This part of the book is fully devoted to the physics of (1 + 1)-dimensional quantum or twodimensional classical models. As usual, I shall distinguish between classical and quantum systems only if it is necessary. Those models where the difference between quantum and classical is not relevant will be called ‘two-dimensional’; those where for some reason or other I want to emphasize the quantum aspect will be called ‘one-dimensional’. The reason I spend so much time discussing two-dimensional physics is that here one finds a kind of paradise for strong interactions and nonperturbative effects. Those effects which are tricky or even impossible to achieve in higher dimensions appear in almost every two-dimensional model. To our greatest satisfaction there are theoretical tools at our disposal which allow us to solve many of the corresponding problems and describe these phenomena. The availability of these strong mathematical tools is a unique feature of two-dimensional physics. Even if it turns out to be impossible to generalize these tools for higher dimensions, I hope that the reader will be rewarded for the time spent by the pleasure obtained from contemplating their beauty. The world we are about to enter has certain distinct features which are worth mentioning in the introduction. The first one concerns the second-order phase transitions. In (1 + 1)-dimensional systems such transitions may occur only at zero temperature.
This part of the book is fully devoted to the physics of (1 + 1)-dimensional quantum or twodimensional classical models. As usual, I shall distinguish between classical and quantum systems only if it is necessary. Those models where the difference between quantum and classical is not relevant will be called ‘two-dimensional’; those where for some reason or other I want to emphasize the quantum aspect will be called ‘one-dimensional’. The reason I spend so much time discussing two-dimensional physics is that here one finds a kind of paradise for strong interactions and nonperturbative effects. Those effects which are tricky or even impossible to achieve in higher dimensions appear in almost every two-dimensional model. To our greatest satisfaction there are theoretical tools at our disposal which allow us to solve many of the corresponding problems and describe these phenomena. The availability of these strong mathematical tools is a unique feature of two-dimensional physics. Even if it turns out to be impossible to generalize these tools for higher dimensions, I hope that the reader will be rewarded for the time spent by the pleasure obtained from contemplating their beauty. The world we are about to enter has certain distinct features which are worth mentioning in the introduction. The first one concerns the second-order phase transitions. In (1 + 1)-dimensional systems such transitions may occur only at zero temperature.
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.
We have discussed the physical and mathematical aspects of one-dimensional quantum particles with1/ r 2 interactions. On the physics side, we began with the simplest two-body system, and culminated in the exact dynamics of a supersymmetric electronic model. In this way we have tried to draw a comprehensive picture of one-dimensional quantum systems, taking the canonical systems with 1/ r 2 interactions. On the mathematical side, the most important subjects are first the Jack polynomials, and second the Yangians. These two subjects were originally developed independently, but are actually closely interrelated. Our basic viewpoint is that the 1/ r 2 systems provide the most natural working model to synthesize these two subjects, and to sharpen the concepts further. There are many topics, however, which we could not discuss in this book. This is firstly because of our insufficient understanding of the topics, and secondly because of our intention of making a reasonably sized book. We shall mention some of the omitted topics, hoping that they will be covered on another occasion. Examples of other canonical systems, which have escaped our discussion in this book, include the Calogero–Sutherland model with harmonic confinement. The spectrum of the model is equidistant as in the simple harmonic oscillator. Because of this feature the model can be compared with the chiral conformal field theory (CFT) with right-going particles only. We refer to a recent review [88] for the systems. In relation to the CFT, there are active studies to investigate the quasi-particle structure for the general case of the internal symmetry.
Interactions in many-body systems bring about collective phenomena such as superconductivity and magnetism. In many cases, simple mean-field theory provides a basic understanding of these phenomena. In fermion systems in one dimension, however, neither the mean-field theory nor perturbation theory works if it starts from the non-interacting fermions. This is because the interaction effects in one dimension are much stronger than those in higher dimensions. Intuitively speaking, two particles cannot avoid collision in a single-way track in contrast with two and three dimensions. Thus the interaction effects appear in a drastic way in one dimension. Another aspect in one dimension, which overcompensates the difficulty of perturbation and mean-field theories, is that a complete account of interaction effects is possible under certain conditions. The class of systems satisfying such conditions is referred to as exactly solvable. Soon after the establishment of quantum mechanics, Bethe solved exactly the Heisenberg spin model in one dimension [28]. The basic idea of the solution is now called the Bethe ansatz. Since then, theoretical physics in one dimension has developed into a magnificent edifice, including sophisticated mathematical techniques. In many cases, the eigenfunctions derived by the Bethe ansatz consist of plane waves that are defined stepwise for each spatial configuration of particles. Since the coefficients of plane waves depend on the configuration, the property of the wave function cannot be made explicit without detailed knowledge of these coefficients. We mention some of the recent monographs on the Bethe ansatz and its extensions [54, 118, 179]. A comprehensive account on exactly solvable models has recently been given by Sutherland [178].
