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Physical Review Letters

Boson Correlation Energies from Reduced Hamiltonian Interpolation

David A. Mazziotti, Dudley R. Herschbach

Published Dec 20, 1999• 19 citations
Publisher

Abstract

The ground-state energies of interacting bosons are computed beyond the mean-field approximation by a new method which we call reduced Hamiltonian interpolation (RHI). In this interpolation the $N$-particle Hamiltonian is represented through a sequence of $p$-particle expanded and reduced Hamiltonians that give upper and lower bounds on the true energy. A synthesis of ideas from $N$-representability and dimensional interpolation, the RHI interpolates over the number $p$ of quasiparticles (equivalent to spatial dimension) to calculate the $N$-particle energy as the mean of close upper and lower bounds. Application to bosons with harmonic interactions yields more than $99%$ of the correlation energy.

DOI
10.1103/physrevlett.83.5185

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The ground-state energies of interacting bosons are computed beyond the mean-field approximation by a new method which we call reduced Hamiltonian interpolation (RHI). In this interpolation the NNN-particle Hamiltonian is represented through a sequence of ppp-particle expanded and reduced Hamiltonians that give upper and lower bounds on the true energy. A synthesis of ideas from NNN-representability and dimensional interpolation, the RHI interpolates over the number ppp of quasiparticles (equivalent to spatial dimension) to calculate the NNN-particle energy as the mean of close upper and lower bounds. Application to bosons with harmonic interactions yields more than 9999%99 of the correlation energy.
The ground-state energies of interacting bosons are computed beyond the mean-field approximation by a new method known as reduced Hamiltonian interpolation (RHI) [D. A. Mazziotti and D. R. Herschbach, Phys. Rev. Lett. 83, 5185 (1999)]. In the RHI, the N-particle Hamiltonian is represented through a sequence of p-particle expanded and reduced Hamiltonians that give upper and lower bounds on the true energy. Assimilating ideas from N representability and dimensional interpolation, the RHI technique interpolates over the number p of quasiparticles to determine the N-particle energy with close upper and lower bounds. With the Hellmann-Feynman theorem, we extend the RHI to compute the two-particle reduced density matrix (2RDM) as well as the energy. We examine both the computational advantages of RHI in comparison with traditional methods and the possibility of extending the RHI to treat fermion correlation. Applied to bosons with harmonic interactions as well as a two-level system, the RHI technique yields more than 99% of the correlation energy and an accurate correlation correction for the elements of the 2RDM.
We present a simple renormalization-group method of investigating ground-state properties of interacting bosonic systems. Our method reduces the number of particles in a system, which makes numerical calculations possible for large systems. It is conceptually simple and easy to implement, and allows investigation of properties unavailable through mean-field approximations, such as one- and two-particle reduced density matrices of the ground state. As an example, we model a weakly interacting one-dimensional Bose gas in a harmonic trap. Compared to the mean-field Gross-Pitaevskii approximation, our method provides a more accurate description of the ground-state one-particle density matrix. We have also obtained the Hall-Post lower bounds for the ground-state energy of the gas. All results have been obtained by straightforward numerical diagonalization of the Hamiltonian matrix.
A many-body theory for interacting bosons is developed within the framework of minimizing the ground-state energy with respect to the two-particle reduced-density matrix (2-RDM) subject to NNN-representability conditions. The NNN-representability conditions, which ensure that the 2-RDM may be derived from an NNN-particle wave function, are imposed through a hierarchy of positivity conditions where the ppp-positivity conditions restrict the metric matrices for p∕2p∕2p∕2-body operators to be positive semidefinite. Using two-positivity, we minimize the ground-state energies of 5−−100005--10 0005−−10000 harmonically interacting bosons in a harmonic external potential. The energies and 2-RDMs obtained are in agreement with the exact solution except for round-off errors, which implies that for this class of boson interactions two-positivity conditions alone yield exact results for any interaction strength. The ground-state energies obtained at strong interactions are more accurate than many-body perturbative techniques by many orders of magnitude.
The well known and analytically solvable Lipkin–Meshkov–Glick model was devised to test several many-body approximation approaches used in nuclear physics. Here, we use this model to illustrate a method that sets lower bounds for the ground state energy of a quantum Hamiltonian. The idea of the method consists of dividing the Hamiltonian into identical groups of terms that represent subdivisions in the particle population of the system, and then to make an appropriate use of the ground energy of one of these groups. The bounds clearly improve as the size of these groups increases.
The variational determination of the two-boson reduced density matrix is described for a one-dimensional system of N N harmonically trapped bosons interacting via contact interaction, where N N ranges from 2 2 to 10^4 10 4 . The ground-state energies are calculated, and compared to existing methods, including the analytic case (for N=2) N = 2 ) and mean-field approaches such as the one-dimensional Gross-Pitaevskii equation and its variations. Structural properties such as the density and correlation functions are also derived, including the behaviour of the correlation function when boson coordinates coincide. This collectively demonstrates the capacity of the reduced density matrix method to accurately calculate collective ground-state properties of bosonic systems comprising few to many bosons, including the cross-over region between these extremes, across a large range of interaction strengths.
We study a system of AAA identical interacting bosons trapped by an external field by solving ab initio the many-body Schr\"odinger equation. A complete solution by using, for example, the traditional hyperspherical harmonics (HH) basis develops serious practical problems due to the large degeneracy of HH basis. Symmetrization of the wave function, calculation of the matrix elements, etc., become an immensely formidable task as AAA increases. Instead of the HH basis, here we use a new basis, called ``potential harmonics'' (PH) basis, which is a subset of HH basis. We assume that the contribution to the orbital and grand orbital [in $3(A\ensuremath{-}1)$-dimensional space of the reduced motion] quantum numbers comes only from the interacting pair. This implies inclusion of two-body correlations only and disregard of all higher-body correlations. Such an assumption is ideally suited for the Bose-Einstein condensate (BEC), which is required, for experimental realization of BEC, to be extremely dilute. Hence three and higher-body collisions are almost totally absent. Unlike the $(3A\ensuremath{-}4)$ hyperspherical variables in HH basis, the PH basis involves only three active variables, corresponding to three quantum numbers---the orbital lll, azimuthal mmm, and the grand orbital 2K+l2K+l2K+l quantum numbers for any arbitrary AAA. It drastically reduces the number of coupled equations and calculation of the potential matrix becomes tremendously simplified, as it involves integrals over only three variables for any AAA. One can easily incorporate realistic atom-atom interactions in a straightforward manner. We study the ground and excited state properties of the condensate for both attractive and repulsive interactions for various particle number. The ground state properties are compared with those calculated from the Gross-Pitaevskii equation. We notice that our many-body results converge towards the mean field results as the particle number increases.