Dynamics of supersolid state: Normal fluid, superfluid, and supersolid velocities

Abstract

We consider the dynamics of a supersolid state that is based on Leggett's idea of local phase flow around the atomic sites of the solid. This leads to a local microscopic superfluid velocity that is not Galilean and is defined only relative to the solid and a $T=0$ superfluid fraction ${\ensuremath{\rho}}_{S}/\ensuremath{\rho}<1$ (with $\ensuremath{\rho}$ being the macroscopic mass density). Landau's excitation-based argument for superfluids---that at temperature $T=0$ the normal fluid mass density ${\ensuremath{\rho}}_{N}$ is zero---should apply to supersolids. Together these two arguments imply that at low $T$ there is a missing mass density. We attribute this to a supersolid density ${\ensuremath{\rho}}_{L}$, with ${\ensuremath{\rho}}_{L}\ensuremath{\equiv}\ensuremath{\rho}\ensuremath{-}{\ensuremath{\rho}}_{S}\ensuremath{-}{\ensuremath{\rho}}_{N}$ and a momentum-bearing supersolid velocity ${v}_{Li}$. We derive the macroscopic dynamical equations for this system with three momentum-bearing macroscopic velocities $({v}_{Li},{v}_{Ni},{v}_{Si})$ using Onsager's irreversible thermodynamics. For slow spatial variations, ${v}_{Si}$ accelerates along the negative gradient of the chemical potential or mass, like for an ordinary superfluid, and ${v}_{Ni}$ is subject to drag against ${v}_{Li}$ with a characteristic drag time $\ensuremath{\tau}$. There are three contributions to the acceleration of ${v}_{Li}$: drag against ${v}_{Ni}$, along the negative gradient of the chemical potential or mass, and due to the forces of lattice elasticity. For high-frequency motion relative to the drag time $\ensuremath{\tau}$, ${\ensuremath{\rho}}_{L}$ moves independently of ${\ensuremath{\rho}}_{N}$. The normal modes for such a system have a crossover frequency below which the normal fluid velocity ${v}_{Ni}$ is dragged into motion with ${v}_{Li}$ (so the theory reduces to the ${v}_{Si},{v}_{Ni}$ theory of Andreev and Lifshitz) and above which the normal fluid velocity ${v}_{Ni}$ is an independent variable. For an isotropic lattice we study the transverse and longitudinal responses. A frequency-dependent rotational inertia experiment is predicted to show, at frequencies that are low relative to $\ensuremath{\tau}$, that ${\ensuremath{\rho}}_{L}$ and ${\ensuremath{\rho}}_{N}$ move together but that at high frequencies ${\ensuremath{\rho}}_{L}$ moves much more than ${\ensuremath{\rho}}_{N}$.