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Physical Review

Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State

Élliott H. Lieb, Werner Liniger

Published May 15, 1963• 2,707 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.130.1605

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Physical review. B, Condensed matter•Feb 1, 1978•1 citation
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, \ensuremathγ\ensuremath{\gamma}\ensuremathγ. When \ensuremathγ\ensuremath{\gamma}\ensuremathγ 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 \ensuremathγ\ensuremath{\gamma}\ensuremathγ and show that it is analytic for all \ensuremathγ\ensuremath{\gamma}\ensuremathγ, except \ensuremathγ=0\ensuremath{\gamma}=0\ensuremathγ=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.
We continue the analysis of the one-dimensional gas of Bose particles interacting via a repulsive delta function potential by considering the excitation spectrum. Among other things we show that: (i) the elementary excitations are most naturally thought of as a double spectrum, not a single one; (ii) the velocity of sound derived from the macroscopic compressibility is shown to agree with the velocity of sound derived from microscopic considerations, i.e., from the phonon spectrum. We also introduce a distinction between elementary excitations and quasiparticles, on the basis of which we give some heuristic reasons for expecting the double spectrum to be a general feature, even in three dimensions, and not an exception.
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.
We study the ground-state properties and excitation spectrum of the Lieb-Liniger model, i.e. the one-dimensional Bose gas with repulsive contact interactions. We solve the Bethe-Ansatz equations in the thermodynamic limit by using an analytic method based on a series expansion on orthogonal polynomials developed in and push the expansion to an unprecedented order. By a careful analysis of the mathematical structure of the series expansion, we make a conjecture for the analytic exact result at zero temperature and show that the partially resummed expressions thereby obtained compete with accurate numerical calculations. This allows us to evaluate the density of quasi-momenta, the ground-state energy, the local two-body correlation function and Tan’s contact. Then, we study the two branches of the excitation spectrum. Using a general analysis of their properties and symmetries, we obtain novel analytical expressions at arbitrary interaction strength which are found to be extremely accurate in a wide range of intermediate to strong interactions.
The Gaudin integral equation for the ground state of a one-dimensional delta-function attractive spin-1/2 fermions is solved in the form of power series. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities such as the ground state energy are expressed in terms of a single dimensionless parameter gamma =c/D, where c is the coupling constant and D is the number density. The results agree with those obtained from the perturbation calculations, which include the one in the classical electrostatics originally by Kirchhoff. In the strong coupling limit, the connection to the solutions of the Lieb-Liniger integral equation for the ground state of a one-dimensional delta-function repulsive bose gas is shown explicitly.
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.
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.