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Physical Review C

Test of the composite particle representation theory

Kexie Wang, Guangzhou Liu, David F. Collinson, D.H. Feng, Cheng-Li Wu

Published May 1, 1991• 1 citation
Publisher

Abstract

The composite particle representation theory (CPRT) proposed by Wu and Feng is examined by comparing its results to the shell-model calculations of Wildenthal and Chung for the s-d shell nuclei $^{20}\mathrm{O}$, $^{20}\mathrm{F}$, and $^{20}\mathrm{Ne}$. Each identical nucleon pair is treated in the CPRT as a boson. In all cases, the CPRT results are identical to the shell model. It was found that by evaluating the normalizations and the error sums of the CPRT wave functions, the spurious states, a subject of vital importance for any mapping theory, can be and are correctly removed. This shows that the composite particle representation is equivalent to the conventional representation in quantum mechanics. Thus the CPRT can provide an exact theoretical framework to treat particle clusters as ideal bosons (or fermions, depending on the nature of the clusters) in a many-body system, and is particularly useful for any quantum systems where clusterization dominates.

DOI
10.1103/physrevc.43.2268

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The composite particle representation theory (CPRT) proposed by Wu and Feng is examined by comparing its results to the shell-model calculations of Wildenthal and Chung for the s-d shell nuclei 20O^{20}\mathrm{O}20O, 20F^{20}\mathrm{F}20F, and 20Ne^{20}\mathrm{Ne}20Ne. Each identical nucleon pair is treated in the CPRT as a boson. In all cases, the CPRT results are identical to the shell model. It was found that by evaluating the normalizations and the error sums of the CPRT wave functions, the spurious states, a subject of vital importance for any mapping theory, can be and are correctly removed. This shows that the composite particle representation is equivalent to the conventional representation in quantum mechanics. Thus the CPRT can provide an exact theoretical framework to treat particle clusters as ideal bosons (or fermions, depending on the nature of the clusters) in a many-body system, and is particularly useful for any quantum systems where clusterization dominates.
Abstract A generalization of the quantum mechanical representation transformation is presented, in this paper. It is shown that, as an important example, the Boson-expansion method, commonly employed in nuclear physics, corresponds to such a generalized transformation. Using this generalization, we were able to construct a special representation called the “Composite Particle Representation”. In the composite, particle representation, the composite particle degrees of freedom are included, as well as the original particle degrees of freedom. The former is introduced in order that the motion of certain particle clusters can be described as separate entities in a many-body system. This representation is shown to be exactly equivalent to the usual quantum mechanical representation which includes only the original particle degrees of freedom. Many applications of this theory are expected, in particular in the study of hadrons from the quark point of view and the Interacting Boson Model in nuclei.
We establish the quantum mechanics of composite fermions based on the dipole picture initially proposed by Read. It comprises three complimentary components: a wave equation for determining the wave functions of a composite fermion in ideal fractional quantum Hall states and when subjected to external perturbations, a wave-function for mapping a many-body wave function of composite fermions to a physical wave function of electrons, and a microscopic approach for determining the effective Hamiltonian of the composite fermion. The wave equation resembles the ordinary Schrödinger equation but has drift velocity corrections that are not present in the Halperin-Lee-Read theory. The wave-function constructs a physical wave function of electrons by projecting a state of composite fermions onto a half-filled bosonic Laughlin state of vortices. Remarkably, Jain's wave-function can be reinterpreted as the new in an alternative wave-function representation of composite fermions. The dipole picture and the effective Hamiltonian can be derived from the microscopic model of interacting electrons confined in a Landau level, with all parameters determined. In this framework, we can construct the physical wave function of a fractional quantum Hall state deductively by solving the wave equation and applying the wave-function , based on the effective Hamiltonian derived from first principles, rather than relying on intuition or educated guesses. For ideal fractional quantum Hall states in the lowest Landau level, the approach reproduces the well-established results of the standard theory of composite fermions. We further demonstrate that the reformulated theory of composite fermions can be easily generalized for flat Chern bands. Published by the American Physical Society 2024
The validity of the essential ingredients of the microscopic formulation of the interacting boson model: (i) truncation of the shell-model or fermion space and (ii) the procedures for mapping the truncated fermion space on to the boson space, is tested over various (spherical, deformed and transitional) regions both in the simplistic single j-shell, identical boson systems and the more realistic multi j-shell, proton-neutron systems depending on the feasibility of such calculations. In the situations where the truncation and mapping procedures fail to reproduce the exact shell-model results, it is demonstrated that renormalization and higher order mapping procedures cure the associated problems, respectively and provide excellent agreements. In addition, a Hamiltonian of the interacting boson-fermion model which is relevant for a study of odd mass nuclei is derived for a single j-shell case in the shell-model framework, which yields results identical to those in the shell-model for the monopole pairing interaction.
The many-body problem for a system of composite particles is formulated in a way which takes explicit account of the composite nature of these particles and allows a clear separation between interatomic and intraatomic interactions. A second-quantization representation, fully equivalent to the conventional representation in which nuclei and electrons appear explicitly, is developed in terms of atomic annihilation and creation operators satisfying elementary Bose or Fermi commutation relations. All effects of the composite nature of the atoms are exactly contained in the interatomic and intraatomic matrix elements and in certain exchange integrals. An application is made to the problem of Bose condensation of fermion pairs.
The validity of the essential ingredients of the microscopic formulation of the interacting boson model: (i) truncation of the shell-model or fermion space and (ii) the procedures for mapping the truncated fermion space on to the boson space, is tested over various (spherical, deformed and transitional) regions both in the simplistic single j-shell, identical boson sys­ tems and the more realistic multi j-shell, proton-neutron systems depending on the feasibility of such calculations. In the situations where the truncation and mapping procedures fail to reproduce the exact shell-model results, it is demonstrated that renormalization and higher order mapping procedures cure the associated problems, respectively and provide excellent agreements. In addition, a Hamiltonian of the interacting boson-fermion model which is relevant for a study of odd mass nuclei is derived for a single j-shell case in the shell-model framework, which yields results identical to those in the shell-model for the monopole pairing interaction.