Perturbation theory of liquid helium-4 at zero temperature

Abstract

The formal pertubation theory of a Bose liquid, due to Beliaev, Hugenholtz, Pines, Gavoret, and Nozieres, is carefully examined in an effort to surmount the limitations of the Brueckner-Sawada theory of liquid He 4. In order to make the formal results as directly relevant as possible for Brueckner-theoretic calculations, all of the basic results are rederived by time-independent and number-conserving methods, starting from a linked-cluster expansion for bosons which is the exact analog of the Goldstone expansion for normal fermion systems. The analogy between the ground state and collective excitations of liquid He 4, and those of a normal Fermi liquid, is emphasized and explored in detail. Several aspects of the difficult problem of convergence are discussed. A specific partial summation procedure is recommended, the compact-cluster scheme, to deal with the short-range correlations within clusters of more than two particles. It is then argued that the very strong depletion of the condensate would lead to an extremely slow convergence for the most obvious class of approximations, and a remedy for this difficulty is proposed. The possibility of establishing a fairly detailed correspondence between the perturbation-theoretic wavefunction and the Jastrow wavefunction, including Feenberg-CBF extensions of the latter, is also pointed out.