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arXiv:0708.0003

One-dimensional quantum wires: A pedestrian approach to bosonization

Sebastian Eggert

Published Aug 1, 2007• 33 citations
arXivHTMLPDF

Abstract

In these lecture notes we will consider systems in which the motion of electrons is confined to one dimension (1D). In these so-called quantum wires electron-electron interaction effects play an important role because the restricted dimensions enhance the scattering between the electrons and completely destroy the quasi-particle picture. New density wave excitations appear that are described by bosonic operators. Here, we will develop this bosonic description, following a pedestrian approach which does not require any previous knowledge in field theory methods. These notes therefore serve as a detailed introduction into bosonization by carefully deriving the most fundamental formulas.

DOI
10.48550/arxiv.0708.0003

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The Luttinger liquid

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One-dimensional Fermi systems We will now consider the case of one-dimensional (1D) Fermi systems for which the Landau theory fails. The way it fails is quite instructive since it reveals that in one dimension these systems are generally at a (quantum) critical point, and it will also teach us valuable lessons on quantum criticality. It will also turn out that the problem of 1D Fermi systems is closely related to the problem of quantum spin chains. This is a problem that has been discussed extensively by many authors, and there are several excellent reviews on the subject (Emery 1979; Haldane, 1981). Here I follow in some detail the discussion and notation of Carlson and coworkers (Carlson et al ., 2004). One-dimensional (and quasi-1D) systems of fermions occur in several experimentally accessible systems. The simplest one to visualize is a quantum wire . A quantum wire is a system of electrons in a semiconductor, typically a GaAs–AlAs heterostructure built by molecular-beam epitaxy (MBE), in which the electronic motion is laterally confined along two directions, but not along the third. An example of such a channel of length L and width d (here shown as a two-dimensional (2D) system) is seen in Fig. 6.1. Systems of this type can be made with a high degree of purity with very long (elastic) mean free paths, often tens of micrometers or even longer. The resulting electronic system is a 1D electron gas (1DEG). In addition to quantum wires, 1DEGs also arise naturally in carbon nanotubes, where they are typically multi-component (with the number of components being determined by the diameter of the nanotube).