The first volume (General Theory) differs from most textbooks as it emphasizes the mathematical structure and mathematical rigor, while being adapted to the teaching the first semester of an advanced course in Quantum Mechanics (the content of the book are the lectures of courses actually delivered.). It differs also from the very few texts in Quantum Mechanics that give emphasis to the mathematical aspects because this book, being written as Lecture Notes, has the structure of lectures delivered in a course, namely introduction of the problem, outline of the relevant points, mathematical tools needed, theorems, proofs. This makes this book particularly useful for self-study and for instructors in the preparation of a second course in Quantum Mechanics (after a first basic course). With some minor additions it can be used also as a basis of a first course in Quantum Mechanics for students in mathematics curricula. The second part (Selected Topics) are lecture notes of a more advanced course aimed at giving the basic notions necessary to do research in several areas of mathematical physics connected with quantum mechanics, from solid state to singular interactions, many body theory, semi-classical analysis, quantum statistical mechanics. The structure of this book is suitable for a second-semester course, in which the lectures are meant to provide, in addition to theorems and proofs, an overview of a more specific subject and hints to the direction of research. In this respect and for the width of subjects this second volume differs from other monographs on Quantum Mechanics. The second volume can be useful for students who want to have a basic preparation for doing research and for instructors who may want to use it as a basis for the presentation of selected topics.
In this chapter, we take a deeper look at a number of interesting and important models that contain one or two particles. We focus on examples, from a variety of settings and fields, with the goal of not only finding relevant physical phenomena occurring in these systems, but also giving the reader a place where they can actually practise much of what has been said previously throughout the book. In some cases, straight application of the methods explained earlier suffices, while in some others slightly different (yet related) methods are needed. Most of the models studied in this chapter are justified in chapters 4 and 5, sometimes in full. Other times, the justification from chapters 4 and 5 may be only partial, and the interested reader is encouraged to consult more specialized literature in the subject.
This PhD thesis is concerned with the investigation of Extended Hubbard Models on one-dimensional lattices (chains); the approach used to such investigation is that of determining exact results. The extended Hubbard models are models of correlated electrons on lattices; they are the extensions of a model proposed by Hubbard in \\cite{HUB} to explain the metal-insulator transition in d-transition compounds as the effect of electronic correlations, usually neglected in band theories. Later on, a need for extensions of such model was first felt within the context of high-temperature superconductivity; indeed, the different behaviour of doped cuprates with respect to conventional superconductors suggested that a microscopic theory of this phenomenon could be formulated in terms of electron-electron interaction, without explicitly invoking the phonon coupling, like in BCS theory. Although an exhaustive microscopic formulation for cuprate superconductivity has not been achieved yet, the Extended Hubbard Models have been in the meanwhile applied to other problems in condensed matter; they are actually considered the reference paradigm for treating all those systems in which electronic correlations cannot be neglected. \\\\As observed above, this thesis focuses in particular on 1-D chains: indeed in last years the progress in technological synthesization and small-scale observations has shown that there are many classes of materials exhibiting a strong one-dimensional character, such as organic (super)conductors, linear-chain polymers, quantum wires, quantum-dot chains, one-dimensional cuprates. Many experimental results confirm that such materials exhibit a quite rich behaviour, with different phases, charge-spin separation, as well as a strong electronic correlations. However, a sufficient understanding of these phenomena on a theoretical point of view has not been reached yet. This is due to the fact the usual theoretical methods can be shown to fail in D=1, and theoreticians are thus forced to develop new schemes for approaching the problem of many interacting fermions in such dimension. The latter therefore represents a fascinating challenge. Some progress has been made and many aspects are nowadays more clear; nevertheless, due to the lack of traditional techniques, the need for a comparison with exact results is perhaps more urgent in D=1 than in higher dimensions. The aim of this thesis is to provide some results to the research in this field. The present work is organized as follows: the two first chapters are introductory, while chapters 3 and 4 mostly contain the author's original contribution. More explicitly, {\\bf chapter 1} is an introduction to the Hubbard models (both the ordinary and the Extended) and is meant for the reader who is non-familiar with these models. We tried to emphasize the motivation that historically