T-Matrix perturbation theory of solid helium

Abstract

A detailed t-matrix theory is developed for the ground state of solid helium, based on partial summations of the Brueckner-Goldstone expansion. Self-consistent Wannier orbitals are used to define the single-particle basis. This raises some theoretical and practical questions which are carefully examined. (The suitability of this type of basis for Mott insulators is also pointed out.) We draw attention to the existence of a degenerate version of the Goldstone linked-cluster expansion. This can be used (1) to handle the degeneracy problem arising from the possibility of multiply occupied sites in the unperturbed wavefunction, and (2) to justify and derive a model Hamiltonian of the Heisenberg form, in order to describe the spin interactions in solid 3He. A specific partial-summation scheme is then developed. Detailed arguments show that this should provide very rapid convergence, for both the long-range attractive and the short-range repulsive aspects of v. By carefully including the ladders of particle-hole and hole-hole interactions, we obtain a wave equation for the two-body correlations (Bethe-Goldstone equation) which differs from those of previous t-matrix theories. Perturbative comparisons are made with these earlier theories, and some discrepancies between the previous calculations and experiment are qualitatively resolved. The problem of the exchange energy of solid 3He is carefully examined, and several corrections to previous theories are found. Our formula for the exchange energy is manifestly antiferromagnetic, no matter how strong the short-range repulsion may be. The previous t-matrix calculations have all approximated the normally occupied orbitals by Gaussian functions. Our theory satisfies a many-body variational principle, and this is shown to provide a useful criterion for optimizing the size parameter α 2 in such Gaussian-orbital calculations. A linked-cluster expansion is developed for the boson system of solid 4He, and we demonstrate that all to the foregoing results can be readily adapted to this Bose system. Wherever possible, comparisons are made with the results obtained from the Jastrow variational approach. The problems of phonon correlations (real and virtual) are briefly discussed.