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arXiv:cond-mat/9809153 · Physical Review A

Ground-state energy of the spinor Bose-Einstein condensate

Wen-Jui Huang, Shih-Chuan Gou

Published Jun 1, 1999• 22 citations
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Abstract

It is known that for a weakly interacting Bose-Einstein condensate (BEC), the assumption of a two-body $\ensuremath{\delta}$ interaction described by a constant coupling strength gives rise to a divergent ground-state energy. A similar divergence occurs in the spinor condensate in which the spin-spin interaction is included in addition to the repulsive $\ensuremath{\delta}$ interaction. In this paper, we examine, in the standard Bogoliubov approximation, the ground-state energy of a homogeneous spinor BEC with hyperfine spin $f=1.$ The renormalized coupling constants are calculated and expressed in terms of the bare ones using the standard second-order perturbation method. With these renormalized coupling constants, we show that the ultraviolet divergence of the ground-state energy can be exactly eliminated.

DOI
10.1103/physreva.59.4608

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It is known that for a weakly interacting Bose-Einstein condensate (BEC), the assumption of a two-body \ensuremathδ\ensuremath{\delta}\ensuremathδ interaction described by a constant coupling strength gives rise to a divergent ground-state energy. A similar divergence occurs in the spinor condensate in which the spin-spin interaction is included in addition to the repulsive \ensuremathδ\ensuremath{\delta}\ensuremathδ interaction. In this paper, we examine, in the standard Bogoliubov approximation, the ground-state energy of a homogeneous spinor BEC with hyperfine spin f=1.f=1.f=1. The renormalized coupling constants are calculated and expressed in terms of the bare ones using the standard second-order perturbation method. With these renormalized coupling constants, we show that the ultraviolet divergence of the ground-state energy can be exactly eliminated.
The standard calculations of the ground-state energy of a homogeneous Bose gas rely on approximations which are physically reasonable but difficult to control. Lieb and Yngvason ( Phys. Rev. Lett. 80 (1998) 2504) have proved rigorously that the commonly accepted leading-order term of the ground-state energy is correct in the zero-density limit. Here, strong indications are given that also the next-to-leading term is correct. It is shown that the first terms obtained in a perturbative treatment provide contributions which are lost in the Bogoliubov approach.
This study investigates the polaron states formed by an impurity immersed in a spin-1 spinor Bose-Einstein condensate. Employing the Fröhlich Hamiltonian framework, we systematically calculated the ground-state energy and effective mass of polarons in both weak-coupling and strong-coupling regimes. For the weakly interacting regime, the analysis was conducted using second-order perturbation theory. In the strong-coupling regime, we constructed an analytical approximation for the polaron states and performed variational calculations to determine their energy and effective mass. Furthermore, we discussed the transition from the mobile state to the self-trapped state of the polaron in the strong-coupling region. We also estimate the signatures of polaron effects in spinor BEC for the future experiments.
The textbook calculation of the ground-state energy of a dilute gas of Bose particles is examined in detail, and certain mathematical inconsistencies are pointed out. On the basis of this analysis, a refined approach suitable for soft interaction potentials which lend themselves to a low-order Born approximation is developed. This procedure emphasizes the low-density character of the resulting formula for the ground-state energy, and avoids all divergent expressions at intermediate stages of the computation. It is stressed that the standard Bogoliubov approximation, if not augmented by some additional device, leads to an error which manifests itself already in the lowest order of the density. - PACS 03.75.Hh, 05.30.Jp, 03.65.Nk.
We generalize the perturbation theory of weakly interacting bosons to Bose-Einstein condensates with hyperfine spin F=1.F=1.F=1. Analytical expressions of second-order self-energies beyond the mean-field approximation are derived at zero temperature. We further extend the perturbation theory to the finite-temperature case, and derive the corresponding self-energies. We evaluate these self-energies near the poles of the first-order (mean-field) Green's functions within the on-shell approximation, and expand the results in powers of wave number in the long-wavelength limit. Excitation spectra for each mode are obtained analytically at low temperatures. We also derive the ground-state energy and thermodynamic quantities such as free energy of the system.
A method is proposed for a self-consistent evaluation of the coupling constant in the Gross-Pitaevskii equation without involving a pseudopotential replacement. A renormalization of the coupling constant occurs due to medium effects and the trapping potential, e.g., in quasi-1D or quasi-2D systems. It is shown that a simplified version of the Hartree-Fock-Bogoliubov approximation leads to a variational problem for both the condensate and a two-body wave function describing the behavior of a pair of bosons in the Bose-Einstein condensate. The resulting coupled equations are free of unphysical divergences. Particular cases of this scheme that admit analytical estimations are considered and compared to the literature. In addition to the well-known cases of low-dimensional trapping, crossover regimes can be studied. The values of the kinetic, interaction, external, and release energies in low dimensions are also evaluated and contributions due to short-range correlations are found to be substantial.