For many problems in nuclear, high-energy and solid-state physics a crucial step toward a solution involves a diagonalization or simplification of the Hamiltonian of a many-body quantum-mechanical system. Problems of interest include renormalization of the Hamiltonian to accommodate excluded basis spaces, decoupling of a model space region of special interest from irrelevant configurations, or near-diagonalization to invoke perturbative treatments. In such applications, the Similarity Renormalization Group (SRG) approach (or flow equations) can provide solutions where more conventional methods fail. We show that when SRG employs a near symmetry of a system, utilizing the Casimir invariant operators, it evolves the Hamiltonian of the system toward a (block-)diagonal form that can be realized in terms of a finite number of integrity basis operators. In particular, for atomic nuclei, an SRG flow equation that utilizes the SU(3) symmetry, which often dominates the nuclear dynamics, is found to naturally evolve a nucleon-nucleon (NN) interaction toward a block-diagonal form. A remarkable consequence is that a renormalized SU(3)-preserving many-body Hamiltonian expressed in terms of only the SU(3) integrity basis realizes energy spectra identical to the ones of the original realistic NN interaction, which reflects the low-energy symmetries as well as symmetry-breaking patterns of quantum chromodynamics. Such a scheme is especially suitable for ab initio SU(3) symmetry-adapted shell model calculations, which can further allow for perturbative treatments of the nuclear interaction. Moreover, the use of symmetries dramatically reduces the size of the problem, as the flow equation can be rewritten in terms of SU(3) reduced matrix elements of the NN interaction. This is crucial for maintaining the unitarity of the SRG transformations that generate many-body interactions. © Published under licence by IOP Publishing Ltd.
Publication Source (Journal or Book title)
Journal of Physics: Conference Series
Sviratcheva, K., Draayer, J., & Dytrych, T. (2011). Group-theoretical approach to interaction renormalization for ab initio solutions of the nuclear problem. Journal of Physics: Conference Series, 284 (1) https://doi.org/10.1088/1742-6596/284/1/012056