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Oxford University Press eBooks

SIGNIFICANT OTHERS

Thierry Giamarchi

Published Dec 18, 2003• 0 citations
Publisher

Abstract

This chapter looks at less usual realizations of Luttinger liquids than the ones coming from fermionic systems. The first realization is provided by interacting bosons. Indeed, in one dimension it is impossible to decouple completely the statistics from the interactions, and one- and quasi-one-dimensional bosons are also described by Luttinger liquids. The discussion shows how the techniques used to tackle one dimension can be used in higher dimensions as well. An important class of problems examined in this chapter is the case of impurities coupled with a three-dimensional electron gas. Josephson junction arrays and Bose condensates are also considered, along with X-ray, Kondo, and multi-channel Kondo problems. Applications of bosonization to the study of quantum impurities in three-dimensional Fermi liquids are analyzed.

DOI
10.1093/acprof:oso/9780198525004.003.0011

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This chapter looks at less usual realizations of Luttinger liquids than the ones coming from fermionic systems. The first realization is provided by interacting bosons. Indeed, in one dimension it is impossible to decouple completely the statistics from the interactions, and one- and quasi-one-dimensional bosons are also described by Luttinger liquids. The discussion shows how the techniques used to tackle one dimension can be used in higher dimensions as well. An important class of problems examined in this chapter is the case of impurities coupled with a three-dimensional electron gas. Josephson junction arrays and Bose condensates are also considered, along with X-ray, Kondo, and multi-channel Kondo problems. Applications of bosonization to the study of quantum impurities in three-dimensional Fermi liquids are analyzed.
This volume provides a detailed account of bosonization. The first part of the book examines the technical aspects of bosonization including one-dimensional fermions, the Gaussian model, the structure of Hilbert space in conformal theories, Bose-Einstein condensation in two dimensions, non-Abelian bosonization, and the Ising and WZNW models. The second part presents applications of the bosonization technique to realistic models including the Tomonaga-Luttinger liquid, spin liquids in one dimension and the spin-1/2 Heisenberg chain with alternative exchange. The third part addresses the problems of quantum impurities. Chapters cover potential scattering, the X-ray edge problem, impurities in Tomonaga-Luttinger liquids and the multi-channel Kondo problem.
This chapter discusses the technique of bosonization for studying systems of interacting fermions in one dimension. It reviews the low-energy properties of Fermi and Luttinger liquids and presents some of the relations between bosonic and fermionic operators in one dimension. The chapter explores these relations to calculate the correlation functions and the renormalization group properties of various operators for a system of spinless fermions. It explains 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 basic idea of bosonization is that there are certain objects which can be calculated either in a fermionic theory or in a bosonic theory, and the two calculations give the same answer. The chapter also discusses another application of bosonization, which is to study transport, in particular the DC conductivity in one-dimensional wires of interacting fermions.
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.
This chapter provides an overview of the basic ideas of interacting electrons and the peculiarities of one dimension. Interactions among particles are what endows the one-dimensional world with its special properties. Indeed, there are little differences between, for example, one-dimensional free electrons and their higher-dimensional counterparts. Before discussing one-dimensional systems, the effects of interactions in higher dimensions are briefly reviewed. The basic concepts and ideas of Fermi liquids are then explained, along with why one-dimensional systems are so different. This chapter also looks at the fermionic techniques used to tackle one dimension.
The bosonization technique is useful for solving the interacting problem and for extracting a certain number of properties such as power law behavior of the correlation functions. However, this derivation is, strictly speaking, based on a linearization of the spectrum close to the Fermi energy. One can thus wonder what will remain of the results of the previous chapter when it is impossible to stay confined close to the Fermi level, for example, when the interactions become strong. In order to answer this question, it is necessary to rederive the bosonization formulas, in a more phenomenological but more general way. This chapter demonstrates that a similar concept as Fermi liquids exists in one dimension. This concept, called Luttinger liquid, encompasses most of the physics of simple one-dimensional systems and will be the starting point for the study more complex situations. The concept of Luttinger liquids is explained using very low level techniques.
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.