Similarity Renormalization Group Evolution of Many-Body Forces in a One-Dimensional Model

A one-dimensional system of bosons with short-range repulsion and mid-range attraction is used as a laboratory to explore the evolution of many-body forces by the Similarity Renormalization Group (SRG). The free-space SRG is implemented for few-body systems in a symmetrized harmonic oscillator basis using a recursive construction analogous to no-core shell model implementations. This approach, which can be directly generalized to three-dimensional nuclei, is fully unitary up to induced A-body forces when applied with an A-particle basis (e.g., A-body bound-state energies are exactly preserved). The oscillator matrix elements for a given A can then be used in larger systems. Errors from omitted induced many-body forces show a hierarchy of decreasing contribution to binding energies. An analysis of individual contributions to the growth of many-body forces demonstrates such a hierarchy and provides an understanding of its origins.