Velocity-dependent interactions and sum rule in bcc He

Recent neutron scattering experiments [T. Markovich, E. Polturak, J. Bossy, and E. Farhi, Phys. Rev. Lett. 88, 195301 (2002)] on solid ${}^4\mathrm{He},$ discovered opticlike mode in the bcc phase. This excitation was predicted by a recently proposed model that describes the correlated atomic zero-point motion in bcc Helium in terms of dynamic electric dipole moments. Modulations of the relative phase of these dipoles between different atoms describes the anomalously soft $T_1\mathrm{}\left(110\right)\mathrm{}$ phonon and two new opticlike modes, one of which was recently found in the neutron scattering experiments. In this work we show that the correlated dipolar interactions can be written as velocity-dependent interactions. This formulation then results in a modified sum rule for the $T_1\mathrm{}\left(110\right)\mathrm{}$ phonon, in good agreement with recent (unpublished) data. This is a striking example of velocity-dependent interactions appearing on a microscopic level in a simple condensed-matter system.