In our discussion of one-dimensional physics, more than once we have encountered situations when some relevant interaction scales to strong coupling. In such a situation the original description becomes inapplicable at low energies and must be replaced by some other description. In all previous cases I have restricted the discussion by some qualitative analysis, promising to provide more details later. Now the time is ripe to fulfil the promise. In fact, almost all our understanding of strong coupling physics comes from exact solutions of just a few models. The most ubiquitous among them is the sine-Gordon model, which we have already encountered in this book many times. Anyone who thoroughly understands this model and possibly a few others (such as the O(3) nonlinear sigma model and the off-critical Ising model) may consider themself an expert in the area of strongly correlated systems. The sine-Gordon model belongs to a category of integrable field theories. This means that it has an infinite number of constants of motion. On the one hand this fact makes it possible to solve this model exactly and describe its thermodynamics and correlation functions; on the other hand it makes the results less general than one might wish. First of all, not all interesting models are integrable, and one may wonder how and in what respect their behaviour may differ from the behaviour of integrable models. Integrable models possess some exceptional physical properties (such as ballistic transport) which are destroyed when the integrability is violated. There are several important review articles which one can read to familiarize oneself with the subject.
In our discussion of one-dimensional physics, more than once we have encountered situations when some relevant interaction scales to strong coupling. In such a situation the original description becomes inapplicable at low energies and must be replaced by some other description. In all previous cases I have restricted the discussion by some qualitative analysis, promising to provide more details later. Now the time is ripe to fulfil the promise. In fact, almost all our understanding of strong coupling physics comes from exact solutions of just a few models. The most ubiquitous among them is the sine-Gordon model, which we have already encountered in this book many times. Anyone who thoroughly understands this model and possibly a few others (such as the O(3) nonlinear sigma model and the off-critical Ising model) may consider themself an expert in the area of strongly correlated systems. The sine-Gordon model belongs to a category of integrable field theories. This means that it has an infinite number of constants of motion. On the one hand this fact makes it possible to solve this model exactly and describe its thermodynamics and correlation functions; on the other hand it makes the results less general than one might wish. First of all, not all interesting models are integrable, and one may wonder how and in what respect their behaviour may differ from the behaviour of integrable models. Integrable models possess some exceptional physical properties (such as ballistic transport) which are destroyed when the integrability is violated. There are several important review articles which one can read to familiarize oneself with the subject.
This book provides a detailed description of the duality between two integrable systems: the 1+1-dimensional Sine-Gordon model and the 1+1-dimensional Thirring model. While of great importance per se, this duality is only part of the target of the book. In order to reach an understanding of the subtleties involved in the duality, one has to take a journey through the properties of quantum integrable systems, building from the ground up the theory of exact S-matrices and familiarising oneself with the mathematical concept of a quantum group. The book therefore becomes an opportunity for a focussed study of integrability in its wider breadth of interest, always maintaining a clear ultimate purpose in mind: understanding the duality between bosons and fermions in 1+1 dimensions. This should make going through the book from the point of view of the reader/early-career researcher a live enterprise, as opposed to a more passive learning exercise. Key features • The book fills a gap in the existing literature, collating a vast amount scattered literature, with uniformised notation and conventions • The book’s format is ideal for use in PhD Schools and advanced graduate level mini courses, or for beginner PhD students as an introductory reference • Originates for an invited set of lectures • Contains a significant number of worked examples and problems, with solutions. It also presents several open-ended suggestions for literature reviews and guided learning through references
In the first three chapters we have considered problems related to the physical behaviour of one or a few particles (or some other physical objects which can be effectively described by one or a few (quasi)particles as, e.g., the random walk model in polymer physics considered in chapter 1). Though quantum field theory describes systems with an arbitrary number of degrees of freedom (an arbitrary number of (quasi)particles), in chapter 3 we actually applied it to systems with a restricted and small number of particles (e.g., for the description of the scattering process for a few particles). Another way to express this fact is to say that in chapter 3 we have considered field theories at zero temperature. However, the majority of realistic systems contain many identical (indistinguishable) particles such as atoms, electrons, photons etc. An attempt to describe these systems in terms of the individual trajectories of all particles is absolutely hopeless. Instead, we are interested in the collective behaviour of systems and describe them in terms of partition functions, mean values, correlation functions, etc. The methods of derivation, analysis and calculation of such collective characteristics constitute the subject of classical and quantum statistical mechanics.
