Variational theory of bulk4Hewith shadow wave functions: Ground state and the phonon-maxon-roton spectrum

We apply an efficient optimization scheme to shadow wave functions (SWF’s) for the ground state of liquid and solid ${}^4\mathrm{He}$. Results improve on previous variational energies in both phases. In the liquid, the gain over a wave function with only pair and triplet correlations increases with density, providing a quantitative estimate of the increasing effect of higher-order correlations. The discrepancy with the exact ground-state energy is nearly constant over a wide range of densities, yielding excellent estimates for the equilibrium, freezing, and melting densities. The optimal SWF’s can be represented with high precision by density-independent sets of variational parameters for the liquid and the solid phases. An extensive study of ground-state properties demonstrates the uniformly good quality of the variational description afforded by the optimized SWF’s. Based on such an accurate representation of the ground state, and including the correct long-range correlations due to the zero-point motion of phonons, we compute the excitation spectrum and the strength of the single quasiparticle excitation peak. While the use of optimized SWF’s confirms the accuracy of previous studies in the maxon-roton region, the proper treatment of long-range correlations allows us to extend to the phonon region the agreement between theory and experiment. We simulate relatively large systems in order to explore the long-wavelength regime in detail. Even down to wave vectors where the excitation energy exhibits a highly linear dispersion, which would suggest a harmonic phonon mode, the kinetic and potential energies of the excitation are far from verifying equipartition. This result is supported by additional diffusion Monte Carlo simulations within the fixed node approximation. We locate the onset of the harmonic regime for the long-wavelength excitations at an extremely small value of the wave vector, $k\lesssim 0.05\hspace{0.5em}{\mathrm{\AA }}^{-1}$.