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arXiv:1202.5355 · Physical Review A

Improved-Gaussian-approximation calculation of the Bogoliubov mode in a one-dimensional bosonic gas

Qiong Li, Daoguang Tu, Dingping Li

Published Mar 6, 2012• 3 citations
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Abstract

In this paper, we study the homogeneous one-dimensional bosonic gas interacting via a repulsive contact potential by using the improved Gaussian approximation. We obtain the gapless excitation spectrum of the Bogoliubov mode. Our result is in good agreement with the exact numerical calculation based on the Bethe ansatz. We speculate that the improved Gaussian approximation could be a quantitatively good approximation for higher-dimensional systems.

DOI
10.1103/physreva.85.033609

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In this paper, we study the homogeneous one-dimensional bosonic gas interacting via a repulsive contact potential by using the improved Gaussian approximation. We obtain the gapless excitation spectrum of the Bogoliubov mode. Our result is in good agreement with the exact numerical calculation based on the Bethe ansatz. We speculate that the improved Gaussian approximation could be a quantitatively good approximation for higher-dimensional systems.
We investigate the Dirac time-dependent variational method using a Gaussian trial functional for an infinite one-dimensional system of bosons interacting through a repulsive contact interaction. The method produces a set of nonlinear time-dependent equations for the variational parameters. By solving the static equations we have calculated the ground state energy per particle. We have also considered small oscillations about the equilibrium and obtain mode equations which lead us to a gapless dispersion relation. The existence of an exact numerical solution for the ground-state energy and excitations obtained by Lieb allow us to compare with the Gaussian results. We can also, as the system becomes less dilute, see the improvement of the results as compared with the Bogoliubov scheme.
We investigate ground-state and excitation spectrum of a system of non-relativistic bosons in one-dimension interacting through repulsive, two-body contact interactions in a self-consistent Gaussian mean-field approximation. The method consists in writing the variationally determined density operator as the most general Gaussian functional of the quantized field operators. There are mainly two advantages in working with one-dimension. First, the existence of an exact solution for the ground-state and excitation energies. Second, neither in the perturbative results nor in the Gaussian approximation itself we do not have to deal with the three-dimensional patologies of the contact interaction . So that this scheme provides a clear comparison between these three different results.
We study bosons in a one-dimensional hard-wall box potential. In the case of contact interaction, the system is exactly solvable by the Bethe ansatz, as first shown by Gaudin in 1971. Although contained in the exact solution, the boundary energy in the thermodynamic limit for this problem is only approximately calculated by Gaudin, who found the leading order result at weak repulsion. Here we derive an exact integral equation that enables one to calculate the boundary energy in the thermodynamic limit at an arbitrary interaction. We then solve such an equation and find the asymptotic results for the boundary energy at weak and strong interactions. The analytical results obtained from the Bethe ansatz are in agreement with the ones found by other complementary methods, including quantum Monte Carlo simulations. We study the universality of the boundary energy in the regime of a small gas parameter by making a comparison with the exact solution for the hard rod gas.
The lowest order correction term to the Bogoliubov spectrum is calculated analytically for the case of one-dimensional bosons with δ-function interaction. We obtain the spectrum ε(p) = |p|√p2+4nc+(2nc/π)γ1/2·|p|/√p2+4nc+O(γnc), (n = density, ℏ= 2m = 1, strength of δ-function = 2c, γ= c/n). This is compared with Lieb's exact excitation spectrum for this case. Our numerical calculation shows that it is the correct γ1/2 expansion of the spectrum. However this expansion up to second order is not a good approximation for the low wave number part and gives an incorrect expansion of the sound velocity up to second order.
The one-dimensional gas of bosons interacting via a repulsive contact potential was solved long ago via Bethe's ansatz by Lieb and Liniger [Phys. Rev. {\bf 130}, 1605 (1963)]. The low energy excitation spectrum is a Luttinger liquid parametrized by a conformal field theory with conformal charge c=1c=1c=1. For higher energy excitations the spectral function displays deviations from the Luttinger behavior arising from the curvature terms in the dispersion. Adding a corrective term of the form of a mobile impurity coupled to the Luttinger liquid modes corrects this problem. The ``impurity'' term is an irrelevant operator, which if treated non-perturbatively, yields the threshold singularities in the one-particle and one-hole hole Green's function correctly. We show that the exponents obtained via the finite size corrections to the ground state energy are identical to those obtained through the shift function.
We study the correlation effects in a one-dimensional Bose gas with repulsive delta-function interaction. The correlation effects are described by a local-field correction which takes into account the short-range correlations. We find that the ground state energy is in good agreement with the exact result up to intermediate coupling strengths, showing an improvement over the Bogoliubov and other related approximations. The velocity of sound, the static structure factor, and pair-correlation function are also calculated within the present approximation.