Dispersion of Phonons inHe4

Abstract

Although the hydrodynamic Hamiltonian is widely used to predict properties of liquid ${\mathrm{He}}^4$, it has been felt to be inherently unsatisfactory because it was believed that the phonon spectrum at 0°K predicted by this Hamiltonian is $\epsilon =cp$, contrary to experiment. However, a self-energy calculation shows that the predicted spectrum is in fact dispersive, of the form $\epsilon =cp(1-\gamma p^2)$. Lowest order pertrubation theory gives the proper order of magnitude for $\gamma$.