led to their introduction and the physical contexts in which they are applied, not necessarily in one-dimension. \\\\{\\bf Chapter 2} is instead devoted to general problematics of one dimensional physics. We have tried to stress that there is a number of physically interesting materials that can be regarded as actual 1-D systems; we give a brief account of the difficulties leading the usual theoretical approaches, such as mean-field and Fermi Liquid, to fail in D=1. While briefly describing the important results of the proposed alternative formulations (Luttinger Liquid, Renormalization Group), we also point out the points that are still unsatisfactory, and emphasize the importance that exactly integrable models have in this field. \\\\Matching the subjects of chapter 1 and 2, the subsequent ones contain the results of our investigation of exactly solvable Extended Hubbard model on 1-D chains. \\\\We wish to emphasize to the reader non familiar with integrable systems that the literature on this subject is often strongly concerned with the mathematical aspects of the involved methods, leaving the determination of the physical features of the derived integrable models as a secondary issue. On the contrary, we believe that both aspects of the research deserve a relevant role in the study. For these reasons, while chapter 3 contains new results about the general formulation of integrability for fermionic systems, chapter 4 is devoted to the investigation of the physical features for some models whose integrability has been proved in the previous chapter. \\\\More explicitly, in {\\bf chapter 3} (apart from a preliminary introduction of sec.3.1) the theory of integrable {\\it fermionic} systems has been developed within the frame of the Quantum Inverse Scattering Method (QISM). The latter is a technique for obtaining integrable models, traditionally formulated with success for spin systems; its more recent application to electron systems was leaving many unsatisfactory aspects. To overcome this weaknesses, in sec.3.2 we both clarify the theory of matrix representation for fermionic operators, and describe a method for obtaining integrable models of fermions (straightforwardly readable in terms of fermion operators), starting from the the solutions of a mathbbC-number equation (known as Yang-Baxter Equation). In sec.3.3 we first work out a general method of search for solutions of such equation; secondly we explicitly find 96 solutions, corresponding to 96 integrable Extended Hubbard models (most of which are new) for which a set of mutually commuting operators is proved to exist. In sec.3.4 we formulate a general theorem concerning the symmetries of the integrable models in QISM, and widely discuss the symmetries and supersymmetries for the class of extended Hubbard Models.\\\\ {\\bf Chapter 4} is devoted to the investigation of the physical properties of the integrable Extended Hubbard models found in chapter 3. We introduce a technique, called Sutherland's species technique, that, when applicable, yields an effective reduction of the Hilbert space and considerably simplifies the problem of diagonalizing the Hamiltonian. In fact, such technique is proved to be applicable to all the 96 models found above, and is used to determine the exact ground state phase diagram and the thermodynamics of the models. In particular we both recover in a quite straightforward way the phase diagram of a known model, and that of two new models (Model 2 and Model 3) that were not known before. \\\\The phase diagram exhibit structured forms with respect to the filling (density of electrons) and the on-site Coulomb repulsion (U); the pressure and the compressibility in the ground state are also calculated. Model 2 exhibits a superconducting phase particularly enhanced towards positive values of U, as well as a band-controlled insulator-superconductor transition at half filling; model 3 is the first example of exactly solved model with an asymmetric shape with respect to half-filling, due to the absence of particle-hole invariance of the Hamiltonian; a filling controlled metal-insulator transition is found to occur for this model. Generalizations to the other cases are also discussed. \\\\As to the thermodynamics, we compute the exact partition function of Model 3, which we believe to be particular worth of interest due to the above asymmetry. We then derive the exact specific heat of the model as a function of the temperature, which exhibits an interesting two-peak behaviour as well as a nearly universal crossing point. These results are discussed and compared with similar ones present in the literature. The chapters are of course linked each to another according to the above scheme; nevertheless, it has been a purpose of the author to make them as independently readable as possible. In particular, the reader who is more interested in the physical aspects of the models rather than in the general formalism of fermionic QISM can skip chapter 3 (at least in a first-sight reading) and pass directly to chapter 4; apart from some references to previous formulas the understanding should not be affected.\\\\{\\it \\noindent \\underline{Notational Remark:}} The customary symbol to denote the electron filling is n; this has in fact been used throughout the two first introductory chapters. Nevertheless, in order to avoid possible misunderstanding with other extensive quantities, in chapter 3~and~4~the~symbol~has been changed into rho