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Progress of Theoretical Physics

Energy Spectrum of One-Dimensional Many Boson System

Shosuke Sasaki

Published Jun 1, 1989• 4 citations
Publisher

Abstract

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 ).

DOI
10.1143/ptp.81.1158

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Progress of Theoretical Physics•Apr 1, 1981•6 citations
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 analysis for a one-dimensional many boson system with a repulsive delta-function potential is continued. It is shown that the expectation value of the number a0*a0 of bare bosons with zero momentum for the exact ground state ∣Ψ0,0,…,0> is negligibly small compared with the total number n in the limit of an infinitely large coupling constant g; i.e., ≪Ψ0,…,0∣a0*a0∣Ψ0,…,0>/n→0(for n→∞, g→∞). This result shows a striking contrast to the result ≪Ψ0,…,0∣A0*A0∣Ψ0,…,0>/n=1 for an arbitrary g, where A0*A0 indicates the number operator of exactly dressed bosons with zero momentum. It is clarified that the energy of a dressed boson strongly depends upon the number distribution {nq} of the other dressed bosons in the system. The excitation energy of a dressed boson has phonon character when n0/n ≠0, where n0 denotes the number of dressed bosons with zero momentum. When n0/n tends to zero, the phonon character disappears in a drastic way. Some characteristic phenomena in liquid helium II are discussed on the basis of the results of our previous papers and the present paper.
A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.
A unitary operator, which transforms free states into the exact eigenstates for a one-dimensional many boson system with repulsive delta-function potential, is explicitly constructed. By the unitary operator, original bare creation and annihilation operators are transformed into new operators which are shown to create or annihilate the exactly dressed bosons with the interaction cloud. The total Hamiltonian, total number and total momentum in the system are expressed in the diagonalized form in terms of the dressed operators. It is shown that the exact ground state is the condensed state of all exactly dressed bosons with zero momentum.
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 ground state of a one-dimensional system of many bosons interacting through a repulsive $\ensuremath{\delta}$-function potential is studied when the coupling parameter $\ensuremath{\gamma}$ of the system is small. The procedure is based on the variational analysis in the Bijl-Dingle-Jastrow wave-function space. The ground-state energy, determined in a series form in powers of ${\ensuremath{\gamma}}^{\frac{1}{2}}$ and truncated after the third leading term, is found to be in fair agreement with exact results of Lieb and Liniger over a range of the coupling parameter extending up to $\ensuremath{\gamma}=6$. Two slightly different approaches are considered, and they are shown to be equivalent.