Quantum field theory (QFT) is a rich and robust part of mathematical and theoretical physics, with a wide range of applications in statistical physics, condensed matter physics, particle physics and string theory. Many (if not most) of the applications in four space-time dimensions require some sort of systematic approximations. In contrast, quantum field theories in lower dimensions often display special properties which open the door, at least in principle, to their exact solution. In many cases, the methods used to obtain exact solutions are peculiar to low-dimensional space-times, but the exact solutions thus obtained can teach us a great deal about more realistic QFTs in higher dimensions. Thus, lower-dimensional QFTs provide excellent models for exploring deep theoretical and computational issues, in addition to their direct applications to physical processes in reduced dimensions, which include polymers, boundary fluctuations, spin chains, and quantum Hall systems. Such models are also vital testing grounds for fundamental theoretical ideas such as integrability, phase transitions, collective and non-perturbative quantum behaviour, and non-equilibrium statistical mechanics. We should also mention the effective dimensional reduction which occurs in the high temperature limit, or is induced by disorder. Low-dimensional quantum field theory thus sits at a unique intersection point of contemporary mathematics and physics, allowing many of these issues to be addressed rigorously and analytically. Surprising inter-relations continue to arise, from the ubiquitous nature of the Bethe ansatz and Calogero models, to the link between integrable spin chains and the anomalous dimensions of operators in gauge theories. Aspects of the latter connection have been known at least since the 1990s; they have recently received tremendous attention in connection with N =4 supersymmetric Yang--Mills theory (in four space-time dimensions) and the AdS/CFT correspondence. One unifying idea is the Bethe ansatz, whose 75th anniversary we celebrate this year, and whose wide-ranging impact in diverse branches of modern mathematical physics is made abundantly clear in this special issue. In addition, new ideas and techniques of significance to both mathematics and physics have emerged: the recent and spectacular progress on geometrical aspects of phase transitions made possible through the ideas of stochastic (Schramm) Loewner evolution, or SLE, is a particularly noteworthy example. All papers in this special issue are invited, refereed contributions from leading experts. Many are pedagogical reviews, capturing the current state of knowledge in a particular subject, putting the subject in context and outlining open challenges. Others are research papers, reflecting the rapid flux of new ideas and results in this rich and exciting field. Together, they represent a broad sample of the entire field. We would like to take this opportunity to thank all of the authors for their efforts in making this special issue possible, and Kazuhiro Hikami for his help in the early stages of this project. We hope it inspires many new developments!
Quantum field theory (QFT) is a rich and robust part of mathematical and theoretical physics, with a wide range of applications in statistical physics, condensed matter physics, particle physics and string theory. Many (if not most) of the applications in four space-time dimensions require some sort of systematic approximations. In contrast, quantum field theories in lower dimensions often display special properties which open the door, at least in principle, to their exact solution. In many cases, the methods used to obtain exact solutions are peculiar to low-dimensional space-times, but the exact solutions thus obtained can teach us a great deal about more realistic QFTs in higher dimensions. Thus, lower-dimensional QFTs provide excellent models for exploring deep theoretical and computational issues, in addition to their direct applications to physical processes in reduced dimensions, which include polymers, boundary fluctuations, spin chains, and quantum Hall systems. Such models are also vital testing grounds for fundamental theoretical ideas such as integrability, phase transitions, collective and non-perturbative quantum behaviour, and non-equilibrium statistical mechanics. We should also mention the effective dimensional reduction which occurs in the high temperature limit, or is induced by disorder. Low-dimensional quantum field theory thus sits at a unique intersection point of contemporary mathematics and physics, allowing many of these issues to be addressed rigorously and analytically. Surprising inter-relations continue to arise, from the ubiquitous nature of the Bethe ansatz and Calogero models, to the link between integrable spin chains and the anomalous dimensions of operators in gauge theories. Aspects of the latter connection have been known at least since the 1990s; they have recently received tremendous attention in connection with N =4 supersymmetric Yang--Mills theory (in four space-time dimensions) and the AdS/CFT correspondence. One unifying idea is the Bethe ansatz, whose 75th anniversary we celebrate this year, and whose wide-ranging impact in diverse branches of modern mathematical physics is made abundantly clear in this special issue. In addition, new ideas and techniques of significance to both mathematics and physics have emerged: the recent and spectacular progress on geometrical aspects of phase transitions made possible through the ideas of stochastic (Schramm) Loewner evolution, or SLE, is a particularly noteworthy example. All papers in this special issue are invited, refereed contributions from leading experts. Many are pedagogical reviews, capturing the current state of knowledge in a particular subject, putting the subject in context and outlining open challenges. Others are research papers, reflecting the rapid flux of new ideas and results in this rich and exciting field. Together, they represent a broad sample of the entire field. We would like to take this opportunity to thank all of the authors for their efforts in making this special issue possible, and Kazuhiro Hikami for his help in the early stages of this project. We hope it inspires many new developments!
