Partial Dynamical Symmetry in a Fermion System

Abstract

The relevance of the partial dynamical symmetry concept for an interacting fermion system is demonstrated. Hamiltonians with partial SU(3) symmetry are presented in the framework of the symplectic shell-model of nuclei and shown to be closely related to the quadrupole-quadrupole interaction. Implications are discussed for the deformed light nucleus 20 Ne.PACS numbers: 21.60Fw, 21.60.Cs, 27.30+t Symmetries play an important role in dynamical systems. They provide labels for the classification of states, determine selection rules, and simplify the relevant Hamiltonian matrices. Algebraic, symmetry-based models offer significant simplifications when the Hamiltonian under consideration commutes with all the generators of a particular group ('exact symmetry') or when it is written in terms of the Casimir operators of a chain of nested groups ('dynamical symmetry') [1]. In both cases basis states belonging to inequivalent irreducible representations (irreps) of the relevant groups do not mix, the Hamiltonian matrix has block structure, and all properties of the system can be expressed in closed form. An exact or dynamical symmetry not only facilitates the numerical treatment of the Hamiltonian, but also its interpretation and thus provides considerable insight into the physics of a given system. Naturally, the application of exact or dynamical symmetries to realistic situations has its limitations. Usually the assumed symmetry is only approximately fulfilled, and imposing certain symmetry requirements on the Hamiltonian might result in constraints which are too severe and incompatible with experimentally observed features of the system. The standard approach in such situations is to break the symmetry. Partial Dynamical Symmetry (PDS) [2] corresponds to a particular symmetry-breaking for which the Hamiltonian is not invariant under the symmetry group and hence various irreps are mixed in its eigenstates, yet it possess a subset of 'special' solvable states which respect the symmetry. This new scheme has recently been introduced in bosonic systems and has been applied to the spectroscopy of deformed nuclei [3] and to the study of mixed systems with coexisting regularity and chaos [4]. It is the purpose of this Letter to demonstrate the relevance of the partial dynamical symmetry concept to fermion systems. More specifically, in the framework of the symplectic shellmodel of nuclei [5], we will prove the existence of a family of fermionic Hamiltonians with partial SU(3) symmetry. The PDS Hamiltonians are rotationally invariant and closely related to the quadrupole-quadrupole interaction; hence our study will shed new light on this important interaction. We will compare the spectra and eigenstates of the quadrupole-quadrupole and PDS Hamiltonians for the deformed light nucleus 20 Ne.The quadrupole-quadrupole interaction is an important ingredient in models that aim at reproducing quadrupole collective properties of nuclei. (11) 1m =L m , the orbital angular momentum operator). A fermion realization of these generators...

Partial dynamical symmetry and odd-even staggering in deformed nuclei

“…The essential idea is to relax the stringent conditions of complete solvability, so that only part of the eigenspectrum retains all the DS quantum numbers. Various types of bosonic and fermionic PDS are known to be relevant to nuclear spectroscopy [4][5][6][7][8][9][10][11][12][13]. In the present contribution we demonstrate the relevance of PDS to the odd-even staggering in the γ-band of 156 Gd [13].…”

Linking partial and quasi dynamical symmetries in rotational nuclei

“…This situation, referred to as partial dynamical symmetry (PDS) [4], was shown to be relevant to specific nuclei [4][5][6][7][8][9][10][11][12] and molecules [13]. In parallel, the notion of quasi dynamical symmetry (QDS) was introduced and discussed in the context of nuclear models [14][15][16][17][18][19][20][21].…”

Partial dynamical symmetry in an interacting fermion system

“…To illustrate that the PDS Hamiltonians discussed here are physically relevant, applications to realistic nuclear systems have to be considered. For 2~ and 12C such studies were carried out in [2] and [6], respectively: Energy spectra and eigen- HQ.Q 4-c3Xs + c4f(4 are shown. HIpDS has families of pure (and nearly pure) SU(3) eigenstates which can be organized into rotational bands; they are indicated in the figure.…”

“…Partial dynamical symmetry (PDS) describes an intermediate situation in which some eigenstates exhibit a symmetry which the associated Hamiltonian does not share. The objective of this approach is to remove undesired constraints from the theory while preserving the useful aspects of a dynamical Symmetry, such as solvability, for a subset of eigenstates [1,2]. Here an example of a PDS in an interacting fermion system is presented.…”