Bounds for the phonon-roton dispersion in superfluid 4He

TL;DR

This paper derives and calculates upper bounds for the phonon-roton dispersion in superfluid helium-4 using a sum rule approach, improving upon previous approximations and exploring bounds involving current correlations.

Contribution

It introduces explicit upper bounds for the phonon-roton dispersion in superfluid helium-4, utilizing a sum rule approach and including new results for the current correlation function.

Findings

Bounds significantly improve upon Feynman approximation

Explicit calculations at different densities

New insights into current correlation functions

Abstract

The sum rule approach is used to derive upper bounds for the dispersion law $\omega_0(q)$ of the elementary excitations of a Bose superfluid. Bounds are explicitly calculated for the phonon-roton dispersion in superfluid $^4$He, both at equilibrium ($\rho=0.02186$ \AA$^{-3}$) and close to freezing ($\rho=0.02622$ \AA$^{-3}$). The bound $\omega_0(q) \le 2S(q)\mid\chi(q)\mid^{-1}$, where $S(q)$ and $\chi(q)$ 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 $\omega_F(q)= q^2(2mS(q))^{-1}$. 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.