Abstract This book provides a thorough introduction to the fascinating world of phase transitions as well as many related topics, including random walks, combinatorial problems, quantum field theory and S-matrix. Fundamental concepts of phase transitions, such as order parameters, spontaneous symmetry breaking, scaling transformations, conformal symmetry, and anomalous dimensions, have deeply changed the modern vision of many areas of physics, leading to remarkable developments in statistical mechanics, elementary particle theory, condensed matter physics and string theory. This self-contained book provides an excellent introduction to frontier topics of exactly solved models in statistical mechanics and quantum field theory, renormalization group, conformal models, quantum integrable systems, duality, elastic S-matrix, thermodynamics Bethe ansatz and form factor theory. The clear discussion of physical principles is accompanied by a detailed analysis of several branches of mathematics, distinguished for their elegance and beauty, such as infinite dimensional algebras, conformal mappings, integral equations or modular functions. Besides advanced research themes, the book also covers many basic topics in statistical mechanics, quantum field theory and theoretical physics. Each argument is discussed in great detail, paying attention to an overall coherent understanding of physical phenomena. Mathematical background is provided in supplements at the end of each chapter, when appropriate. The chapters are also followed by problems of different levels of difficulty. Advanced undergraduate and graduate students will find a rich and challenging source for improving their skills and for accomplishing a comprehensive learning of the many facets of the subject.
For most of the last century, condensed matter physics has been dominated by band theory and Landau's symmetry breaking theory. In the last twenty years, however, there has been an emergence of a new paradigm associated with fractionalization, emergent gauge bosons and fermions, topological order, string-net condensation, and long range entanglements. These new physical concepts are so fundamental that they may even influence our understanding of the origin of light and electrons in the universe. This book is a pedagogical and systematic introduction to the new concepts and quantum field theoretical methods in condensed matter physics. It discusses many basic notions in theoretical physics which underlie physical phenomena in nature, including a notion that unifies light and electrons. Topics covered include dissipative quantum systems, boson condensation, symmetry breaking and gapless excitations, phase transitions, Fermi liquids, spin density wave states, Fermi and fractional statistics, quantum Hall effects, topological/quantum order, and spin liquid and string-net condensation. Methods discussed include the path integral, Green's functions, mean-field theory, effective theory, renormalization group, bosonization in one- and higher dimensions, non-linear sigma-model, quantum gauge theory, dualities, projective construction, and exactly soluble models beyond one-dimension.
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.
Abstract This book is an introduction to statistical field theory, which is an important subject within theoretical physics and a field that has seen substantial progress in recent years. The book covers fundamental topics in great detail and includes areas like conformal field theory, quantum integrability, S-matrices, braiding groups, Bethe ansatz, renormalization groups, Majorana fermions, form factors, the truncated conformal space approach and boundary field theory. It also provides an introduction to lattice statistical models. Many topics are discussed at a fairly advanced level but via a pedagogical approach. In particular, the book presents in a clear way non-perturbative methods of quantum field theories that have become decisive tools in many different areas of statistical and condensed matter physics, and which are currently an essential foundation of the working knowledge of a modern theoretical physicist.
