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arXiv:1506.05189

Planar limit of 1D many-body system

Fen Zuo, Yi-hong Gao

Published Jun 17, 2015• 0 citations
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Abstract

We review one dimensional matrix theory and its variations, collective field theory and quantum phase space description. In the planar limit, these theories become classical and can be easily analyzed. With these descriptions, one dimensional interacting many-body system can be solved exactly when the particle number goes to infinity. As an example, bosonic and two-component fermionic systems with a $δ$-function interaction are analyzed in detail.

DOI
10.48550/arxiv.1506.05189

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In this article a survey is given of the present status of the quantum theory of many-particle systems with special emphasis on the underlying principles. Very little is said about applications. A brief introduction to some models, which are of interest for an approximate or qualitative description of actual physical systems, is followed by a discussion of two important phenomena in many-particle systems: quasi-particles and collective excitations. Next, an outline is given of mathematical methods which have been used for a precise treatment of such phenomena. A rather thorough discussion of the Bogoliubov model for interacting bosons, and the BCS model for fermions, serves as an illustration. Some new features, typical of many-particle systems, are discussed on the basis of these models. Also another approach to the many-body problem is discussed which is more logical and mathematically more rigorous. This approach differs from standard procedures in that the limit Ω-> infinity, N -> infinity is not taken at the end of the calculations but in the beginning. This quantum theory of infinite systems leads, among other things, to a better understanding of the models mentioned above.
In this work we develop a complete variational many-body theory for a system of NNN trapped bosons interacting via a general two-body potential. The many-body solution of this system is expanded over orthogonal many-body basis functions (configurations). In this theory both the many-body basis functions and the respective expansion coefficients are treated as variational parameters. The optimal variational parameters are obtained self-consistently by solving a coupled system of noneigenvalue---generally integro-differential---equations to get the one-particle functions and by diagonalizing the secular matrix problem to find the expansion coefficients. We call this theory multiconfigurational Hartree theory for bosons or MCHB(M)\mathrm{MCHB}(M)MCHB(M), where MMM specifies explicitly the number of one-particle functions used to construct the configurations. General rules for evaluating the matrix elements of one- and two-particle operators are derived and applied to construct the secular Hamiltonian matrix. We discuss properties of the derived equations. We show that in the limiting cases of one configuration the theory boils down to the well-known Gross-Pitaevskii and the recently developed multi-orbital mean fields. The invariance of the complete solution with respect to unitary transformations of the one-particle functions is utilized to find the solution with the minimal number of contributing configurations. In the second part of our work we implement and apply the developed theory. It is demonstrated that for any practical computation where the configurational space is restricted, the description of trapped bosonic systems strongly depends on the choice of the many-body basis set used, i.e., self-consistency is of great relevance. As illustrative examples we consider bosonic systems trapped in one- and two-dimensional symmetric and asymmetric double well potentials. We demonstrate that self-consistency has great impact on the predicted physical properties of the ground and excited states and show that the lack of self-consistency may lead to physically wrong predictions. The convergence of the general MCHB(M)\mathrm{MCHB}(M)MCHB(M) scheme with a growing number MMM is validated in a specific case of two bosons trapped in a symmetric double well.