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.
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 \ensuremathϵk\ensuremath→ 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 \ensuremathρk\ensuremath→ 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.
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.
Presently there are two distinct classes of microscopic theories which describe a system of many identical spinless bosons. One is the Bogoliubov theory, which places special emphasis on the occupation of the zero-momentum state and as such depends explicitly on the number of particles in this state, N0. The other depends on density fluctuations to describe the fluid. In this work, we derive a modified version of the Bogoliubov theory which results in a number-conserving Hamiltonian with no explicit dependence on N0. The ground- and first-excited-state energies evaluated through second order in N211 are equivalent to the results of a variational-perturbative procedure based on the method of correlated basis functions in the uniform limit.
A general system composed of many weakly interacting bosons and/or fermions is considered. The object is to develop a procedure for renormalizing the single-particle creation operators, so as to remove the interactions to successively higher orders of perturbation. The basic idea is that creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ must satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where H is the Hamiltonian and ${\ensuremath{\omega}}_{i}$ are the particle energies. For the many-weakly-interacting-particle system, the zeroth-order boson and fermion creation operators satisfy these equations to zeroth order. It is shown that if the creation operators are renormalized so as to satisfy the Hamiltonian commutator equations to order m, and also to satisfy the appropriate boson commutators and fermion anticommutators to order m, then the problem is solved to order m. In particular, the vectors formed by operating on the ground state with the renormalized creation operators, according to the usual boson and fermion occupation-number representation, are eigenvectors of H and are orthonormal, all to order m. A procedure is otained for finding the (m+1)-order contributions to the particle creation energies, in terms of the m-order operators. Explicit first-order calculations of these general results are provided for a system of bosons and a system of fermions, and these first-order results are shown to include similar results of Rayleigh-Schr\"odinger perturbation theory, the random-phase approximation, and the method of thermodynamic Green's functions. The problem of anharmonic lattice dynamics is studied in detail, and a method of undetermined coefficients is used to renormalize the phonon creation operators to first order. The phonon energies are calculated to second order, and this calculation shows that the interactions between renormalized phonons cannot be removed in second order. Statistical averages of the phonon energies give the energy shifts and lifetimes which have been calculated previously by various propagator techniques. In addition, the renormalized energies are used to calculate the temperature-dependent part of the Helmholtz free energy correct to second order. As a final example, electron-phonon interactions in a normal metal are studied. The electron and phonon creation operators are renormalized to first order, the particle energies are calculated to second order, and statistical averages of the particle energies recover the usual thermodynamic Green's-function results for energy shifts and lifetimes. These examples show the simplicity by which the renormalization procedure obtains a great amount of detailed information about the single-particle nature of a many-particle system.
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) 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 f0 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.
This article reviews recent advances in the theory of the three-dimensional dilute homogeneous Bose gas at zero and finite temperature. Effective-field-theory methods are used to formulate a systematic perturbative framework that can be used to calculate the properties of a system at T=0. The perturbative expansion of these properties is essentially an expansion in the gas parameter na3, where a is the s-wave scattering length and n is the number density. In particular, the leading quantum corrections to the ground-state energy density, the condensate depletion, and long-wavelength collective excitations are rederived in an efficient and economical manner. Nonuniversal effects are also discussed. These effects are higher-order corrections that depend on properties of the interatomic potential other than the scattering length, such as the effective range. The article critically examines various approaches to the dilute Bose gas in equilibrium at finite temperature. These include the Bogoliubov approximation, the Popov approximation, the Hartree-Fock-Bogoliubov approximation, the \ensuremath{\Phi}-derivable approach, optimized perturbation theory, and renormalization-group techniques. The article ends with a look at recent calculations of the critical temperature of the dilute Bose gas, which include 1/N techniques, lattice simulations, self-consistent calculations, and variational perturbation theory.
For a Bose system interacting through a two-body potential there is a large occupation number in the zero-momentum state with small fluctuations. Thus, we can express the chemical potential and low-lying excitation energy in terms of the density fluctuation only. The formulas derived are shown to coincide with well-established results in the low-density Bose gas. The use of experimental neutron scattering data is suggested to improve the numerical answer. A particular application is made to the He4 case. If we utilize the Feynman variational excited state as the (approximate) true state vector, which is correct experimentally for low-momentum excitations, we get an explicit phonon excitation for actual He4.