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Physical Review A

Interacting one-dimensional Bose gas: Beyond Bogolyubov

Prabasaj Paul, Bill Sutherland

Published Oct 10, 2000• 2 citations
Publisher

Abstract

The properties of a one-dimensional system of bosons interacting via a repulsive delta-function potential are studied here using a modified Bogolyubov method. The motivation for the modification is twofold: First, to cure an internal inconsistency that predicts a single-particle ground-state occupancy of minus infinity when the Bogolyubov scheme is applied to this model. Second, to abandon the initial assumption---at odds with the exact result---of a macroscopically occupied single-particle ground-state. Whereas the Bogolyubov scheme uses the lowest-energy-single-particle state as a reservoir for the other states, our method uses, for each single-particle state, all single-particle states of lower energy collectively as a reservoir. This not only removes the inconsistency but also allows us to estimate the ground-state energy in much better agreement with the exact result.

DOI
10.1103/physreva.62.055601

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The properties of a one-dimensional system of bosons interacting via a repulsive delta-function potential are studied here using a modified Bogolyubov method. The motivation for the modification is twofold: First, to cure an internal inconsistency that predicts a single-particle ground-state occupancy of minus infinity when the Bogolyubov scheme is applied to this model. Second, to abandon the initial assumption---at odds with the exact result---of a macroscopically occupied single-particle ground-state. Whereas the Bogolyubov scheme uses the lowest-energy-single-particle state as a reservoir for the other states, our method uses, for each single-particle state, all single-particle states of lower energy collectively as a reservoir. This not only removes the inconsistency but also allows us to estimate the ground-state energy in much better agreement with the exact result.
The method of self-consistent-field approximation is applied for the study of the ground-state properties of a one-dimensional system of bosons, of density $\ensuremath{\rho}$, first interacting through finite rigid rods, of length lll, and again via a repulsive $\ensuremath{\delta}$-function potential. We obtain the energy spectrum and the sound velocity. The results are close to the exact one for $\ensuremath{\rho}l<0.5$. The energy spectrum obtained shows a rotonlike region which is qualitatively similar to the two- and three-dimensional versions. For a Bose gas interacting through a repulsive $\ensuremath{\epsilon}$ potential of strength 2c2c2c, we derive the energy spectrum, the sound velocity, and the ground-state energy. The results are compared with Girardeau and Lieb's exact expressions. In the small-coupling limit our calculations show the same results as obtained by Bogoliubov's perturbation theory. The ground-state energy obtained for small $\ensuremath{\gamma}=\frac{c}{\ensuremath{\rho}}$ is found to agree closely with several different works found in the literature. Unfortunately our theory diverges for the ground-state energy in the limit $\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\infty}$. The sound velocity in this limit is finite but twice as small as from Girardeau's exact results.
A gas of one-dimensional Bose particles interacting via a repulsive delta-function potential has been solved exactly. All the eigenfunctions can be found explicitly and the energies are given by the solutions of a transcendental equation. The problem has one nontrivial coupling constant, $\ensuremath{\gamma}$. When $\ensuremath{\gamma}$ is small, Bogoliubov's perturbation theory is seen to be valid. In this paper, we explicitly calculate the ground-state energy as a function of $\ensuremath{\gamma}$ and show that it is analytic for all $\ensuremath{\gamma}$, except $\ensuremath{\gamma}=0$. In Part II, we discuss the excitation spectrum and show that it is most convenient to regard it as a double spectrum---not one as is ordinarily supposed.
A simple model of a boson system is considered for which the exact Schr\"odinger equation reduces to a difference equation which can be solved numerically. Application of the Bogoliubov method and comparison of its results with the numerical results then yield some insight into the validity of the Bogoliubov approximations. The simple model considered has, in the absence of interactions, only a zero energy state and two states of unit energy available to each boson. Initially there is assumed to be no interaction between bosons in the degenerate excited states and all existing interactions are assumed to be repulsive. The ground-state energy is calculated in the Bogoliubov approximation, with and without the depletion effect considered, and in another numerical approximation. These results and the results of the numerical solution of the exact Schr\"odinger equation converge to the Bogoliubov result with depletion ignored when the number of particles in the system approaches infinity. It is surprising to note that, for the interaction strengths considered, the Bogoliubov result is within 3% of the numerical result for as few as 32 particles in the system, and within 10% for only 4 particles. A modification of the above system is considered in which there exists an additional two-body interaction between particles in the degenerate excited states which may be attractive or repulsive. It is shown that the ground-state energy, with this additional interaction present, deviates from the Bogoliubov value linearly with the strength of the added interaction, i.e., $W={W}_{\mathrm{Bog}}+\ensuremath{\alpha}F$ (with FFF the strength parameter of the interaction). The factor $\ensuremath{\alpha}$ is found to be small and constant over a wide range of FFF and in this range the deviation of WWW from the Bogoliubov value is small. There is, however, a transition point FT{F}_{T}FT​, beyond which $\ensuremath{\alpha}$ is constant and large and the usual Bogoliubov approximation is invalid for interactions more attractive than that characterized by FT{F}_{T}FT​. A Bogoliubov-like approximation is shown to be quite accurate in this region. It is then shown that $F_{T}^{\ensuremath{-}1}$ is a linear function of the number of particles in the system.
We continue the study of the integrodifferential equation proposed previously for the evaluation of the ground-state energy of an imperfect Bose gas. We apply it here to the one-dimensional delta-function gas where the exact result is known for all values of the coupling constant $\ensuremath{\gamma}$. The results are: (i) For small $\ensuremath{\gamma}$, the equation gives the correct first two terms in an asymptotic series; (ii) a numerical solution of the equation shows that the maximum relative error occurs for $\ensuremath{\gamma}=\ensuremath{\infty}$ in which case it is 19%; (iii) for $\ensuremath{\gamma}=\ensuremath{\infty}$ we are able to compare the exact two-particle distribution function with that given by the equation. The agreement is quite good.
This is a book about Lieb's Simplified approach to the Bose gas, which is a family of effective single-particle equations to study the ground state of many-body systems of interacting Bosons. It was introduced by Lieb in 1963, and recently found to have some rather intriguing properties. One of the equations of the approach, called the Simple equation, has been proved to make a prediction for the ground state energy that is asymptotically accurate both in the low- and the high-density regimes. Its predictions for the condensate fraction, two-point correlation function, and momentum distribution also agree with those of Bogolyubov theory at low density, despite the fact that it is based on ideas that are very different from those of Bogolyubov theory. In addition, another equation of the approach called the Big equation has been found to yield numerically accurate results for these observables over the entire range of densities for certain interaction potentials. This book is an introduction to Lieb's Simplified approach, and little background knowledge is assumed. We begin with a discussion of Bose gases and quantum statistical mechanics, and the notion of Bose-Einstein condensation, which is one of the main motivations for the approach. We then move on to an abridged bibliographical overview on known theorems and conjectures about Bose gases in the thermodynamic limit. Next, we introduce Lieb's Simplified approach, and its derivation from the many-body problem. We then give an overview of results, both analytical and numerical, on the predictions of the approach. We then conclude with a list of open problems.