Cambridge University Press eBooks•Feb 28, 2013•3 citations
In these lecture notes we will consider systems in which the motion of electrons is confined to one dimension (1D). In these so-called quantum wires electron-electron interaction effects play an important role because the restricted dimensions enhance the scattering between the electrons and completely destroy the quasi-particle picture. New density wave excitations appear that are described by bosonic operators. Here, we will develop this bosonic description, following a pedestrian approach which does not require any previous knowledge in field theory methods. These notes therefore serve as a detailed introduction into bosonization by carefully deriving the most fundamental formulas.
Due to progress in nanotechnology high-quality quantum wires can nowadays be fabricated. The behavior of particles in one dimension differs significantly from that in three-dimensional (3D) systems, yet the physics of such low-dimensional systems is generally not very well represented in standard undergraduate or graduate curricula. For instance, the Fermi liquid paradigm, working well for electrons in 3D systems, breaks down in 1D and has to be replaced by the Tomonaga-Luttinger liquid (TLL) description. However, most of the introductory papers on the TLL restrict themselves to a summary of results to be compared with experiments, while review papers are often fraught with technical details only accessible to the theorist. The present paper provides an elementary discussion of some phenomena distinguishing quantum wires and quasi-one-dimensional systems from their 3D counterparts targeting experimentalists and graduate students. It aims to convey the basic ideas of TLL theory and to relate it to alternative phases, such as charge and spin density waves. The important role of fluctuations in quasi-1D systems is pointed out and the connection is made to the problem of high-Tc superconductivity. The discussion is kept on a level, which should be accessible with a basic knowledge of quantum mechanics.
The Chapter 2 is devoted to the study of the transport properties of a quantum wire, described by the Tomonaga-Luttinger model, in the presence of a backscattering potential provided by several point-like or extended time-dependent impurities (barriers) and find regimes of conductance enhancement. In Chapte 3 we study a set of crossed 1D systems, which are coupled with each other via tunnelling at the crossings. Concentrating on the case of a single crossing, we were able to explain recent experiments on crossed metallic and semiconducting nanotubes, which showed the presence of localized states in the region of crossing. The Chapter 4 treats the effects of disorder and interactions in a quantum Hall ferromagnet. We study the problem by projecting the original fermionic Hamiltonian into magnon states, which behave as bosons in the vicinity of the ferromagnetic ground state. The approach permits the reformulation of a strongly interacting model into a non-interacting one. The latter is a non-perturbative scheme that consists in treating the two-particle neutral excitations of the electron system as a bosonic single-particle. Indeed, the employment of bosonization facilitates the inclusion of disorder in the study of the system. It has been shown previously that disorder may drive a quantum phase transition in the Hall ferromagnet. However, such studies have been either carried out in the framework of the nonlinear sigma model, as an effective low-energy theory, or included the long-range Coulomb interaction in a quantum description only up to the Hartree-Fock level. Here, we establish the occurrence of a disorder-driven quantum phase transition from a ferromagnetic 2DEG to a spin glass phase by taking into account interactions between electrons up to the random phase approximation level in a fully quantum description. In the Chapter 5 we derive and discuss an experimentally realistic model describing ultracold atoms in an optical lattice including a commensurate, but staggered, Zeeman field. The resulting band structure is quite exotic; fermions in the third band have an unusual rounded picture-frame Fermi surface (essentially two concentric squircles), leading to imperfect nesting. We develop a generalized theory describing the spin and charge degrees of freedom simultaneously, and show that the system can develop a coupled spin-charge-density wave order. This ordering is absent in studies of the Hubbard model that treat spin and charge density separately.
This dissertation is devoted to theoretical studies of strongly interacting one-dimensional and quasi one-dimensional electron systems. The properties of one-dimensional electron systems can be studied within the bosonization technique, which presents fermions as collective bosonic density excitations. The power of this approach is the ability to treat electron-electron interaction exactly in the low-energy limit. The approach predicts the failure of Fermi liquid and an absence of long-range order in one-dimensions. The low-energy description of one-dimensional interacting systems is called the Tomonaga-Luttinger liquid theory. For example, the edges of quantum Hall systems are one-dimensional and described by a chiral Tomonaga-Luttinger liquid. Another example is a quantum spin Hall system with helical edge states, which are also described by a Tomonaga-Luttinger liquid. In our first work, a study of magnetized edge states of quantum spin-Hall system is presented. A magnetic field dependent signature of such edges is obtained, which can be verified in a Coulomb drag experiment. The second part of the dissertation is devoted to quasi-one dimensional antiferromagnetic lattices. A spatially anisotropic lattice antiferromagnet can be viewed as an array of one dimensional spin chains coupled in a way to match the lattice symmetry. This allows to use the non-Abelian bosonization technique to describe the low-energy physics of spin chains and study the inter-chain interactions perturbatively. The work presented in the dissertation studies the effect of Dzyaloshinskii-Moriya interaction on the magnetic phase diagram of the spatially anisotropic kagome antiferromagnet. We predict a Dzyaloshinskii-Moriya interaction driven phase transition from Neel to Neel+dimer state. In the third work, a novel model of the fractional quantum Hall effect is given. Wave functions of two-dimensional electrons in strong and quantizing magnetic field are essentially one-dimensional. That invites one to use the one-dimensional phenomenological bosonization to describe the density fluctuations of the two-dimensional interacting electrons in magnetic field. Remarkably, the constructed trial bosonized fermion operator describing the electron states with a fixed Landau gauge momentum is effectively two-dimensional.
