Long-range correlations in quantum solids

Nearly 40 years after the admirable paper [Phys. Rev. 155, 88 (1967)] in which Reatto and Chester suggested a ground state wave function for liquid He-4 capturing long-range correlations related to the zero-point motion of compressional waves (longitudinal phonons), we investigate the possibility of mimicking their approach in the case of solid He-4. Starting from elasticity theory, via a 'quantization' procedure, we propose a way to describe correlations among particles related to transverse long wavelength oscillations of the crystal lattice. The idea of exploring new kinds of long-range correlations in variational models arises from the evidence that some correlations are missing even in the most accurate models, like the Shadow Wave Function, which are not able to describe the long distance behaviour of the one-body density matrix in the solid phase computed in a perfect crystal: no variational model of the ground state is known which is able to reproduce qualitatively the long-range behaviour of this function, which turns out to be an exponential decaying function of the distance when computed via exact finite temperature and zero temperature Path Integral methods. In this paper we derive a functional form for the contribution of the zero-point motion of transverse phonons to the ground state microscopic wave function. Such an approach opens many possibilities of defining variational models, and we propose a first attempt, consisting of a modified Shadow Wave Function.