Simplified approach to the repulsive Bose gas from low to high densities and its numerical accuracy

In 1963, a simplified approach was developed to study the ground-state energy of an interacting Bose gas with a purely repulsive potential. It consists in the derivation of an equation, which is not based on perturbation theory and which gives the exact expansion of the energy at low densities. This equation is expressed directly in the thermodynamic limit and only involves functions of three variables, rather than 3N. Here, we revisit this approach, introduce two more equations, and show that these yields accurate predictions for various observables for all densities for repulsive potentials with positive Fourier transform. Specifically, in addition to the ground-state energy, we have shown that the simplified approach gives predictions for the condensate fraction, two-point correlation function, and momentum distribution. We have carried out a variety of tests by comparing the predictions of the equations with quantum Monte Carlo calculations for exponential interaction potentials as well as a different, finite range potential of positive type, and have found remarkable agreement. We thus show that the simplified approach provides an alternative theoretical tool to understand the behavior of the many-body Bose gas, not only in the small and large density ranges, which have been studied before, but also in the range of intermediate density, for which much less is known.