Conformal field theory (CFT) has proven to be one of the richest and deepest subjects of modern theoretical and mathematical physics research, especially as regards statistical mechanics and string theory. It has also stimulated an enormous amount of activity in mathematics, shaping and building bridges between seemingly disparate fields through the study of vertex operator algebras, a (partial) axiomatisation of a chiral CFT. One can add to this that the successes of CFT, particularly when applied to statistical lattice models, have also served as an inspiration for mathematicians to develop entirely new fields: the Schramm–Loewner evolution and Smirnov's discrete complex analysis being notable examples. When the energy operator fails to be diagonalisable on the quantum state space, the CFT is said to be logarithmic. Consequently, a logarithmic CFT is one whose quantum space of states is constructed from a collection of representations which includes reducible but indecomposable ones. This qualifier arises because of the consequence that certain correlation functions will possess logarithmic singularities, something that contrasts with the familiar case of power law singularities. While such logarithmic singularities and reducible representations were noted by Rozansky and Saleur in their study of the U (1|1) Wess–Zumino–Witten model in 1992, the link between the non-diagonalisability of the energy operator and logarithmic singularities in correlators is usually ascribed to Gurarie's 1993 article (his paper also contains the first usage of the term 'logarithmic conformal field theory'). The class of CFTs that were under control at this time was quite small. In particular, an enormous amount of work from the statistical mechanics and string theory communities had produced a fairly detailed understanding of the (so-called) rational CFTs. However, physicists from both camps were well aware that applications from many diverse fields required significantly more complicated non-rational theories. Examples include critical percolation, supersymmetric string backgrounds, disordered electronic systems, sandpile models describing avalanche processes, and so on. In each case, the non-rationality and non-unitarity of the CFT suggested that a more general theoretical framework was needed. Driven by the desire to better understand these applications, the mid-1990s saw significant theoretical advances aiming to generalise the constructs of rational CFT to a more general class. In 1994, Nahm introduced an algorithm for computing the fusion product of representations which was significantly generalised two years later by Gaberdiel and Kausch who applied it to explicitly construct (chiral) representations upon which the energy operator acts non-diagonalisably. Their work made it clear that underlying the physically relevant correlation functions are classes of reducible but indecomposable representations that can be investigated mathematically to the benefit of applications. In another direction, Flohr had meanwhile initiated the study of modular properties of the characters of logarithmic CFTs, a topic which had already evoked much mathematical interest in the rational case. Since these seminal theoretical papers appeared, the field has undergone rapid development, both theoretically and with regard to applications. Logarithmic CFTs are now known to describe non-local observables in the scaling limit of critical lattice models, for example percolation and polymers, and are an integral part of our understanding of quantum strings propagating on supermanifolds. They are also believed to arise as duals of three-dimensional chiral gravity models, fill out hidden sectors in non-rational theories with non-compact target spaces, and describe certain transitions in various incarnations of the quantum Hall effect. Other physical applications range from two-dimensional turbulence and non-equilibrium systems to aspects of the AdS/CFT correspondence and describing supersymmetric sigma models beyond the topological sector. We refer the reader to the reviews in this collection for further applications and details. More recently, our understanding of logarithmic CFT has improved dramatically thanks largely to a better understanding of the underlying mathematical structures. This includes those associated to the vertex operator algebras themselves (representations, characters, modular transformations, fusion, braiding) as well as structures associated with applications to two-dimensional statistical models (diagram algebras, eg. Temperley–Lieb; quantum groups). Not only are we getting to the point where we understand how these structures differ from standard (rational) theories, but we are starting to tackle applications both in the boundary and bulk settings. It is now clear that the logarithmic case is generic, so it is this case that one should expect to encounter in applications. We therefore feel that it is timely to review what has been accomplished in order to disseminate this improved understanding and motivate further applications. We now give a quick overview of the articles that constitute this special issue. Adamović and Milas provide a detailed summary of their rigorous results pertaining to logarithmic vertex operator (super)algebras constructed from lattices. This survey discusses the C2-cofiniteness of the (p, p') triplet models (this is the generalisation of rationality to the logarithmic