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Journal of Mathematical Physics

Density and Current Density as Coordinates for a System of Interacting Bosons at Absolute Zero

B. Tsu-Shen Chang

Published Jun 1, 1971• 8 citations
Publisher

Abstract

The quantum field Hamiltonian expressed in terms of density and current density variables has been employed together with the equal-time commutation relations among these variables to find the ground state energy and the density fluctuation excitation spectrum of a system of interacting bosons at T = 0°K. The approximation involved consists in assuming that the density fluctuation in space is small compared with the average density. The results easily obtained in the lowest-order approximation agree with those of Bogoliubov. However, in our treatment no condensation of particles in zero-momentum state is assumed or apparent. A connection between the present treatment and the quantum hydrodynamic approach to the irrotational flow of a Bose liquid has been made.

DOI
10.1063/1.1665691

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Perturbation Theory Based on Currents for Interacting Bosons at Absolute Zero

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As a continuation of a recent paper, using density and current density as coordinates for interacting bosons at absolute zero, this work deals with the perturbation theory. The zerothorder Hamiltonian is the one shown to yield the Bogoliubov energy spectrum. First- or secondorder perturbation calculations have been made of the three-phonon and the four-phonon vertices for the energy corrections of the low-lying excited states. The results in the long-wavelength limit differ, although only slightly, from the quantum hydrodynamic calculations, and indicate that the perturbed excitation energy ${\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ is an odd-power series in $k$ if the Fourier coefficients of the two-body potential concerned can be approximated by a constant or an evenpower series in $k$. However, whether this remains so in all higher orders of various perturbations is an open question. Two methods were used in the calculation. The first involves functional representations of the coordinates in the ${\ensuremath{\rho}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ representation recently reported. The second method is new, and employs "occupation-number" representations of the current algebra by introducing creation and annihilation operators for "phonons" from density fluctuations. These latter representations enable us to benefit from the advantages of the canonical field theory.

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Physical Review A•Jul 1, 1970•8 citations
The quantum field Hamiltonian expressed in terms of density and current density variables has been employed together with the equal-time commutation relations among these variables to find the ground state energy and the density fluctuation excitation spectrum of a system of interacting bosons at T = 0°K. The approximation involved consists in assuming that the density fluctuation in space is small compared with the average density. The results easily obtained in the lowest-order approximation agree with those of Bogoliubov. However, in our treatment no condensation of particles in zero-momentum state is assumed or apparent. A connection between the present treatment and the quantum hydrodynamic approach to the irrotational flow of a Bose liquid has been made.
As a continuation of a recent paper, using density and current density as coordinates for interacting bosons at absolute zero, this work deals with the perturbation theory. The zerothorder Hamiltonian is the one shown to yield the Bogoliubov energy spectrum. First- or secondorder perturbation calculations have been made of the three-phonon and the four-phonon vertices for the energy corrections of the low-lying excited states. The results in the long-wavelength limit differ, although only slightly, from the quantum hydrodynamic calculations, and indicate that the perturbed excitation energy ${\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ is an odd-power series in kkk if the Fourier coefficients of the two-body potential concerned can be approximated by a constant or an evenpower series in kkk. However, whether this remains so in all higher orders of various perturbations is an open question. Two methods were used in the calculation. The first involves functional representations of the coordinates in the ${\ensuremath{\rho}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ representation recently reported. The second method is new, and employs "occupation-number" representations of the current algebra by introducing creation and annihilation operators for "phonons" from density fluctuations. These latter representations enable us to benefit from the advantages of the canonical field theory.
An exact expression for the single-particle reduced density matrix of a translationally invariant system of bosons is obtained, as an average of a functional of the density and current operators. An approximation scheme, based on small fluctuations in density and current, is used in its lowest order, to approximate the single-particle reduced density matrix and the condensate density by a functional of the structure factor, which is a measurable quantity. This is done for the ground state and for finite temperatures. The resulting expressions for the condensate density are discussed. For the ideal gas the result is correct below the condensation temperature. For the case of an interacting liquid, Bose-Einstein condensation exists, in the ground states for two- and three-dimensional systems, and for finite temperatures it does exist only in three dimensions. The present experimental knowledge of the structure factor enables only a crude estimate of the condensed fraction which is found to be 0.1-0.2.
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.
It is now well established that current and density may be used as coordinates in describing a nonrelativistic many-body system. The current and density operators form a closed algebra as they obey a set of equal-time commutation relations. A particular choice of the algebra containing only the longitudinal component of the current describes a fluid in a simple way. For Bose systems, we show that there exists a special choice of this longitudinal current which is an irreducible representation of the algebra, but expressed in two different ways leading to the formulations of Bogoliubov and Zubarev (BZ) and that of Sunakawa and his coworkers (S). This demonstration proves the formal equivalence of the two formulations. The BZ formalism leads to a non-Hermitian Hamiltonian for which use of a certain mathematical technique recently proposed by us yields completely divergence-free results for Bose systems. A temperature-dependent matrix Green's-function theory is developed and the self-energy matrix calculated. From this, we also deduce the structure factor in a straightforward way. These results are applied here to compute numerically the excitation spectrum of the superfluid liquid helium using the experimental structure factor as the only input into the computation. Comparison of this with other calculations and with experiment is discussed in detail in this paper. The problem of a charged Bose gas is considered as a testing ground for all theories of Bose fluids since certain exact results are known for this system. We have applied our method using the BZ Hamiltonian to this problem and we are in complete agreement with the exact results. It thus appears that the current-algebra approach can be quite successfully applied in elucidating properties of interacting Bose systems.
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.