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Physical Review

Ground-State Energy and Excitation Spectrum of a System of Interacting Bosons

Nicolaas Marinus Hugenholtz, David Pines

Published Nov 1, 1959• 806 citations
Publisher

Abstract

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 $E(\mathrm{k})$ 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 $n$ is the density and ${f}_{0}$ is the scattering length for the assumed two-body interaction between the bosons. The coefficients $a$ and $b$ are independent of the shape of the interaction, and are the only terms thus far calculated. The coefficient $b$ 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.

DOI
10.1103/physrev.116.489

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a Theoretical Study of a System of Interacting Bosons at Low Temperatures.

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•Jan 1, 1969•0 citations
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 (E0\ensuremathΩ)=12n2f0[1+a(nf03)12+b(nf03)lnnf03+c(nf03)+d(nf03)32ln(nf03)+\ensuremath⋯],(\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}],(\ensuremathΩE0​​)=21​n2f0​[1+a(nf0​3)21​+b(nf0​3)lnnf0​3+c(nf0​3)+d(nf0​3)23​ln(nf0​3)+\ensuremath⋯], 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.
We investigate the properties of a dilute boson-fermion mixture at zero temperature. The ground-state energy of the system is calculated to second order in the Bose-Fermi coupling constant. The Green's function formalism is applied to obtain results for the effect of Bose-Fermi interactions on the spectrum of phonon excitations, the momentum distribution, the condensate fraction, and the superfluid density of the bosonic component. A quantitative discussion of these effects is presented in the case of mixtures with attractive interspecies interaction.
The problem of a system of bosons at zero temperature, interacting via a strong short-ranged repulsive interaction and a weak long-ranged attraction, is treated by the methods of quantum-field theory. It is shown that if the parameters of the attractive part of the interaction are suitably restricted, an approximation to the self-energy parts of the Green's functions, similar to that used by Beliaev for the case of purely repulsive interactions, leads to an excitation spectrum which in the limit of small momenta represents phonons. The approximation also yields an NNN-body bound state corresponding to zero pressure. Apart from minor differences, these results are in agreement with an earlier calculation of Huang.
The ground state of a many-particle boson system is studied for two closely related limits: the uniform limit and the weak coupling limit. The former is defined by $\ensuremath{\alpha}=1\ensuremath{-}g(0)\ensuremath{\ll}1$ and the latter by $\ensuremath{\beta}=\frac{(N\ensuremath{-}{N}_{0})}{N}\ensuremath{\ll}1$, where g(r)g(r)g(r) is the radial distribution function and N0{N}_{0}N0​ is the mean occupation number of the zero-momentum state. In the uniform limit the variation-perturbation approach based on (a) the method of correlated basis functions and (b) the series expansion in powers of $\ensuremath{\alpha}$ is found to be equivalent to the field-theoretic treatment given by Brueckner (for the charged-boson gas) in the weak coupling limit. In particular, it is shown that the variation-perturbation energy obtained for the uniform limit in the momentum representation is identical through second order to the ground-state energy evaluated for $\ensuremath{\beta}\ensuremath{\ll}1$ by summing one- and two-ring diagrams in the Bogoliubov occupation-number representation. The charged-boson gas and the one-dimensional boson system with a $\ensuremath{\delta}$-function interaction are considered to examine some of the interesting features of the uniform-limit procedure.
The properties of a Bose-Einstein gas with repulsive interparticle interactions are determined at low temperature. It is shown that the contribution to the energy arising from particle excitation with small momentum transfers computed in conventional perturbation theory is divergent, but that this difficulty can be avoided by an alternative procedure. An exact method is developed for dealing with that part of the Hamiltonian which gives rise to the perturbation divergence and an exact solution is obtained. The treatment of the region of large momentum transfers is then carried out by introducing the concept of the scattering length of the interaction. The energy is given by the well-known result of the optical theorem corrected by a series in powers of ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$, where $\ensuremath{\rho}$ is the density and aaa the scattering length. Examination of the processes contributing to the energy shows that 95.8% of the correction of order ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$ results from a simple alteration of the perturbation energy denominators which takes into account the interaction energy of excited pairs with the unexcited pairs of the medium. The remaining 4.2% arises from multiple particle excitation. These methods also predict the existence of phonon excitation with phonon energies that are linear in the phonon momentum for small excitation and approach q22M\frac{{q}^{2}}{2M}2Mq2​ for large momentum. The detailed consideration of this spectrum forms part of the following paper.The results are compared with the theory of fermion systems with strong interactions, where again the largest correction to the first-order "optical" energy arises from alteration of the two-particle propagator which is required to take into account interactions with the many unexcited particles of the medium. The propagator corrections in the fermion and boson systems are similar except for a characteristic difference arising from the statistics. The theory for bosons is shown to be essentially identical with that given by Brueckner and Levinson in their general formulation of the theory of many-body systems with strong interactions.Systems with attractive but saturating interactions are also considered and some new difficulties are shown to arise which cannot be treated by a straightforward application of the methods of this paper.
The neutral system consisting of charged-bosons and -fermions has been investigated in terms of density parameter expansion at the high-density limit. The ground state energy of this system is calculated and accords with the result obtained in our previous work. The elementary excitations in the system determined from the poles of the boson Green's function are the two collective excitations in the low momentum region : acoustic- and optical-phonon like. The life-time of the acoustic phonon approaches infinite as s/v_f →0 where s and v_f arc the sound velocity and Fermi velocity, respectively, while that of the optical pbonon is infinite.
A self consistent field approach is developed for truncating the infinite set of equations obtained in apply­ ing the equation of motion method to interacting systems. The operator which is assumed to create elementary excita­ tions in the system is expanded in terms of simpler opera­ tors and both the coefficients in the expansion and relations independent of the coefficients are found. The method is applied to a system of Bosons at low temperatures in the approximation that the interaction is both tempera­ ture and density independent. Corrections to the pairing approximation are obtained due both to the inclusion of more terms in the Hamiltonian and the inclusion of collective excitations. It is shown that in the long wavelength limit these corrections can be included by renormalizing both the single particle and pairing energies. The renormalization can be carried out by adding a term to the single particle energy and an extra interaction to the pairing part of the Hamiltonian. This interaction is attractive over part of momentum space and obeys the criterion for a simple Bose condensation. It is shown that no single particle like zero momentum excitation exists. A collective excitation is con­ sidered for the zero momentum excitation and it is found v vi that its energy must be non-zero if it exists at all. The application to superfluid He11 is discussed.