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Physical Review C

Fermion-boson interactions in a solvable model

Osvaldo Civitarese, M. Reboiro

Published Nov 1, 1998• 10 citations
Publisher

Abstract

The interplay between fermionic and bosonic degrees of freedom, in the context of many-body systems and with reference to applications in nuclear and elementary particle physics, is revisited. Starting from the formulation due to Geyer, Hahne, and co-workers, bifermion operators which obey bosonic transformation rules are introduced. Fermion-boson coupling terms are included in the original Hamiltonian. Exact solutions to this Hamiltonian are compared to approximate ones. The occurrence of fermion and boson condensates is discussed.

DOI
10.1103/physrevc.58.2787

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The interplay between fermionic and bosonic degrees of freedom, in the context of many-body systems and with reference to applications in nuclear and elementary particle physics, is revisited. Starting from the formulation due to Geyer, Hahne, and co-workers, bifermion operators which obey bosonic transformation rules are introduced. Fermion-boson coupling terms are included in the original Hamiltonian. Exact solutions to this Hamiltonian are compared to approximate ones. The occurrence of fermion and boson condensates is discussed.
Introduction In many cases in physics, one has to deal simultaneously with collective and single-particle excitations of the system. The collective excitations are usually bosonic in nature while the single-particle excitations are often fermionic. One is therefore led to consider a system which includes bosons and fermions. In this book we discuss applications of a general algebraic theory of mixed Bose- Fermi systems to atomic nuclei. The collective degrees of freedom here can be described in terms of a system of interacting bosons as discussed in a previous book (Iachello and Arima, 1987), henceforth referred to as Volume 1. The single-particle degrees of freedom represent the motion of individual nucleons in the average nuclear field. They are described in terms of a system of interacting fermions. The coupling of fermions and bosons leads to the interacting boson-fermion model which has been used extensively in recent years to discuss the properties of nuclei with an odd number of nucleons. The interacting boson-fermion model was introduced by Arima and one of us in 1975 (Arima and Iachello, 1975). It was subsequently expanded by Iachello and Scholten (1979) and cast into a form more readily amenable to calculations. As in the corresponding case of even-mass systems, the algebra of creation and annihilation operators can be realized in several ways. One of these is the Hoistein-Primakoff realization which leads to a slightly different version of the interacting boson-fermion model called the truncated quadrupole phonon-fermion model (Paar, 1980; Paar and Brant, 1981), based on the boson realization introduced by Janssen, Jolos and Donau in 1974 and discussed in Sect. 1.4.6 of Volume 1.
In this thesis the method of bosonization of fermionic many-body systems in any number of dimensions is developed. After an introduction to the problem in chapter 1 and to the method of bosonization in chapter 2 (with its application to the non-interacting problem) the Fermi liquid behavior of interacting fermions in dimensions higher than one is discussed in chapter 3. In chapter 4 I rederive the well-known bosonization of the Tomonaga-Luttinger model in one dimension within the framework discussed in the preceding chapters. In chapter 5 I present the bosonization of elementary excitations of exactly solvable one dimensional systems with special emphasis on the Hubbard chain. Chapter 6 contains the conclusions.
A new formalism of the nuclear many-body problem is established in which one ean work with various physical state vector spaces consisting of both Fermions and Bosons, all being equivalent to the original one consisting only of Fermions. With the help of the usual commutation relations and anticommutation relations between the annihilation and creation operators of Boson and Fermion, a generalized state vector space is established, which contains all the physical spaces each being equivalent to the original physical space in terms of pure Fermion operators. Transformation between the state vectors in various equivalent physical spaces are constructed. Basic operators in the original state space are transformed into effective operators in the new physical spaces.
Generalized many-fermion versions of the independent-boson model with fixed or variable fermion number are solved exactly. In particular, the linear-response characteristics of these systems are discussed. Exact results for physical quantities, such as the partition function and the two-time correlation function for single-fermion operators, as needed in the Kubo formula, are given and discussed. These nontrivial models may serve as simple model systems for interacting many-particle systems and as a testing ground for approximation schemes for more complex coupled many-particle systems.