Model of supersolid without defects
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
The recent surge of interest in supersolids 1 makes it important to reach a clearer understanding of the mechanical properties of such materials. In particular why a supersolid behaviour is observed in a rotating experiment, whilst, as in a ordinary solid no constant mass flux is driven by pressure gradient?
In 1994, with Yves Pomeau 2, we proposed a fully explicit model of supersolid where many properties can be discussed in details. We thought timely to reconsider this model, in particular with respect to its properties of elasticity coupled to its ability to carry some sort of superflow in the absence of any defect. Although supersolidity is often related to the presence of defects, vacancies and so forth our model introduces an important difference between ordinary (classical) crystals and supersolids: in perfect classical crystals there is an integer number (or a simple fraction) of atoms per unit cell. Therefore the number density and the lattice parameters are not independent. On the contrary, in our model of supersolid, there is no such relation. The lattice parameters and the average density can be changed independently.
This model is based upon the original Gross-Pitaevksii equation with an integral term with a kernel that can be seen as a two-body potential in a first Born approximation. This model yields the exact spectrum found long ago by Bogoliubov, namely a relation between the energy of the elementary excitations and their momentum depending on the two-body potential. In this framework the roton minimum is a precursor of crystallisation. By increasing the density, crystallization happens through a first order phase transition. As shown in the crystal phase shows a periodic modulation of density in space together with some superfluid-like behaviour under rotation.
The aim of our present work 3 is to show that, besides this behaviour, the system has also solid-like behaviour, at least under small stress. At larger stress, it flows plastically, the plasticity being facilitated by the eventual presence of defects. Moreover, we derive the equation of motion for the average density, the phase and the displacement in the solid. A new propagating mode appears besides the usual longitudinal and transverse phonons in regular crystals. This mode is partly a modulation of the coherent quantum phase, like the phonons in superfluids at zero temperature. We discuss the boundary conditions and how to handle steady rotation and pressure driven flow in this model.
The most important feature of the model predicts a paradoxical behavior of a supersolid: the existence of a non classical rotational inertia fraction in the limit of small rotation speed but no superflow under small (and finite) stress nor external force. The only matter flow for finite stress is due to plasticity.
Finally, the nonclassical rotational inertia fraction (NCRIF) is computed numerically in a two dimensional square box for different values of the compression $Lambda= U_0m a^2\hbar^2 n a^3$ ($U_0$ is a measure of the particle interaction energy, $a$ its range and $n$ is the average number density of the solid). The figure shows a non-zero NCRIF in the limit of small rotation speed. In this limit NCRIF$_0$ is a function (see the inset) of the dimensionless compression $Łambda$ only (and seems to decrease in an exponential way as $Łambdaınfty$). Those measurements are in a qualitative agreement with the recent experiments by Kim and Chan.
1) E. Kim and M.H.W. Chan, Nature (London) 427, 225 (2004); Science 305, 1941 (2004); Phys. Rev. Lett. 97, 115302 (2006).\\
2) Y. Pomeau and S. Rica, Phys. Rev. Lett. 72, 2426 (1994).\\
3) C. Josserand, Y. Pomeau and S. Rica, ``Coexisting ordinary elasticity and superfluidity in a model of defect-free supersolid'', to appear Phys. Rev. Lett. (2007).