Green's functions of phonons for many-boson systems at high density are obtained from the phonon Hamiltonian given in a previous paper on the basis of the coherent-state representation. A dispersion equation derived from these Green's functions gives rise to the excitation spectrum leading to the Feynman energy in the low-momentum limit. It is shown that the dynamic structure function obtained here satisfies the sum rules in the low-momentum limit. As an illustrative example, the one-dimensional Lieb-Liniger model is considered. The relationships to the previous theories, in particular, to the velocity field approach of Yamasaki, Kebukawa and Sunakawa and to our density and phase operator approach are elucidated by the use of a unitary transformation. Discussion will be made on the D-wave portion of the roton-roton interaction in relation to the resonance state of roton pair studied in one of our previous papers. Finally, it is pointed out that the zeroth Fourier transform of the phase operator is related to the superfluid velocity in the two-fluid theory of liquid helium II.
In this paper we suggest a new ground-state wave function and low-temperature density matrix for a strongly interacting system of bosons. Our basic assumption is that the system can support long-wavelength phonons and that these can propagate independently of any other mode of motion. We therefore write the ground-state function as the product of two factors. One factor arises from the zero-point motion of the phonons, and we show that it has the form ${\ensuremath{\Pi}}_{i<j}f({r}_{\mathrm{ij}})$, where
lnf(r) has an infinite range. The other factor is assumed to have this same form but with
lnf(r) of finite range; it takes into account the short-range correlations arising from the strong repulsive part of the interparticle potential. The function we have chosen to represent the short-range correlations is not new; functions of this kind were first introduced by Bijl and later by Jastrow. At finite temperatures we use a density matrix for an ensemble of excited phonon states. We find that for small wave vectors
k, the structure factor
S(k) is equal to $\frac{\ensuremath{\hbar}k}{2mc}$, a result that was first derived by Feynman. At a finite temperature
T,
S(k) tends to the constant value
mc2kBT, where
m is the mass of the particles and
c the velocity of propagation of the phonons. The momentum distribution
nk has a ${k}^{\ensuremath{-}1}$ singularity at absolute zero and a stronger, ${k}^{\ensuremath{-}2}$ singularity at finite temperatures. The small-
k behavior of both
S(k) and
nk is completely controlled by the correlations introduced by the phonons. Both the ground-state function and the density matrix imply that there is a finite fraction of particles in the zero-momentum state in three dimensions; this fraction does not seem to be appreciably affected by the infinite-range correlations introduced by the phonons. We find, however, that these correlations imply that a one-dimensional Bose system does not exhibit Bose-Einstein condensation at any temperature, while a two-dimensional system exhibits Bose-Einstein condensation only at absolute zero.
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 spectrum of phonon-like collective excitations in the system of Bose-atoms in optical lattice (more generally, in the system of quantum particles described by the Bose-Hubbard model) is investigated.Such excitations appear due to displacements of particles with respect to their local equilibrium positions.The two-level model taking into account the transitions of bosons between the ground state and the first excited state in potential wells, as well as interaction between them, is used.Calculations are performed within the random phase approximation in the hard-core boson limit.It is shown that excitation spectrum in normal phase consists of the one exciton-like band, while in the phase with BE condensate an additional band appears.The positions, spectral weights and widths of bands strongly depend on chemical potential of bosons and temperature.The conditions of stability of a system with respect to the lowering of symmetry and displacement modulation are discussed.
An alternative approach to the quantum statistics of interacting particles is proposed. It consists of calculating the equilibrium thermodynamical quantities of the many-body system via the pair distribution function with the assumption that the particles interact with each other only through pair central forces. The proposed approach has some advantage over the usual treatment via the partition function in that the pair distribution function is easier to deal with than the partition function in certain circumstances particularly when the collective motion description of the system is desirable. This is because only the pair distribution function can be expressed directly in terms of the collective interaction which is closely connected with collective elementary excitation, such as a plasmon in the electron gas and a phonon in the hard-sphere Boson gas. The equation of state as well as the internal energy are obtained in the form of integrals of the pair distribution function. The close analogy between the pair distribution function and the two-body propagator, which appears in the quantum field theory, makes it possible to analyze the former by the use of Feynman diagrams identical with those usually introduced for the latter. The collective interaction, which is defined by the sum of the direct and the indirect interactions, is introduced as a particular partial sum of the perturbation series of the pair distribution function. This is used in rewriting the pair distribution function in terms of the collective interaction.It is shown that, while the simple chain approximation to the collective interaction in the electron gas is responsible for the transfer of a plasmon, the same approximation to the collective pseudo-interaction in the hard-sphere Boson system has a relation similar to that of transfer of a phonon, both cases occuring at low temperatures. The explicit calculation of the pair distribution functions for these systems at the absolute zero temperature is carried out up to the first order in the collective interaction (simple chain approximation.) These results are used to calculate the ground-state energies. For the electron gas the energy thus obtained confirms Gell-Mann-Bruckner's calculation of the correlation energy. For the hard-sphere Bosons the calculated energy reproduces the result of Lee, Huang, and Yang. The extension of the calculation to the finite temperature case is also indicated. In particular the classical Debye-Huckel equation of state for the electron gas is briefly discussed.