Exploration of relativistic symmetry by the similarity renormalization group

Abstract

The similarity renormalization group is used to transform Dirac Hamiltonian into a diagonal form, which the upper(lower) diagonal element becomes an operator describing Dirac (anti)particle. The eigenvalues of the operator are checked in good agreement with that of the original Hamiltonian. Furthermore, the pseudospin symmetry is investigated. It is shown that the pseudospin splittings appearing in the non-relativistic limit are reduced by the contributions from these terms relating the spin-orbit interactions, added by those relating the dynamical terms, and the quality of pseudospin symmetry origins mainly from the competition of the dynamical effects and the spin-orbit interactions. The spin symmetry of antiparticle spectrum is well reproduced in the present calculations.PACS numbers: 21.10. Hw,21.10.Pc,03.65.Pm,05.10.Cc Many years ago a quasidegeneracy was observed in heavy nuclei between single-nucleon doublets with quantum numbers (n, l, j = l + 1/2) and (n − 1, l + 2, j = l + 3/2) where n, l, and j are the radial, the orbital, and the total angular momentum quantum numbers, respectively [1,2]. The quasidegenerate states were suggested to be pseudospin doublets j =l ±s with the pseudo orbital angular momentuml = l + 1, and the pseudospin angular momentums = 1/2, and have explained a number of phenomena in nuclear structure. Because of these successes, there have been comprehensive efforts to understand the origin of this symmetry. Until 1997, it was identified as a relativistic symmetry [3]. Nevertheless, there is still a large amount of attention on this symmetry. The pseudospin symmetry (PSS) of nuclear wave functions was tested in Refs. [4,5] with conclusion supporting the claim in Ref. [3]. The existence of broken PSS was checked in Refs. [6,7], where the quasidegenerate pseudospin doublets were confirmed to exist near the Fermi surface for spherical and deformed nuclei. The isospin dependence of PSS was investigated in Ref. [8], where it is found that PSS is better for exotic nuclei with a highly diffuse potential. PSS was shown to be approximately conserved in medium-energy nucleon scattering from even-even nuclei [9][10][11]. In combination with the analytic continuation method, the resonant states were exposed to hold the PSS in Refs. [12,13]. In Ref.[14], the conditions which originate the spin and pseudospin symmetries in the Dirac equation were shown to be the same that produce equivalent energy spectra of relativistic spin-1/2 and spin-0 particles in the presence of vector and scalar potentials. Furthermore, the symmetries and super-symmetries of the Dirac Hamiltonian were checked for particle moving in the spherical or axially-deformed scalar and vector potentials [15]. More reviews on the PSS can be found in the literature [16] and the references therein. Recently, a perturbation method was adopted to investigate the spin and pseudospin symmetries by dividing the Dirac Hamiltonian into the part of possessing the exact (pseudo)spin symmetry and that of breaking the symmetry [17].Despite the...

Pseudospin symmetry in nuclear structure and its supersymmetric representation

“…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].…”

“…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.…”

Further investigation of relativistic symmetry with the similarity renormalization group

“…To extract the different components from H D , the SRG is used to transform the Dirac Hamiltonian into a block-diagonal form. The details can be referred to the literature [36]. The diagonal Dirac Hamiltonian is written as…”

“…Recently, we have examined the PSS by use of the similarity renormalization group and shown explicitly the relativistic origin of this symmetry [36]. We have also studied the relativistic effect of this symmetry [37].…”

“…In this paper, we explore the dependence of the PSS on the shape of the potential and the quantum numbers of the states by using this Hamiltonian obtained in Ref. [36] in order to disclose the influence of every component, especially those relating the dynamical effect and the spin-orbit interactions.…”