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arXiv:1306.6201

Random Phase Approximation without Bogoliubov Quasi-particles

Lu Jia

Published Jun 26, 2013• 0 citations
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Abstract

A new version of random phase approximation is proposed for low-energy harmonic vibrations in nuclei. The theory is not based on the quasi-particle vacuum of the BCS/HFB ground state, but on the pair condensate determined in Ref. [4]. The current treatment conserves the exact particle number all the time. As a first test the theory is considered in two special cases: the degenerate model (large pairing limit) and the vanished-pairing limit.

DOI
10.48550/arxiv.1306.6201

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A new version of random phase approximation is proposed for low-energy harmonic vibrations in nuclei. The theory is not based on the quasi-particle vacuum of the BCS/HFB ground state, but on the pair condensate determined in Ref. [4]. The current treatment conserves the exact particle number all the time. As a first test the theory is considered in two special cases: the degenerate model (large pairing limit) and the vanished-pairing limit.
In the theoretical investigation of nonrelativistic feimion systems, several characteristic approximation schemes play a decisive role. Examples are the BCS theory of superconductivity, the Hartree-Fock method in atomic and molecular physics and its generalization by means of the Bogoliubov theory (containing also BCS as a special case) used in nuclear physics. Furthermore, the Random-Phase-Approximation (RPA) is of fundamental importance in plasma physics as well as in the explanation of the well known vibrational states of nuclei. In this connection, it should be pointed out that the socälled nuclear models, which to a certain extent brought nuclear physics into miscredit, represent in fact nothing else but a more or less intuitive substitute for special well defined approximation schemes: the introduction of an average potential in heavy nuclei stands for Hartree-Fock, whereas the so-called surface vibrations visualize RPA. In addition, the possibility to describe elementary excitations by the naive concept of independent particle motion is justified theoretically by the Bogoliubov transformation which leads to the so-called quasi-particles.
A previous study of the pairing interaction Hamiltonian by means of the equations of motion is extended. In the prior note, ground-state and single-quasiparticle properties were treated by an approximation method which was strictly particle conserving. According to the general theory, further improvement of the ground-state energies and single-particle occupation probabilities requires information about excited states of seniority zero and unity. A nonlinear, number-conserving random-phase approximation is developed for this purpose. Numerical calculations are carried out in a linearized version which is almost equivalent to the usual quasi-boson approximation. As part of the approximation, the conservation-of-particles condition is relaxed for the excited states. Results of the numerical study establish the consistency of the assumptions made and yield improved ground-state and single-particle properties. By comparison with the few models for which exact numerical results are available, the results of this paper are seen to be comparable with or superior to those of other improved approximation methods so far reported, especially so as the number of particles increases. Finally, it is pointed out that just as the results of the previous paper provide the input data for the present work, the results reported here can serve in like manner for more elaborate approximations, which are proposed.
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.
A new microscopic theory for nuclear pairing is proposed through the generalized density matrix formalism. Instead of the variation principle, we start from the Heisenberg equations of motion for density matrix operators ${K}_{12}\ensuremath{\equiv}{a}_{2}{a}_{1}$. The theory generalizes the well-known particle-particle random-phase approximation. Its analytical equations are as simple as that of the BCS theory and can be solved within a similar computer time if the model space is not too large. The current theory conserves the exact particle number and is valid at arbitrary pairing strength (including those below the BCS critical strength). It is also of interest to other mesoscopic systems such as ultrasmall metallic grains.
The properties of the pairing interaction in the shell model framework are considered with the aid of an exact numerical solution utilizing the quasispin symmetry. We emphasize the features which are out of reach for the usual approximate techniques based on the BCS approach supplemented by the random phase approximation treatment of pair vibrations, especially in the region of weak pairing where the BCS plus random-phase-approximation theory fails. Chaotic aspects of the mixing generated by the pairing interaction are studied. The level repulsion and large information entropy of the eigenstates coexist with the absence of thermalization of single-particle motion. The full spectrum of pair vibration on average displays a spin dependence similar to that for a rigid rotor.