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Progress of Theoretical Physics

Excitation Spectrum of a One-Dimensional Many-Boson System

Makoto Konagai Takahashi

Published Jan 1, 1976• 0 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.55.33

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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.
An energy spectrum is exactly examined for a one-dimensional many boson system where bosons are interacting to each other through a delta-functional repulsive potential. After a unitary transformation, the Hamiltonian is diagonalized to be the following compact form ∑ p ( p2 /2 m ) np + (π\hbar/2 mL ) ×∑ p , q | p - q | npnq + (1/6 m )(π\hbar/ L ) 2 [(∑ pnp ) 3 - ∑ pnp )] at the infinitely large limit of the coupling constant. This form is non-linear with respect to the numbers np of quasi-particles. The non-linear terms are Galilean-invariant and produce a phonon-like spectrum. In a case of a finite coupling contant g , the total energy is expanded to power series of (1/ g ).
The Bogoliubov-Zubarev method of collective coordinates is used to derive the energy spectrum of elementary excitations in a weakly interacting many-boson system through second-lowest order in a weak coupling expansion. The obtained result of the spectrum is shown to be equivalent to that evaluated by means of the variation-perturbation procedure based on the method of correlated basis functions in the uniform limit. Comparison of the excitation spectrum is also made briefly with two other results calculated using the Bogoliubov-Zubarev Hamiltonian in somewhat related but different approaches.
The validity of Sunakawa, Yamasaki and Kebukawa's Hamiltonian and that of Bogoliubov and Zubarev's Hamiltonian are examined. Perturbational expansion of the ground state energy by these Hamiltonians disagrees with the exact solution of Lieb and Liniger for one-dimensional Bose system with repulsive delta-function interaction. This fact suggests that these Hamiltonians are not microscopic descriptions of the many-Boson system. Mathematical inconsistency in Bogoliubov and Zubarev's theory is also pointed out. Moreover analytic expression of high density expansion for the ground state energy density e0 is found out to be e0n-3=γ-(4/3π)γ3/2+(1/6-1/π2)γ2+O(γ5/2), γ≡c/n, for one-dimensional Bose system with delta function interaction (density n, strength 2c, ħ=2m=1) by the use of the correlated basis function method.
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.
We study the integrable model of one-dimensional bosons with contact repulsion. In the limit of weak interaction, we use the microscopic hydrodynamic theory to obtain the excitation spectrum. The statistics of quasiparticles changes with the increase of momentum. At lowest momenta good quasiparticles are fermions, while at higher momenta they are Bogoliubov bosons, in accordance with recent studies. In the limit of strong interaction, we analyze the exact solution and find exact results for the spectrum in terms of the asymptotic series. Those results undoubtedly suggest that fermionic quasiparticle excitations actually exist at all momenta for moderate and strong interaction and also at lowest momenta for arbitrary interaction. Moreover, at strong interaction we find highly accurate analytical results for several relevant quantities of the Lieb-Liniger model.