Bounds for the phonon-roton dispersion in superfluid 4He

Abstract

The sum rule approach is used to derive upper bounds for the dispersion law $ω_0(q)$ of the elementary excitations of a Bose superfluid. Bounds are explicitly calculated for the phonon-roton dispersion in superfluid $^4$He, both at equilibrium ($ρ=0.02186$ Å$^{-3}$) and close to freezing ($ρ=0.02622$ Å$^{-3}$). The bound $ω_0(q) \le 2S(q)\midχ(q)\mid^{-1}$, where $S(q)$ and $χ(q)$ are the static structure factor and density response respectively, is calculated microscopically for several values of the wavevector $q$. The results provide a significant improvement with respect to the Feynman approximation $ω_F(q)= q^2(2mS(q))^{-1}$. A further, stronger bound, requiring the additional knowledge of the current correlation function is also investigated. New results for the current correlation function are presented.