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Cambridge University Press eBooks

The Bosonization Method in Nuclear Physics

FRANCESCO IACHELLO

Published Apr 6, 1995• 0 citations
Publisher

Abstract

The bosonization method developed in nuclear physics in the last 20 years is briefly reviewed. Introduction In the last 20 years, a considerable amount of work has gone into the development of bosonization methods for highly correlated finite fermion systems. This work has been stimulated by the phenomenological successes of the interacting boson model introduced in 1974 in which atomic nuclei composed of nucleons (fermions) are treated in terms of an interacting system of bosons (fermion pairs). In this article the logic scheme of the bosonization method in nuclear physics will be briefly reviewed. The purpose here is to provide the basic references upon which the review is built and sources where further references can be found. Bosonization methods were introduced much earlier than 1974 in connection with infinite Fermi systems. The method developed in nuclear physics has some similarities with those developed in other areas of physics, but it also has major differences. In nuclear physics one maps not only operators but also states and seeks a description not only of the ground state but also of the entire excitation spectrum. Furthermore, number projection plays a very important role contrary to the case of infinite systems where number projection is not relevant. These differences, and others, will be briefly remarked upon in this article. Strongly Interacting Finite Fermion Systems Fermions and the Shell Model Quite often in physics one has to deal with a system of interacting fermions.

DOI
10.1017/cbo9780511524240.019

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FRANCESCO IACHELLO, P. Van Isacker

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.

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From Interacting Nucleons to Hamiltonian Lattice Gauge Theories: A Unified Many-Body Treatment

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Comparing and contrasting nuclei and cold atomic gases

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The experimental revolution in ultracold atomic gas physics over the past decades has brought tremendous amounts of new insight to the world of degenerate quantum systems. Here we compare and contrast the developments of cold atomic gases with the physics of nuclei since many concepts, techniques, and nomenclatures are common to both fields. However, nuclei are finite systems with interactions that are typically much more complicated than those of ultracold atomic gases. The similarities and differences must therefore be carefully addressed for a meaningful comparison and to facilitate fruitful crossdisciplinary activity. We first consider condensates of bosonic and paired systems of fermionic particles with the mean-field description, but take great care to point out potential problems in the limit of small particle numbers. Along the way we review some of the basic results of Bose–Einstein condensate (BEC) and Bardeen–Cooper–Schrieffer (BCS) theory, as well as the BCS–BEC crossover and the Fermi gas in the unitarity limit, all within the context of ultracold atoms. Subsequently, we consider the specific example of an atomic Fermi gas from a nuclear physics perspective, comparing degrees of freedom, interactions, and relevant length and energy scales of cold atoms and nuclei. Next we address some attempts in nuclear physics to transfer the concepts of condensates in nuclei that can in principle be built from bosonic alpha-particle constituents. We also consider Efimov physics, a prime example of nuclear physics transferred to cold atoms, and consider which systems are more likely to show interesting bound state spectra. Finally, we address some recent studies of the BCS–BEC crossover in light nuclei and compare them to the concepts used in ultracold atomic gases. While many-body concepts such as BEC or BCS states are applicable in both subfields, we find that the interactions and finite particle numbers in nuclei can obscure the clear meaning they have in cold atoms. On the other hand, universal results from atomic physics should have an impact on certain limits of the nuclear domain. In particular, with advances in the trapping of few-body atomic systems we expect a more direct exchange of ideas and results.

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In order for a boson-expansion theory to be useful for practical purposes, it must satisfy at least two requirements: It must be in the form of a linked-cluster expansion, and the pure (ideal) boson states must be usable as basis states. Previously, we constructed such a boson theory and used it successfully for many realistic calculations. This construction, however, lacked mathematical rigor. In the present paper, we develop an entirely new approach, which results in the same boson expansions obtained earlier, but now in a mathematically rigorous fashion. The achievement of the new formalism goes beyond this. Its framework is much more general and flexible than was that of the earlier formalism, and it allows us to extend the calculations beyond what had been done in the past.NUCLEAR STRUCTURE Boson expansion theories, commutator method, Marumori-Yamamura-Tokunaga method, fermion system, norm matrix, linked-cluster expansion, ideal-boson state, physical boson state, even-even nucleus, odd-$A$ nucleus, nuclear collectivity.

