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Journal of Physics G Nuclear Physics

Constrained Hartree approximation for systems with many interacting bosons

M. C. Cambiaggio, J. Dukelsky, Guillermo R. Zemba

Published Sep 1, 1985• 1 citation
Publisher

Abstract

An extension to bosons of a constrained Hartree variational method is developed in which an angular momentum constraint is imposed. The method is applied to a soluble bosonic model, the Q-consistent Hamiltonian of Warner and Casten (1982). The results are compared not only with the exact ones but also with those obtained from the cranked Hartree variational approximation.

DOI
10.1088/0305-4616/11/9/005

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An extension to bosons of a constrained Hartree variational method is developed in which an angular momentum constraint is imposed. The method is applied to a soluble bosonic model, the Q-consistent Hamiltonian of Warner and Casten (1982). The results are compared not only with the exact ones but also with those obtained from the cranked Hartree variational approximation.
A rotating bosonic many-body system in a harmonic trap is studied with the three-dimensional cranked Hartree-Fock-Bogoliubov method at zero temperature, which has been applied to nuclear many-body systems at high spin. This method is a variational method extended from Hartree-Fock theory, which can treat the pairing correlations in a self-consistent manner. An advantage of this method is that a finite-range interaction between constituent particles can be used in the calculation, unlike the original Gross-Pitaevskii approach. To demonstrate the validity of our method, we present a calculation for a toy model---that is, a rotating system of ten bosonic particles interacting through the repulsive quadrupole-quadrupole interaction in a harmonic trap. It is found that the yrast states, the lowest-energy states for the given total angular momentum, do not correspond to the Bose-Einstein condensate, except for a few special cases. One such case is a vortex state, which appears when the total angular momentum LLL is twice the particle number NNN (i.e., L=2NL=2NL=2N).
The third quantization (3rd Q) for bosons provides the exact steady-state solution of the Lindblad equation with quadratic Hamiltonians. By decomposing the interaction of the Bose-Hubbard model (BHM) according to Hartree approximation, we present a self-consistent formalism for solving the open-system bosonic Lindblad equation with weak interactions in the steady state. The 3rd Q with Hartree approximation takes into account the infinite Fock space of bosons, while its demand of resource scales polynomially with the system size. We examine the method by analyzing three examples of the BHM, including the uniform chain, interaction-induced diode effect, and Su-Schrieffer-Heeger Hubbard model. When compared with the simulations with capped boson numbers for small systems, the 3rd Q with Hartree approximation captures the qualitative behavior and suggests an upper bound of the steady-state value. Finite-size scaling confirms the results from the 3rd Q with Hartree approximation converging towards the thermodynamic limit. Thus, the manageable method allows us to characterize and predict large-system behavior of quantum transport in interacting bosonic systems relevant to cold-atom experiments.
The problem of N bosons interacting with each other via repulsive delta-function interactions and with an external, attractive, delta-function potential is solved, within the Hartree approximation, exactly. If the interparticle interactions are above a certain value, there is no bound state. Thus the bound state does not just expand to compensate for the increase in the repulsive Hartree potential. Moreover as the interaction strength is increased to that value, the ground-state wavefunction develops a pole at the position of the attractive potential.
We develop the multi-layer multi-configuration time-dependent Hartree method for bosons (ML-MCTDHB), a variational numerically exact ab initio method for studying the quantum dynamics and stationary properties of general bosonic systems. ML-MCTDHB takes advantage of the permutation symmetry of identical bosons, which allows for investigations of the quantum dynamics from few to many-body systems. Moreover, the multi-layer feature enables ML-MCTDHB to describe mixed bosonic systems consisting of arbitrary many species. Multi-dimensional as well as mixed-dimensional systems can be accurately and efficiently simulated via the multi-layer expansion scheme. We provide a detailed account of the underlying theory and the corresponding implementation. We also demonstrate the superior performance by applying the method to the tunneling dynamics of bosonic ensembles in a one-dimensional double well potential, where a single-species bosonic ensemble of various correlation strengths and a weakly interacting two-species bosonic ensemble are considered.