate theories has also been made. The density-density response function has also been calculated in an approximation in which the particle-hole interaction in the polarization bubble is taken into account to all orders. Results of this dynamic theory are compared with those of other approximations.