Probing the Symmetries of the Dirac Hamiltonian with Axially Deformed Scalar and Vector Potentials by Similarity Renormalization Group

Abstract

Symmetry is an important and basic topic in physics. The similarity renormalization group theory provides a novel view to study the symmetries hidden in the Dirac Hamiltonian, especially for the deformed system. Based on the similarity renormalization group theory, the contributions from the nonrelativistic term, the spin-orbit term, the dynamical term, the relativistic modification of kinetic energy, and the Darwin term are self-consistently extracted from a general Dirac Hamiltonian and, hence, we get an accurate description for their dependence on the deformation. Taking an axially deformed nucleus as an example, we find that the self-consistent description of the nonrelativistic term, spin-orbit term, and dynamical term is crucial for understanding the relativistic symmetries and their breaking in a deformed nuclear system.

Pseudospin symmetry in nuclear structure and its supersymmetric representation

“…[130,131] for the spherical case and Ref. [132] for the axially deformed case. For the spherical case, the effective Hamiltonian for the nucleons in the Fermi sea expanded up to the (1/M 3 )-th order reads [130] …”

“…For that, one of the possible ways is to investigate the Dirac equation with the similarity renormalization group evolution, which has been done in Ref. [132] for the deformed systems. The supersymmetric representation that follows is probably nontrivial, because it involves multidimensional supersymmetric quantum mechanics [163,164].…”

“…In this Subsection, we will first introduce the basic idea of similarity renormalization group for the Dirac Hamiltonian [130,131,132], then present the perturbative nature of pseudospin symmetry by combining the supersymmetric quantum mechanics, the similarity renormalization group, and the perturbation calculations [136,137].…”

“…[130,131,132] bridged the gap between the perturbation calculations and the SUSY description of PSS by using the SRG technique. The idea of SRG [133,134,135] is to drive the Hamiltonian toward a band-diagonal form via the so-called flow equation and unitary transformations that suppress offdiagonal matrix elements.…”

“…Recent works by Guo and coauthors [130,131,132] bridged the gap between the perturbation calculations and the supersymmetric description of pseudospin symmetry by using the similarity renormalization group (SRG) [133,134,135] for transforming the Dirac Hamiltonian into a diagonal form. The effective Hamiltonian expanded in a series of 1/M is Hermitian, which makes the perturbation calculations feasible.…”