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Progress of Theoretical Physics

Property of Many-Phonon Norm Matrix

T. Suzuki, Masahiko Fuyuki, K. Matsuyanagi

Published Apr 1, 1979• 15 citations
Publisher

Abstract

It is shown that the many-phonon states built up from collective quasiparticle pairs satisfy the orthogonality condition in a good approximation, if the approximation scheme recently proposed by Holzwarth, Janssen and Jolos is applied to the intrinsic space in the quasiparticle state space, which is orthogonal to the pairing rotational and vibrational excitations.

DOI
10.1143/ptp.61.1082

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It is shown that the many-phonon states built up from collective quasiparticle pairs satisfy the orthogonality condition in a good approximation, if the approximation scheme recently proposed by Holzwarth, Janssen and Jolos is applied to the intrinsic space in the quasiparticle state space, which is orthogonal to the pairing rotational and vibrational excitations.
An operator A† that satisfies [H, A†] = h/ωA† converts a stationary-state eigenfunction of the Hamiltonian H into another eigenfunction with energy eigenvalue increased by h/ω. Such operators describe collective excitations of many-particle systems, and their properties can be used to construct an intrinsic Hamiltonian that is dynamically independent of the collective degrees of freedom, without introducing subsidiary conditions. The procedure developed by Lipkin, valid when h/ω is real and positive, is extended to make possible the construction of an intrinsic Hamiltonian when h/ω vanishes and A† is Hermitian, and also when h/ω is complex. The nuclear cranking model is shown to be a special case of the proposed general method for vanishing h/ω in which effective moments of inertia occur as eigenvalues of linear equations. Several examples are worked out in detail, all dealing with an interacting phonon-electron system in the random-phase approximation. Results derived are the explicit screened Coulomb interaction resulting from electronic plasma excitations, a verification of the renormalized phonon frequency spectrum and phonon-electron interaction derived in the adiabatic approximation, and the resulting screened Coulomb and phonon-induced electronic interactions obtained when plasma and phonon excitations are treated simultaneously.
A nonlinear generalization of the quasiparticle random phase approximation is constructed with the aim of describing anharmonic effects in the low-lying spectra of vibrational nuclei. Angular momentum zero pairing correlations of the ground state are described by a generalized BCS calculation based on a variational method. The dynamics of the vibrational states are described by the Heisenberg equations of motion for fermion pair operators; matrix elements of products of pair operators are treated by sum rule methods. Kinematical constraints are expressed by the algebra of the pair operators and by boson-like polynomial expansions for the quasiparticle conserving multipole operators in terms of the quasiparticle pair changing operators, these expansions enforcing the content of the Pauli principle. The physical basis for the choice of approximations is described. The final algorithm is organized so as to involve little more than repeated BCS-like and RPA-like calculations, converging to a self-consistent solution of the full nonlinear scheme. Detailed discussion is carried out for a modified form of the pairing plus quadrupole interaction Hamiltonian, but the method is not confined to this model.NUCLEAR STRUCTURE Approximate many-body methods applied to modified pairing plus quadrupole model. Description of vibrational spectrum of single closed shell nuclei. Equations of motion and algebraic methods.
We discuss a variational approach to collective excitations in a boson formalism based on quasiparticles. Bosons are defined in correspondence with pairs of quasiparticles and boson images of fermion operators are constructed by means of a mapping procedure of Marumori-type. Phonons of the type used within the random phase approximation (RPA) are introduced as Bogoliubov transformations of these bosons. The variables entering into the definition of these phonons as well as of the quasiparticle operators are fixed simultaneously by minimizing the expectation value of the boson Hamiltonian in the vacuum of the phonons. The approach is tested within an exactly solvable two-level model which is characterized by a pairing Hamiltonian. A quite good agreement is found for the energies of the ground state and of the first 0+{0}^{+}0+ excited state. The comparison with the Bardeen-Cooper-Schrieffer method and the quasiparticle RPA as well as with some recent self-consistent RPA-type approaches is discussed.
The Hamiltonian most often used to describe a system of electrons interacting with lattice vibrations has the asymmetry that the electrons are represented by the creation and annihilation operators which arise from quantum field theory whereas the lattice vibrations are represented by the one-particle ladder operators of the one-dimensional harmonic-oscillator problem. Although the equal spacing of the harmonic-oscillator energy levels and the commutation relations satisfied by the oscillator ladder operators have led to an interpretation of this formalism in terms of the "creation and annihilation" of phonons, the phonons thus defined are not the quasiparticles which are obtained by a field-theoretical formulation of the lattice Hamiltonian in the harmonic approximation. A Hamiltonian is derived in this paper in which both the electrons and the lattice (in the harmonic approximation) are represented in field-theoretical formalism. A lattice creation operator in this formulation actually creates a vibrational mode (in a given state) from the physical vacuum, in contrast to a ladder operator which merely generates transitions between two states of an already existing vibrational mode. As a consequence, the number operator for the lattice quasiparticles commutes with the Hamiltonian, a property which is useful in certain applications. It is shown that even though our Hamiltonian is formally different from the ordinary one, it yields equivalent results for the state vectors of physical interest.
Collective excitations are one of the most common and interesting features of many-body systems. Of particular interest are those collective modes which can be interpreted in terms of vibrations. Nuclei show a large variety of such modes, both low-lying and high-lying. In particular the giant dipole resonance is due to the coherent motion of protons against neutrons. Its analogue in metal clusters is the dipole plasmon excitation, due to the oscillation of electrons against the positive ions. The Random Phase Approximation (RPA) is extensively used as a microscopic theory to study the basic properties of these collective excitations. In the derivation of RPA use is made of the Quasi Boson Approximation (QBA). It consists in replacing the expectation values in the correlated ground state with those in the uncorrelated (Hartree-Fock) one. Strictly related to QBA is the feature of RPA of predicting a harmonic spectrum. On the other hand, the existence of anharmonicities in the multiphonon spectra of nuclei and their relevance in various physical processes are well established. Overcoming QBA has represented the starting point of many attempts aiming at improving RPA. A recently developed approach in this line will be presented and discussed. A very nice property of RPA is to satisfy Energy Weighted Sum Rules (EWSR). However, some violations are present in several extensions of RPA. We will show how to cure this shortcoming.
A many•body technique called the dynamical nuclear field theory is developed by taking the example of pair rotation.The time-dependent one-body Hamiltonian contains the collective coordinates; the magnitude and phase aIigleof the energy gap.Effective forms of the original Hamiltonian and other operators are derived for a direct-product space of the collective states and a restricted set of the particle states defined by the driving Hamiltonian in the intrinsic frame.Pair rotations of even-and odd-A systems are treated in the lowest order approximation.The particle-rotor picture and the cranking model are shown to be equivalent.*) Brief report on DNFT in the Tamm-Dancoff approximation can be found in Ref. 13).