Bose crystal as a standing sound wave

Abstract

A new class of solutions for Bose crystals with a simple cubic lattice consisting of N atoms is found. The wave function (WF) of the ground state takes the form \Psi_0=e^{S_{w}^{l}+S_{b}}*\prod_j [\sin{k_{l_x}x_{j}}\sin{k_{l_y}y_{j}}\sin{k_{l_z}z_{j}}], where e^{S_{b}} is the ground-state WF of a fluid, and \textbf{k}_l=(\pi/a_l, \pi/a_l, \pi/a_l) (a_l is the lattice constant). The state with a single longitudinal acoustic phonon is described by the WF \Psi_k=[\rho_{-k}+corrections + 7 permutations]\Psi_0, where the permutations give the terms with different signs of components of vector k. The structure of \Psi_k is such that the excitation corresponds, in fact, to the replacement of \textbf{k}_l in some triple of sines from \Psi_0 by \textbf{k}. Such a structure of \Psi_0 and \Psi_k means that the crystal is created by sound: the ground state of a cubic crystal is formed by N identical three-dimensional standing waves similar to a longitudinal sound. It is also shown that the crystal in the ground state has a condensate of atoms with \textbf{k}=\textbf{k}_l. The nonclassical inertia moment observed in crystals He-4 can be related to the synchronous tunneling of condensate atoms.