To a casual ostrich the world of quantum physics in one dimension may sound a little one-dimensional, suitable perhaps for those with an unhealthy obsession for the esoteric. Nothing of course could be further from the truth. The field is remarkably rich and broad, and for more than fifty years has thrown up innumerable challenges. Theorists, realising that the role of interactions in 1D is special and that well known paradigms of higher dimensions (Fermi liquid theory for example) no longer apply, took up the challenge of developing new concepts and techniques to understand the undoubted pecularities of one-dimensional systems. And experimentalists have succeeded in turning pipe dreams into reality, producing an impressive and ever increasing array of experimental realizations of 1D systems, from the molecular to the mesoscopic---spin and ladder compounds, organic superconductors, carbon nanotubes, quantum wires, Josephson junction arrays and so on. Many books on the theory of one-dimensional systems are however written by experts for experts, and tend as such to leave the non-specialist a touch bewildered. This is understandable on both fronts, for the underlying theoretical techniques are unquestionably sophisticated and not usually part of standard courses in many-body theory. A brave author it is then who aims to produce a well rounded, if necessarily partial, overview of quantum physics in one dimension, accessible to a beginner yet taking them to the edge of current research, and providing en route a thorough grounding in the fundamental ideas, basic methods and essential phenomenology of the field. It is of course the brave who succeed in this world, and Thierry Giamarchi does just that with this excellent book, written by an expert for the uninitiated. Aimed in particular at graduate students in theoretical condensed matter physics, and assumimg little theoretical background on the part of the reader (well just a little), Giamarchi writes in a refreshingly relaxed style with infectious enthusiasm for his subject, and readily combines formal instruction with physical insight. The result is a serious, pedagogical yet comprehensive guide to the fascinating and important field of one-dimensional quantum systems, for which many a graduate student (and not a few oldies) will be grateful. The first half of the book, chapters 1--5, is devoted to a coherent presentation of the essential concepts and theoretical methods of the field. After a basic introduction to the unique behaviour of interacting electrons in one dimension, and to early fermionic approaches to the problem, Giamarchi turns to the technique of bosonization, introducing chapter 3 with a Marxist quote: `A child of five would understand this. Send for a child of five.' This most powerful technique is presented in a step by step fashion, and serious perusal of the chapter will benefit all ages since bosonization is used extensively throughout the rest of the book. The same is true of chapter 3 where a phenomenological and physically insightful introduction is given to the Luttinger liquid---the key concept in the low-energy physics of one-dimensional systems, analogous to the Fermi liquid in higher dimensions. Chapter 4 deals with what the author calls `refinements', or complications of the sort theorists in particular welcome; such as how the Luttinger liquid description is modified by the presence of long-ranged interactions, the Mott transition (`we forgot the lattice Gromit'), and the effects of breaking spin rotational invariance on application of a magnetic field. Finally chapter 5 describes various microscopic methods for one dimension, including a brief discussion of numerical techniques but focussing primarily on the Bethe ansatz---the famous one-dimensional technique others seek to emulate but whose well known complexity necessitates a relatively brief discussion, confined in practice to the spin-1/2 Heisenberg model. In the second half of the book, chapters 6--11, a range of different physical realizations of one-dimensional quantum physics are discussed. According to taste and interest, these chapters can be read in essentially any order. Spin systems are considered in chapter 6, beginning with spin chains---Jordan--Wigner, the bosonization solution---before moving to frustration, the spin-Peierls transition, and spin ladders; and including experimental examples of both spin chain and ladder materials. Chapters 7 and 8 deal with interacting lattice fermions, the former with single chain problems, notably the Hubbard, t-J and related models; and the latter with coupled fermionic chains, from finite to infinite, including a fulsome discussion of Bechgaard salts (organic conductors) as exemplars of Luttinger liquid behaviour. The effect of disorder in fermionic systems is taken up in chapter 9, and here the reader may react: interacting systems are tough enough, why make life harder? But disorder is always present to some degree in real systems---quantum wires, for example, discussed briefly in the chapter---and its effects particularly acute in one dimension. It simply cannot be avoided, even if the problem of interacting, disordered one-dimensional systems is still a long way off being solved. The penultimate chapter deals with the topical issues of boundaries, isolated impurities and constrictions, with a primary focus on mesoscopic examples of Luttinger liquids, notably carbon nanotubes and edge states in the quantum Hall effect. Finally `significant other' examples of Luttinger liquids, namely interacting one-dimensional bosons, are considered in chapter 11; which concludes with a discussion of bosonization techniques in the context of quantum impurities in Fermi liquids---the x-ray, Kondo and multichannel Kondo problems. The quality of the product attests to the fact that writing this impressive tome was a labour of love for the author. Anyone with a serious interest in getting to grips with one-dimensional quantum systems simply needs the book on their shelves---and will have great fun reading it too.
Table of contents 1. Introduction 2. Non-Fermi-liquid features of Fermi liquids: 1D physics in higher dimensions 3. Dzyaloshinskii-Larkin solution of the Tomonaga-Luttinger model 4. Renormalization group for interacting fermions 5. Single impurity in a 1D system: scattering theory for interacting electrons 6. Bosonization solution 7. Transport in quantum wires 7.1 Conductivity and conductance 7.2 Dissipation in a contactless measurement 7.3 Conductance of a wire attached to reservoirs 7.4 Spin component of the conductance 7.5 Thermal conductance: Fabry-Perrot resonances of plasmons 8. Appendices
One-dimensional Fermi systems We will now consider the case of one-dimensional (1D) Fermi systems for which the Landau theory fails. The way it fails is quite instructive since it reveals that in one dimension these systems are generally at a (quantum) critical point, and it will also teach us valuable lessons on quantum criticality. It will also turn out that the problem of 1D Fermi systems is closely related to the problem of quantum spin chains. This is a problem that has been discussed extensively by many authors, and there are several excellent reviews on the subject (Emery 1979; Haldane, 1981). Here I follow in some detail the discussion and notation of Carlson and coworkers (Carlson et al ., 2004). One-dimensional (and quasi-1D) systems of fermions occur in several experimentally accessible systems. The simplest one to visualize is a quantum wire . A quantum wire is a system of electrons in a semiconductor, typically a GaAs–AlAs heterostructure built by molecular-beam epitaxy (MBE), in which the electronic motion is laterally confined along two directions, but not along the third. An example of such a channel of length L and width d (here shown as a two-dimensional (2D) system) is seen in Fig. 6.1. Systems of this type can be made with a high degree of purity with very long (elastic) mean free paths, often tens of micrometers or even longer. The resulting electronic system is a 1D electron gas (1DEG). In addition to quantum wires, 1DEGs also arise naturally in carbon nanotubes, where they are typically multi-component (with the number of components being determined by the diameter of the nanotube).