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American Journal of Physics

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

Steven A. Moszkowski, George L. Strobel

Published Nov 1, 1979• 2 citations
Publisher

Abstract

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.
DOI
10.1119/1.11616

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The ground state of a one-dimensional system of many bosons interacting through a repulsive $\ensuremath{\delta}$-function potential is studied when the coupling parameter $\ensuremath{\gamma}$ of the system is small. The procedure is based on the variational analysis in the Bijl-Dingle-Jastrow wave-function space. The ground-state energy, determined in a series form in powers of ${\ensuremath{\gamma}}^{\frac{1}{2}}$ and truncated after the third leading term, is found to be in fair agreement with exact results of Lieb and Liniger over a range of the coupling parameter extending up to $\ensuremath{\gamma}=6$. Two slightly different approaches are considered, and they are shown to be equivalent.
The properties of a one-dimensional system of bosons interacting via a repulsive delta-function potential are studied here using a modified Bogolyubov method. The motivation for the modification is twofold: First, to cure an internal inconsistency that predicts a single-particle ground-state occupancy of minus infinity when the Bogolyubov scheme is applied to this model. Second, to abandon the initial assumption---at odds with the exact result---of a macroscopically occupied single-particle ground-state. Whereas the Bogolyubov scheme uses the lowest-energy-single-particle state as a reservoir for the other states, our method uses, for each single-particle state, all single-particle states of lower energy collectively as a reservoir. This not only removes the inconsistency but also allows us to estimate the ground-state energy in much better agreement with the exact result.
The method of correlated basis functions is used in the study of the ground-state properties of a one-dimensional system of bosons interacting through a repulsive $\ensuremath{\delta}$-function potential. Ground-state energies are determined numerically for intermediate values of the coupling constant in three steps: (1) a trial form of the radial distribution function generated by the Bijl-Dingle-Jastrow (BDJ) type of wave function is introduced to obtain approximate ground-state energies, (2) the paired-phonon analysis is applied to improve the BDJ description of the ground state, and (3) leading corrections to the improved approximate energies are evaluted from the second-order perturbation energy generated by the three-phonon vertex. The obtained energy values are found to agree closely with the exact results calculated by Lieb and Liniger.
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 system of many bosons interacting via a repulsive s-function potential is carried out through third-order corrections, when the coupling parameter γ of the system is small. The procedures' are basically developed following both the Bogoliubov-Zubarev and the Davison-Feenberg methods. It is seen numerically that a slight difference exists in between, at least, starting from the third-order correction; the former one gives rise to a more improved agreement with the exact results obtained by Lieb and Liniger, and also, the variational contribution in the later is still dominant even in the third-order correction.