Excitation spectrum of a supersolid

Abstract

The quantum-lattice-gas model is used to describe the solid phase of a system consisting of Bose particles. The Hamiltonian is diagonalized in the spin-wave approximation and a two-branch excitation spectrum is obtained. When the system exhibits a Bose-Einstein (BE) condensation (i.e., in the supersolid phase), the excitation is a coupled density---order-parameter oscillation. The lower branch is proportional to $k$ ($\ensuremath{\omega}\ensuremath{\sim}k$) and is mixed with the phonon spectrum; the upper branch has a gap and is proportional to ${k}^{2}$ ($\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\omega}}_{0}+b{k}^{2}$, $b>0$). The magnitude of the gap ${\ensuremath{\omega}}_{0}$ is a pressure-dependent quantity and is estimated to be in the range of 0 (at the superfluid---supersolid transition) to ${10}^{12}$ per sec (at the supersolid---normal-solid transition). In the normal-solid phase (no BE condensation) these two branches do not couple to density oscillations and cannot be observed by neutron scattering experiments. Therefore the existence of the upper branch by neutron scattering experiments may be used as a criterion for the existence of a supersolid.