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arXiv:2211.07799 · Progress of Theoretical and Experimental Physics

Number conservation in odd-particle number random phase approximation and extensions

M. Tohyama

Published Jan 11, 2023• 0 citations
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Abstract

Abstract The number conservation law in the odd-particle number random-phase approximation (oRPA) and its extension (EoRPA) is studied by applying them to a pairing model and 16O. It is found in the application to 16O that the number conservation law is not fulfilled in oRPA or EoRPA and that it is drastically improved in EoRPA due to the inclusion of ground-state correlation effects. PACS numbers: 21.60.Jz

DOI
10.1093/ptep/ptad005

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Abstract The number conservation law in the odd-particle number random-phase approximation (oRPA) and its extension (EoRPA) is studied by applying them to a pairing model and 16O. It is found in the application to 16O that the number conservation law is not fulfilled in oRPA or EoRPA and that it is drastically improved in EoRPA due to the inclusion of ground-state correlation effects. PACS numbers: 21.60.Jz
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.
Coupled equations for even and odd particle number correlation functions are set up via the equation of motion method. For the even particle number case this leads to self-consistent random-phase approximation equations already known from the literature. From the equations of the odd particle number case the single-particle occupation probabilities are obtained in a self-consistent way. This is the essential new procedure of this work. Both even and odd particle number cases are based on the same correlated vacuum and, thus, are coupled equations. Applications to the Lipkin model and to the one-dimensional Hubbard model give very good results.
The self-consistent random-phase approximation (SCRPA) is reexamined within a multilevel-pairing model with double degeneracy. It is shown that the expressions for occupation numbers used in the original version of the SCRPA violate the particle number for nonsymmetric particle-hole (ph) spectra. A renormalization is introduced to restore the particle number, which leads to the expressions of occupation numbers similar to those derived by Hara et al. for the ph case. The results of calculations within the ph symmetric case show that this number-conserving SCRPA yields the energies of the ground state and the first excited state of the system with $\ensuremath{\Omega}+2$ particles relative to the ground state of the system with $\ensuremath{\Omega}$ particles in close agreement with those obtained within the original SCRPA. However, it gives a slightly larger correlation energy.
A method is proposed to improve the treatment of the ground-state correlations in a finite Fermi system compared with the standard random phase approximation (RPA) or earlier suggested renormalized RPA. The correlations lead to nonzero quasiparticle occupancies in the ground state. The method employs modified quasiparticles obtained by a canonical transformation of usual quasiparticles explicitly involving the quasiparticle occupation numbers. A set of equations is derived, which allows one to determine these occupation numbers along with the RPA modes. The formalism is illustrated with the Lipkin-Meshkov-Glick model, and a model for superconducting pairing at a finite temperature. With the new approach, the ground-state correlations are significantly reduced, the energy of the first excited state becomes closer to the exact solution around the region where the RPA collapses, and the superconducting gap monotonously decreases instead of the sharp phase transition. We discuss the effective equivalence of the interaction effects and variation of temperature for the ground-state correlations.
Particle number fluctuations in the quasiparticle random-phase approximation (QRPA) and the renormalized quasiparticle random-phase approximation are investigated. A closed expression for the ground state wave function is used, which allows an exact evaluation of the ground state phonon number and correlation function, being valid to any order in the backward amplitudes. In realistic calculations a sudden increase in the particle number fluctuations is found in both approximations when the particle-particle interaction approaches the value at which the QRPA collapses. This behavior is strongly correlated with the ground state quasiparticle content and can be understood as a signature of the phase transition previously found in simpler exactly solvable models.
A limitation common to all extensions of random-phase approximation including only particle-hole configurations is that they violate to some extent the energy weighted sum rules. Considering one such extension, the improved RPA (IRPA), already used to study the electronic properties of metallic clusters, we show how it can be generalized in order to eliminate this drawback. This is achieved by enlarging the configuration space, including also elementary excitations corresponding to the annihilation of a particle (hole) and the creation of another particle (hole) on the correlated ground state. The approach is tested within a solvable 3-level model.