Various sum rules for the density and particle operators are derived and discussed. We investigate in detail the properties of the particle state, the natural counterpart of the Feynman state describing collective density excitations. An explicit expression for the energy of the particle state is derived in terms of the interatomic potential, the two-body half-diagonal density, and the momentum distribution. Sum rules accounting for the coupling between particle and density excitations are also derived and the role of the Bose-Einstein condensation explicitly pointed out. Finally we discuss the separate contribution to the various sum rules arising from one-phonon and multiparticle excitations.
The spectral properties of a mixture of bosons hybridized with fermion pairs are studied in the superconducting state. Such a model exhibits two limiting behaviors. One is reminiscent of a BCS system, provided that the total concentration n of particles is below a certain critical concentration
nc, in which case bosons exist only in thermally excited states. The other limit for n\ensuremath{\geqslant}
nc describes a situation largely controlled by the condensation of the bosons. The variation of the single-boson as well as collective excitations as a function of n and for the entire temperature regime from T=0 K to
Tc will be discussed, both with and without long-range Coulomb interaction between the charge carriers. The method for calculation applied here is based on the dielectric formalism and the use of Ward identities, currently used in the theory of the interacting Bose gas, which allows us to derive the density and current correlation functions consistent with local charge-current conservation laws. In the absence of a Coulomb interaction the behavior of the system is characterized by the existence of a sound-wave-like branch in the Bose single-particle and collective excitation spectrum. In the BCS regime the sound velocity is proportional to the Fermi velocity while in the Bose regime it is proportional to the strength of the boson-fermion hybridization and decreases with the superfluid density as we approach
Tc. In the presence of a long-range Coulomb interaction and in the Bose regime one observes, besides the usual optical plasmon mode at high frequency and corresponding to in-phase density oscillation of the two subsytems, an out-of-phase collective mode in the two particle continuum just above the gap. Moreover, upon approaching
Tc a relatively well-defined quasiparticle feature starts to emerge within the gap and is seen in both the Bose single-particle excitation and the density oscillation spectrum. In the BCS regime, due to the very small number of occupied bosonic states, the low-energy quasiparticle features are absent. \textcopyright{} 1996 The American Physical Society.
Various sum rules, accounting for the coupling between density and particle excitations and emphasizing in an explicit way the role of Bose–Einstein condensation, are discussed. Important consequences on the fluctuations of the particle operator as well as on the structure of elementary excitations are reviewed. These include a recent generalization of the Hohenberg–Mermin–Wagner theorem holding at zero temperature. Introduction The sum rule approach has been employed extensively in the literature in order to explore various dynamic features of quantum many body systems from a microscopic point of view (see [1] and references therein). An important merit of the method is its explicit emphasis on the role of conservation laws and of the symmetries of the problem. Furthermore, the explicit determination of sum rules is relatively easy and often requires only a limited knowledge of the system. Usually the sum rule approach is, however, employed without giving special emphasis to the possible occurrence of (spontaneously) broken symmetries. For example, the most famous f -sum rule [2] holding for a large class of systems is not affected by the existence of an order parameter in the system. The purpose of this paper is to discuss a different class of sum rules which are directly affected by the presence of a broken symmetry. These sum rules can be used to predict significant properties of the system which are the consequence of the existence of an order parameter.
We critically examine the Hartree-Fock-Bogoliubov (HFB) solution of the equations of motion for condensate fluctuations in a weakly coupled plasma of charged bosons at finite temperature. Analytic expressions are derived for the first two infrared-divergent terms in both the momentum distribution of the noncondensate and the anomalous Bose correlation function at low momenta. Incorporation into the theory of the appropriate form of the Hugenholtz-Pines relation for the chemical potential is needed to cancel an unphysical divergence. Exact cancellation of infrared-divergent terms is demonstrated in the HFB shift of the single-particle excitation energy away from the Bogoliubov value at long wavelengths, with the residual terms raising it towards the plasma frequency at low temperature. Numerical illustrations are presented for a number of properties of the boson plasma as functions of temperature and density in the weak-coupling regime: these are the chemical potential, the condensate fraction, the normal and anomalous momentum distribution functions and the corresponding one-body density matrices, and the dispersion relation of single-particle excitations.
The main focus of this work is on the predictions made by the dielectric formalism in regard to the relationship between single-particle and collective excitation spectra in a gas of pointlike charged bosons at finite temperature
T below the critical region of Bose-Einstein condensation. Illustrative numerical results at weak coupling
(rs=1) are presented within the random phase approximation. We show that within this approach the single-particle spectrum forms a continuum extending from the transverse to the longitudinal plasma mode frequency and leading to a double-peak structure as
T increases, whereas the density fluctuation spectrum consists of a single broad peak. We also discuss the momentum distribution and the superfluidity of the gas.
The momentum distribution ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{n}}_{p}$ of atoms in a Bose-condensed system is computed in the long-wavelength limit $p\ensuremath{\rightarrow}0$. We use frequency-moment sum rules for the single-particle Green's function, including a generalized version of Wagner's sum rule appropriate to hard-core potentials. We show that at finite temperatures ($\mathrm{cp}\ensuremath{\ll}{k}_{B}T$) the correction to ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{n}}_{p}=\frac{{n}_{0}{m}^{2}}{{\ensuremath{\rho}}_{s}{p}^{2}}$ involves the off-diagonal self-energy ${\ensuremath{\Sigma}}_{+\ensuremath{-}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}},\ensuremath{\omega}=0)$. In calculating ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{n}}_{p}$ in the limit $\mathrm{cp}\ensuremath{\gg}{k}_{B}T$, we include first-sound as well as second-sound contributions.