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The bosonization method developed in nuclear physics in the last 20 years is briefly reviewed. Introduction In the last 20 years, a considerable amount of work has gone into the development of bosonization methods for highly correlated finite fermion systems. This work has been stimulated by the phenomenological successes of the interacting boson model introduced in 1974 in which atomic nuclei composed of nucleons (fermions) are treated in terms of an interacting system of bosons (fermion pairs). In this article the logic scheme of the bosonization method in nuclear physics will be briefly reviewed. The purpose here is to provide the basic references upon which the review is built and sources where further references can be found. Bosonization methods were introduced much earlier than 1974 in connection with infinite Fermi systems. The method developed in nuclear physics has some similarities with those developed in other areas of physics, but it also has major differences. In nuclear physics one maps not only operators but also states and seeks a description not only of the ground state but also of the entire excitation spectrum. Furthermore, number projection plays a very important role contrary to the case of infinite systems where number projection is not relevant. These differences, and others, will be briefly remarked upon in this article. Strongly Interacting Finite Fermion Systems Fermions and the Shell Model Quite often in physics one has to deal with a system of interacting fermions.
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.
What has since become known as the coupled cluster method (CCM) was invented some 35 years ago to calculate ground-state energies of closed-shell atomic nuclei.The method has since been extended in many different ways to the point where it is nowadays recognized as providing one of the most universally applicable, most powerful, and numerically most accurate of all ab initio methods in the microscopic quantum many-body theory of finite or infinite systems.We review here the applications within nuclear physics, and thereby show how the CCM provides a unifying approach of almost unique stature.The early calculations on both closed-and open-shell nuclei and on infinite nuclear matter are thus related to the recent extensions which provide both extremely accurate calculations for light nuclei and a framework for handling various problems in nuclear field theory.The latter include pion-nucleon field theory and such Hamiltonian lattice gauge field theories as the Abelian U(l) model and the non-Abelian SU(2) model.In all cases we stress the underlying concepts and the high quality of the results obtained.
We discuss the technique of bosonization for studying systems of interacting fermions in one dimension. After briefly reviewing the low-energy properties of Fermi and Luttinger liquids, we present some of the relations between bosonic and fermionic operators in one dimension. We use these relations to calculate the correlation functions and the renormalization group properties of various operators for a system of spinless fermions. We then apply the methods of bosonization to study the Heisenberg antiferromagnetic spin 1/2 chain, the Hubbard model in one dimension and transport in clean quantum wires and in the presence of isolated impurities.
The experimental revolution in ultracold atomic gas physics over the past decades has brought tremendous amounts of new insight to the world of degenerate quantum systems. Here we compare and contrast the developments of cold atomic gases with the physics of nuclei since many concepts, techniques, and nomenclatures are common to both fields. However, nuclei are finite systems with interactions that are typically much more complicated than those of ultracold atomic gases. The similarities and differences must therefore be carefully addressed for a meaningful comparison and to facilitate fruitful crossdisciplinary activity. We first consider condensates of bosonic and paired systems of fermionic particles with the mean-field description, but take great care to point out potential problems in the limit of small particle numbers. Along the way we review some of the basic results of Bose–Einstein condensate (BEC) and Bardeen–Cooper–Schrieffer (BCS) theory, as well as the BCS–BEC crossover and the Fermi gas in the unitarity limit, all within the context of ultracold atoms. Subsequently, we consider the specific example of an atomic Fermi gas from a nuclear physics perspective, comparing degrees of freedom, interactions, and relevant length and energy scales of cold atoms and nuclei. Next we address some attempts in nuclear physics to transfer the concepts of condensates in nuclei that can in principle be built from bosonic alpha-particle constituents. We also consider Efimov physics, a prime example of nuclear physics transferred to cold atoms, and consider which systems are more likely to show interesting bound state spectra. Finally, we address some recent studies of the BCS–BEC crossover in light nuclei and compare them to the concepts used in ultracold atomic gases. While many-body concepts such as BEC or BCS states are applicable in both subfields, we find that the interactions and finite particle numbers in nuclei can obscure the clear meaning they have in cold atoms. On the other hand, universal results from atomic physics should have an impact on certain limits of the nuclear domain. In particular, with advances in the trapping of few-body atomic systems we expect a more direct exchange of ideas and results.
We discuss the technique of bosonization for studying systems of interacting fermions in one dimension. After briefly reviewing the low-energy properties of Fermi and Luttinger liquids, we present some of the relations between bosonic and fermionic operators in one dimension. We use these relations to calculate the correlation functions and the renormalization group properties of various operators for a system of spinless fermions. We then apply the methods of bosonization to study the Heisenberg antiferromagnetic spin 1/2 chain, the Hubbard model in one dimension, and transport in clean quantum wires and in the presence of isolated impurities.
In order for a boson-expansion theory to be useful for practical purposes, it must satisfy at least two requirements: It must be in the form of a linked-cluster expansion, and the pure (ideal) boson states must be usable as basis states. Previously, we constructed such a boson theory and used it successfully for many realistic calculations. This construction, however, lacked mathematical rigor. In the present paper, we develop an entirely new approach, which results in the same boson expansions obtained earlier, but now in a mathematically rigorous fashion. The achievement of the new formalism goes beyond this. Its framework is much more general and flexible than was that of the earlier formalism, and it allows us to extend the calculations beyond what had been done in the past.NUCLEAR STRUCTURE Boson expansion theories, commutator method, Marumori-Yamamura-Tokunaga method, fermion system, norm matrix, linked-cluster expansion, ideal-boson state, physical boson state, even-even nucleus, odd-AAA nucleus, nuclear collectivity.