Boson correlation energies via variational minimization with the two-particle reduced density matrix: ExactN-representability conditions for harmonic interactions

A many-body theory for interacting bosons is developed within the framework of minimizing the ground-state energy with respect to the two-particle reduced-density matrix (2-RDM) subject to N-representability conditions. The N-representability conditions, which ensure that the 2-RDM may be derived from an N-particle wave function, are imposed through a hierarchy of positivity conditions where the p-positivity conditions restrict the metric matrices for $p∕2$-body operators to be positive semidefinite. Using two-positivity, we minimize the ground-state energies of $5--10 000$ harmonically interacting bosons in a harmonic external potential. The energies and 2-RDMs obtained are in agreement with the exact solution except for round-off errors, which implies that for this class of boson interactions two-positivity conditions alone yield exact results for any interaction strength. The ground-state energies obtained at strong interactions are more accurate than many-body perturbative techniques by many orders of magnitude.