setting), describes Zhu's algebra for (some of) these theories and outlines the difficulties involved in explicitly constructing the modules responsible for their logarithmic nature. Cardy gives an account of a popular approach to logarithmic theories that regards them, heuristically at least, as limits of ordinary (but non-rational) CFTs. More precisely, it seems that any given correlator may be computed as a limit of standard (non-logarithmic) correlators, any logarithmic singularities that arise do so because of a degeneration when taking the limit. He then illustrates this phenomenon in several theories describing statistical lattice models including the n → 0 limit of the O(n ) model and the Q → 1 limit of the Q-state Potts model. Creutzig and Ridout review the continuum approach to logarithmic CFT, using the percolation (boundary) CFT to detail the connection between module structure and logarithmic singularities in correlators before describing their proposed solution to the thorny issue of generalising modular data and Verlinde formulae to the logarithmic setting. They illustrate this proposal using the three best-understood examples of logarithmic CFTs: the (1, 2) models, related to symplectic fermions; the fractional level WZW model on , related to the beta gamma ghosts; and the WZW model on GL(1|1). The analysis in each case requires that the spectrum be continuous; C2-cofinite models are only recovered as orbifolds. Flohr and Koehn consider the characters of the irreducible modules in the spectrum of a CFT and discuss why these only span a proper subspace of the space of torus vacuum amplitudes in the logarithmic case. This is illustrated explicitly for the (1, 2) triplet model and conclusions are drawn for the action of the modular group. They then note that the irreducible characters of this model also admit fermionic sum forms which seem to fit well into Nahmrsquo;s well-known conjecture for rational theories. Quasi-particle interpretations are also introduced, leading to the conclusion that logarithmic C2-cofinite theories are not so terribly different to rational theories, at least in some respects. Fuchs, Schweigert and Stigner address the problem of constructing local logarithmic CFTs starting from the chiral theory. They first review the construction of the local theory in the non-logarithmic setting from an angle that will then generalise to logarithmic theories. In particular, they observe that the bulk space can be understood as a certain coend. The authors then show how to carry out the construction of the bulk space in the category of modules over a factorisable ribbon Hopf algebra, which shares many properties with the braided categories arising from logarithmic chiral theories. The authors proceed to construct the analogue of all-genus correlators in their setting and establish invariance under the mapping class group, i.e. locality of the correlators. Gainutdinov, Jacobsen, Read, Saleur and Vasseur review their approach based on the assumption that certain classes of logarithmic CFTs admit lattice regularisations with local degrees of freedom, for example quantum spin chains (with local interactions). They therefore study the finite-dimensional algebras generated by the hamiltonian densities (typically the Temperley–Lieb algebras and their extensions) that describe the dynamics of these lattice models. The authors then argue that the lattice algebras exhibit, in finite size, mathematical properties that are in correspondence with those of their continuum limits, allowing one to predict continuum structures directly from the lattice. Moreover, the lattice models considered admit quantum group symmetries that play a central role in the algebraic analysis (representation structure and fusion). Grumiller, Riedler, Rosseel and Zojer review the role that logarithmic CFTs may play in certain versions of the AdS/CFT correspondence, particularly for what is known as topologically massive gravity (TMG). This has been a very active subject over the last five years and the article takes great care to disentangle the contributions from the many groups that have participated. They begin with some general remarks on logarithmic behaviour, much in the spirit of Cardyrsquo;s review, before detailing the distinction between the chiral (no logs) and logarithmic proposals for critical TMG. The latter is then subjected to various consistency checks before discussing evidence for logarithmic behaviour in more general classes of grav
The conference brought together leading researchers at the interface between statistical mechanics and quantum field theory, which is, and has been, a very rich, dynamic and useful area of theoretical physics. The following topics that are of very active and current interest were covered: the study of renormalization group flows between field theories using the methods of quantum integrability, S-matrix theory and the thermodynamic Bethe Ansatz; impurity problems approached both from the point of view of conformal field theory and quantum integrability; the Kondo effect and quantum wires; solvable models with 1/r2 interactions (Haldane-Shastri models); Yangian symmetries in 1/r2 models and in conformal field theories; correlation functions in integrable 1+1 field theories; integrability in three dimensions; conformal invariance and the quantum hall effect; supersymmetry in statistical mechanics; and relations to two-dimensional Yang-Mills and QCD.