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Progress of Theoretical Physics

On the Method of “Auxiliary Variables”

Kazuo Hiroike

Published Feb 1, 1959• 8 citations
Publisher

Abstract

The method of “auxiliary variables” (collective coordinates) used by Bogolyubov and Zubarev is formulated in a somewhat different manner. Improvements are made in two points: 1) The Hamiltonian is given in an Hermitian form and 2) the number of the auxiliary variables can be limited, if necessary. We calculate the ground state energy and the excitation energy spectrum of the spinless bosons taking account of phonon-phonon interactions. The expression for the ground state energy is shown to be equivalent to the one derived by Bogolyubov and Zubarev.

DOI
10.1143/ptp.21.327

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Physical Review•Nov 1, 1959•806 citations
The method of “auxiliary variables” (collective coordinates) used by Bogolyubov and Zubarev is formulated in a somewhat different manner. Improvements are made in two points: 1) The Hamiltonian is given in an Hermitian form and 2) the number of the auxiliary variables can be limited, if necessary. We calculate the ground state energy and the excitation energy spectrum of the spinless bosons taking account of phonon-phonon interactions. The expression for the ground state energy is shown to be equivalent to the one derived by Bogolyubov and Zubarev.
By introducing a number of auxiliary variables, quantum-mechanical collective and internal motions in a Bose system are described. In a collective representation obtained here, the wave functions are expressed as functions of the auxiliary variables and the internal coordinates. The energy for the ground state and the excitation energies of collective motion and internal motion are obtained for the case of a long-range interaction potential. The lowest energy obtained proves to be smaller than that afforded by the Hartree-Fock approximation. The excitations of the collective motion with smaller wave numbers than a cut-off wave number kc correspond to Landau's phonons. The excitations of the internal motion resemble Landau's rotons; but they are forbidden for the smaller wave numbers K ≦ kc. When one applies this method to the problem of liquid helium, it becomes necessary to take account of the two-body correlation of individual particles. It is pointed out that the phonon energy depends upon the zero-point motion of the individual particles.
The ground-state energy of an interacting many-boson system obtained by Bogoliubov and Zubarev using the collective-coordinate method is shown to be equivalent through second order to the variation-perturbation energy evaluated in the uniform limit by means of the method of correlated basis functions. It is also shown that the wave function derived with a slight modification of the Bogoliubov-Zubarev approach is equivalent to that determined formally from the Rayleigh-Schr\"odinger perturbation theory in the uniform limit.
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 wave functions of the ground (Ψ 0 ) and the first excited (Ψ k ) states of He II in the second-order approximation, i.e., up to the first two corrections to the corresponding solutions for a weakly nonideal Bose gas, are determined by the collective variable method, which was proposed by Bogolyubov and Zubarev and developed in the studies by Yukhnovskii and Vakarchuk. The functions Ψ 0 and Ψ k = ψ k Ψ 0 are determined as the eigenfunctions of the N -particle Schrödinger equation from a system of coupled equations for Ψ 0 , Ψ k , and the quasiparticle spectrum E ( k ) of helium II. The results consist in the following: (1) these equations are solved numerically for a higher order approximation compared with those investigated earlier (the first-order approximation), and (2) Ψ 0 and ψ k are derived from a model potential of interaction between He 4 atoms (rather than from the structure factor as earlier) in which the potential barrier is joined with the attractive potential found from experiment. The height V 0 of the potential barrier is a free parameter. Except for V 0 , the model does not have any free parameters or functions. The calculated values of the structure factor, the ground-state energy E 0 , and the quasiparticle spectrum E ( k ) of He II are in agreement with the experimental values for V 0 ≈ 100 K. The second-order correction to the logarithm of Ψ 0 significantly affects the value of E 0 and provides the asymptotics E ( k → 0) = ck , while the second-order correction to ψ k slightly affects the E ( k ). The second-order corrections to Ψ 0 and ψ k have a smaller effect on the results compared with the first-order corrections, whereby the theory is in agreement with experiment; therefore, one may assume that the truncated Ψ 0 and ψ k well describe the microstructure of He II. Thus, the series for Ψ 0 and Ψ k can be truncated in spite of the fact that the expansion parameter is not very small (∼1/2).
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.
In this paper properties of a boson gas at zero temperature are investigated by means of field-theoretic methods. Difficulties arising from the depletion of the ground state are resolved in a simple way by the elimination of the zero-momentum state. The result of this procedure when applied to the calculation of the Green's functions of the system is identical to that of Beliaev. It is then shown generally that for a repulsive interaction the energy E(k)E(\mathrm{k})E(k) of a phonon of momentum k, which is found as the pole of a one-particle Green's function, approaches zero for zero momentum, which means that the phonon spectrum does not exhibit an energy gap.The Green's function method is applied to the calculation of the properties of a low-density boson gas. The next order term beyond that calculated by Lee and Yang, and Beliaev for the ground-state energy is obtained and the general form of the series expansion is found to be $(\frac{{E}_{0}}{\ensuremath{\Omega}})=\frac{1}{2}{n}^{2}{f}_{0}[1+a{(n{{f}_{0}}^{3})}^{\frac{1}{2}}+b(n{{f}_{0}}^{3})\mathrm{ln}n{{f}_{0}}^{3}+c(n{{f}_{0}}^{3})+d{(n{{f}_{0}}^{3})}^{\frac{3}{2}}\mathrm{ln}(n{{f}_{0}}^{3})+\ensuremath{\cdots}],$ where nnn is the density and f0{f}_{0}f0​ is the scattering length for the assumed two-body interaction between the bosons. The coefficients aaa and bbb are independent of the shape of the interaction, and are the only terms thus far calculated. The coefficient bbb is in agreement with the hard-sphere gas calculations of Wu and of Sawada.A discussion is given of the intermediate-density calculation of Brueckner and Sawada, and certain possible improvements in the method of summing a selected set of higher-order terms are proposed.