Application of Sum Rules to the Low-Temperature Interacting Boson System

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 ${\mathrm{He}}^4$ 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 $S\left(k\right)\mathrm{}$. This equation can be solved quite easily for the hard-core gas, and we find the low-lying excitation spectrum to have the form ${{\omega }_{\mathrm{hc}}}^2\mathrm{}\left(k\right)=k^2{v_T}^2+{\left(\frac{k^2}{2m}\right)}^2+4\mathrm{}\left(\frac{k^2}{2m}\right)\left[\left(\frac{P}{\rho }-\frac{2}{3}\frac{〈\mathrm{KE}〉}{〈N〉}\right)\left({\mathcal{P}}_1\mathrm{}\right(\mathrm{ka})-{\mathcal{P}}_1\mathrm{}(0\left)\right)+\left(m{v_T}^2-\frac{2P}{\rho }-\frac{2}{3}\frac{〈\mathrm{KE}〉}{〈N〉}\right)\left({\mathcal{P}}_2\mathrm{}\right(\mathrm{ka})-{\mathcal{P}}_2\mathrm{}(0\left)\right)\right],$ where $v_T$, $P$, $\frac{〈\mathrm{KE}〉}{〈N〉}$ and $a$ are, respectively, the sound velocity, pressure, kinetic energy per particle, and core diameter. ${\mathcal{P}}_1$ and ${\mathcal{P}}_2$ 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 $S\left(k\right)\mathrm{}$ 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 $v\left(r\right)\mathrm{}$ and the radial distribution function $S\left(r\right)\mathrm{}$. We apply this relation to the case of liquid ${\mathrm{He}}^4$ and discuss extensions of the single-resonance approximation which can be made with the help of the third sum rule.