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Physical Review A

Comparison between the exact and Hartree solutions of a one-dimensional many-body problem

Francesco Calogero, A. Degasperis

Published Jan 1, 1975• 136 citations
Publisher

Abstract

The exact solution for the ground state of the quantum one-dimensional $N$-boson problem with attractive $\ensuremath{\delta}$-function two-body potentials is compared with the (exact) self-consistent solution of the corresponding variational Hartree problem.

DOI
10.1103/physreva.11.265

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Critiquing variational theories of the Anderson–Hubbard model: real-space self-consistent Hartree–Fock solutions

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arXiv:0808.3899•Journal of Physics Condensed Matter•Aug 1, 2008•3 citations
The exact solution for the ground state of the quantum one-dimensional NNN-boson problem with attractive \ensuremathδ\ensuremath{\delta}\ensuremathδ-function two-body potentials is compared with the (exact) self-consistent solution of the corresponding variational Hartree problem.
There exists a widely believed opinion, that the many-body ground state of a two-dimensional electron system at a completely filled lowest Landau level (the filling factor ν=1ν=1ν=1) is described by the so-called Hartree-Fock wave function, and that this solution is the unique, exact eigenstate of the system at ν=1ν=1ν=1. I show that this opinion is erroneous, construct an infinite number of other variational many-body wave functions, and discuss the properties of a few states which have the energy substantially lower than the energy of the Hartree-Fock state.
We consider two bosons in a one-dimensional harmonic trap, interacting by a contact potential, and compare the exact solution of this problem to a self-consistent numerical solution by using the multiconfigurational time-dependent Hartree (MCTDH) method. We thereby benchmark the predictions of the MCTDH method, and the statement that it is numerically exact, with a few-body problem that has an analytical solution for the most commonly experimentally realized interaction potential in ultracold quantum gases. It is found that exact ground state energy and first order correlations are accurately reproduced by MCTDH up to the intermediate dimensionless coupling strengths corresponding to typical background scattering lengths of magnetically trapped ultracold dilute Bose gases. For larger couplings, established for example by (a combination of) Feshbach resonances and optical trapping, the MCTDH approach overestimates the depth of the trap-induced correlation dip of first order correlations in position space, and underestimates the fragmentation, defined as the average relative occupation of orbitals other than the energetically lowest one. We anticipate that qualitatively similar features in the correlation function may arise for larger particle numbers, paving the way for a quantitative assessment of the accuracy of MCTDH by experiments with ultracold atoms.
A simple and commonly employed approximate technique with which one can examine spatially disordered systems when strong electronic correlations are present is based on the use of real-space unrestricted self-consistent Hartree-Fock wave functions. In such an approach the disorder is treated exactly while the correlations are treated approximately. In this report we critique the success of this approximation by making comparisons between such solutions and the exact wave functions for the Anderson-Hubbard model. Due to the sizes of the complete Hilbert spaces for these problems, the comparisons are restricted to small one-dimensional chains, up to ten sites, and a 4x4 two-dimensional cluster, and at 1/2 filling these Hilbert spaces contain about 63,500 and 166 million states, respectively. We have completed these calculations both at and away from 1/2 filling. This approximation is based on a variational approach which minimizes the Hartree-Fock energy, and we have completed comparisons of the exact and Hartree-Fock energies. However, in order to assess the success of this approximation in reproducing ground-state correlations we have completed comparisons of the local charge and spin correlations, including the calculation of the overlap of the Hartree-Fock wave functions with those of the exact solutions. We find that this approximation reproduces the local charge densities to quite a high accuracy, but that the local spin correlations, as represented by < S_i . S_j >, are not as well represented. In addition to these comparisons, we discuss the properties of the spin degrees of freedom in the HF approximation, and where in the disorder-interaction phase diagram such physics may be important.