Bounds for the phonon-roton dispersion in superfluid 4He.

Abstract

The sum-rule approach is used to derive upper bounds for the dispersion law ${\mathrm{\ensuremath{\omega}}}_{0}$(q) of the elementary excitations of a Bose superfluid. Bounds are explicitly calculated for the phonon-roton dispersion in superfluid $^{4}\mathrm{He}$, both at equilibrium (\ensuremath{\rho}=0.021 86 A${\mathrm{\r{}}}^{\mathrm{\ensuremath{-}}3}$) and close to freezing (\ensuremath{\rho}=0.026 22 A${\mathrm{\r{}}}^{\mathrm{\ensuremath{-}}3}$). The bound ${\mathrm{\ensuremath{\omega}}}_{0}$(q)\ensuremath{\le}2S(q)\ensuremath{\Vert}\ensuremath{\chi}(q)${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{-}}1}$, where S(q) and \ensuremath{\chi}(q) are the static structure factor and density response, respectively, is calculated microscopically for several values of the wave vector q. The results provide a significant improvement with respect to the Feynman approximation ${\mathrm{\ensuremath{\omega}}}_{\mathit{F}}$(q)=${\mathit{q}}^{2}$[2mS(q)${]}^{\mathrm{\ensuremath{-}}1}$. A further, stronger bound, requiring the additional knowledge of the current correlation function is also investigated. Results for the current correlation function are presented.