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.
Major issues arising in the theory of Bose–Einstein condensation are reviewed. These issues, although being principally important, are very often misunderstood, which results in wrong conclusions. The basic point is global gauge symmetry breaking that is a necessary and sufficient condition for Bose–Einstein condensation. Paying no attention to this basic point is a common fallacy leading to a number of confusions. For instance, the attempt of describing Bose condensation without gauge symmetry breaking produces the so-called “grand canonical catastrophe” that actually does not exist in the correct description of Bose condensation accompanied by gauge symmetry breaking. The other common flaw is forgetting to consider the stability of the studied systems. One sometimes accomplishes lengthy calculations and discusses the properties of a system that in reality cannot exist being unstable. In some cases, the seeming instability is caused by the negligence of the simple mathematical reason teaching us that one should not go beyond the approximation applicability. An example of such an artificial instability is related to the appearance of the so-called “thermodynamically anomalous fluctuations” whose arising is due to the use of a second-order approximation for calculating fourth-order terms, in this way distorting the O(2)-class model of a Bose-condensed system to the Gaussian-class model. These and other principal points, important for the correct treatment of Bose-condensed systems, are reviewed, including the resolution of the Hohenberg–Martin dilemma of gapless versus conserving theories for Bose-condensed systems and the problem of statistical ensemble equivalence.
This paper is concerned with the application of sum rules satisfied by the density correlation function to the calculation of properties of the interacting Bose system, particularly liquid
He4 and the hard-core gas. In addition to the two well-known sum rules which have been extensively discussed in the literature, we derive a third useful relation which directly involves the two-body interaction. The assumption that a single excitation exhausts all the sum rules (which is known to lead to the Feynman theory when only the first two sum rules are used) now provides, with the help of the additional sum rule, an explicit integral equation for the liquid structure factor
S(k). This equation can be solved quite easily for the hard-core gas, and we find the low-lying excitation spectrum to have the form ${{\ensuremath{\omega}}_{\mathrm{hc}}}^{2}(k)={k}^{2}{{v}_{T}}^{2}+{\left(\frac{{k}^{2}}{2m}\right)}^{2}+4\left(\frac{{k}^{2}}{2m}\right)\left[\left(\frac{P}{\ensuremath{\rho}}\ensuremath{-}\frac{2}{3}\frac{〈\mathrm{KE}〉}{〈N〉}\right)({\mathcal{P}}_{1}(\mathrm{ka})\ensuremath{-}{\mathcal{P}}_{1}(0))+\left(m{{v}_{T}}^{2}\ensuremath{-}\frac{2P}{\ensuremath{\rho}}\ensuremath{-}\frac{2}{3}\frac{〈\mathrm{KE}〉}{〈N〉}\right)({\mathcal{P}}_{2}(\mathrm{ka})\ensuremath{-}{\mathcal{P}}_{2}(0))\right],$ where
vT,
P,
〈N〉〈KE〉 and
a are, respectively, the sound velocity, pressure, kinetic energy per particle, and core diameter.
P1 and
P2 are sinusoidal functions given in the text. Although the sum rules themselves are exact, there are several difficulties associated with the use, even in the low-
k limit, of the sharp resonance assumption in all such integral relations. In spite of these difficulties, which are discussed briefly, we have pursued the consequences of the sharp-resonance approximation to obtain an integral equation for the liquid structure factor
S(k) together with the above closed-form solution for the hard-core gas excitation spectrum. The speculation that the approximation becomes essentially exact in the low-
k limit provides, through the third sum rule, an expression for the sound velocity in terms of the two-body potential
v(r) and the radial distribution function
S(r). We apply this relation to the case of liquid
He4 and discuss extensions of the single-resonance approximation which can be made with the help of the third sum rule.
We derive and discuss sets of sum rules relating to the density fluctuation operator and to the single-particle creation and annihilation operators in a superfluid of charged bosons. The physical interpretation of the particle - particle and particle - density sum rules hinges on the single-particle excitation spectrum at long wavelengths having a gap equal to the plasma frequency in the presence of the Bose - Einstein condensate. This spectral property is shown to follow from the Hugenholtz - Pines relation for the chemical potential in terms of the half-diagonal two-body density matrix. Data on this density matrix are obtained by quantum Monte Carlo methods and are used to check the self-consistency between the Hugenholtz - Pines relation and the value of the chemical potential calculated from the ground-state energy. We also tabulate the contributions from the plasmon and from multiparticle excitations to various matrix elements and sum rules at long wavelengths.
In this paper we show that a normal total number-of-particle fluctuation can be obtained consistently from the static thermodynamic relation and dynamic compressibility sum rule. In models using the broken U(1) gauge symmetry, in order to keep the consistency between statics and dynamics, it is important to identify the equilibrium state of the system with which the density response function is calculated, so that the condensate particle number
N0, the number of thermal depletion particles $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{N}$, and the number of noncondensate particles
NNC can be unambiguously defined. We also show that the chemical potential determined from the Hugenholtz-Pines theorem should be consistent with that determined from the equilibrium equation of state. The
N4∕3 anomalous fluctuation of the number of noncondensate particles is an intrinsic feature of the broken U(1) gauge symmetry. However, this anomalous fluctuation does not imply the instability of the system. Using the random phase approximation, which preserves the U(1) gauge symmetry, such an anomalous fluctuation of the number of noncondensate particles is completely absent.
Adding a gauge symmetry breaking field $\ensuremath{-}\ensuremath{\nu}\sqrt{V}({a}_{0}+{a}_{0}^{*})$ to the Hamiltonian of some simplified models of an interacting Bose gas we compute the condensate density and the symmetry breaking order parameter in the limit of infinite volume and prove Bogoliubov's asymptotic hypothesis ${\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}⟨{a}_{0}⟩∕\sqrt{V}=\mathrm{sgn}\phantom{\rule{0.2em}{0ex}}\ensuremath{\nu}\phantom{\rule{0.2em}{0ex}}{\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}\sqrt{⟨{a}_{0}^{*}{a}_{0}⟩∕V}$ where the averages are taken in the ground state or in thermal equilibrium states. Letting $\ensuremath{\nu}$ tend to zero in this equation we obtain that Bose-Einstein condensation occurs if and only if the gauge symmetry is spontaneously broken. The simplification consists in dropping the off-diagonal terms in the momentum representation of the pair interaction. The models include the mean field and the imperfect (Huang-Yang-Luttinger) Bose gas. An implication of the result is that the compressibility sum rule cannot hold true in the ground state of the one-dimensional mean-field Bose gas. Our method is based on a resolution of the Hamiltonian into a family of single-mode
(k=0) Hamiltonians and on the analysis of the associated microcanonical ensembles.