Condensed matter theory is a massive field to which no book or books can do full justice. Every chapter in this book is possible material for a book or books. So it is clearly neither my intention nor within my capabilities to give an overview of the entire subject. Instead I will focus on certain techniques that have served me well over the years and whose strengths and limitations I am familiar with. My presentation is at a level of rigor I am accustomed to and at ease with. In any topic, say the renormalization group (RG) or bosonization, there are treatments that are more rigorous. How I deal with this depends on the topic. For example, in the RG I usually stop at one loop, which suffices to make the point, with exceptions like wave function renormalization where you need a minimum of two loops. For non-relativistic fermions I am not aware of anything new one gets by going to higher loops. I do not see much point in a scheme that is exact to all orders (just like the original problem) if in practice no real gain is made after one loop. In the case of bosonization I work in infinite volume from the beginning and pay scant attention to the behavior at infinity. I show many examples where this is adequate, but point to cases where it is not and suggest references. In any event I think the student should get acquainted with these more rigorous treatments after getting the hang of it from the treatment in this book. I make one exception in the case of the two-dimensional Ising model where I pay considerable attention to boundary conditions, without which one cannot properly understand how symmetry breaking occurs only in the thermodynamic limit. This book has been a few years in the writing and as a result some of the topics may seem old-fashioned; on the other hand, they have stood the test of time. Ideally the chapters should be read in sequence, but if that is not possible, the reader may have to go back to earlier chapters when encountering an unfamiliar notion. I am grateful to the Aspen Center for Physics (funded by NSF Grant 1066293) and the Indian Institute of Technology, Madras for providing the facilities to write parts of this book.
Including topics not traditionally covered in literature, such as (1+1)-dimensional QFT and classical 2D Coulomb gases, this book considers a wide range of models and demonstrates a number of situations to which they can be applied. Beginning with a treatise of nonrelativistic 1D continuum Fermi and Bose quantum gases of identical spinless particles, the book describes the quantum inverse scattering method and the analysis of the related Yang–Baxter equation and integrable quantum Heisenberg models. It also discusses systems within condensed matter physics, the complete solution of the sine-Gordon model and modern trends in the thermodynamic Bethe ansatz. Each chapter concludes with problems and solutions to help consolidate the reader's understanding of the theory and its applications. Basic knowledge of quantum mechanics and equilibrium statistical physics is assumed, making this book suitable for graduate students and researchers in statistical physics, quantum mechanics and mathematical and theoretical physics.
As we have seen on several occasions, a great many of the physical phenomena of interest concern collective behaviour or strongly correlated particles. While it is sometimes possible to obtain useful insight using approximation methods, such as perturbation theory, reliable controlled approximations do not exist. In such situations, numerical simulations have become an indispensable tool. We shall not attempt in this chapter to discuss computer programming nor shall we discuss many of the powerful methods, both algorithmic and in data analysis, that have been developed since numerical simulations have become an important research tool. Instead, we shall assume that the reader already has enough familiarity with computer programming to apply the notions which we discuss here. Our goal is to explain enough of the basics of the Monte Carlo method (classical and quantum) to allow the reader to apply it readily to interesting equilibrium statistical physics problems of the type discussed in this book. The problems in this chapter have been tailored with this in mind. The programs needed are relatively simple and the physics illustrates many of the phenomena already seen: phase transitions with and without spontaneous symmetry breaking, critical exponents, diverging susceptibilities, scaling, vortices, superfluids etc. Consequently, some of the problems are somewhat long and may be better considered as mini-projects. Markov chains, convergence and detailed balance Even relatively innocent looking models are impossible to solve exactly. The exact solution of the two-dimensional Ising model is known but not in the presence of an external magnetic field. In three dimensions, even in the absence of a magnetic field, the exact solution is not known. In fact, relatively few models can be solved exactly; see Reference [14] for an excellent review of the subject.
Introduction to Integrable Many-Body systems III This is the third part of a three-volume introductory course about integrable systems of interacting bodies. The emphasis is put onto the method of Thermodynamic Bethe ansatz. Two kinds of integrable models are studied. Systems of itinerant electrons, forming a part of Condensed Matter Physics, involve the Hubbard lattice model of electrons with short-ranged one-site interactions (Sect. 20) and the s-d exchange Kondo model (Sect. 21), describing the scattering of conduction electrons on a spin-s impurity. Methods and basic concepts used in Quantum Field Theory are explained on the integrable (1 + 1)-dimensional sine-Gordon model. We start with the classical description of the model in Sect. 22, analyze its finite energy field configurations (soliton, anti-soliton and breathers) and show its classical integrability. The model is quantized by using two schemes: the conformal (Sect. 23) and Lagrangian (Sect. 24) quantizations. The scattering matrix of the sine-Gordon theory is derived at the full quantum level in the bootstrap scheme and is compared to its classical limit in Sect. 25. The parameters of the scattering matrix are related to those of the Lagrangian by calculating the ground-state energy in an applied magnetic field in two ways: Conformal perturbation theory and Thermodynamic Bethe ansatz (Sect. 26). The relation of the sine-Gordon theory to the XXZ Heisenberg model, which provides a complete solution of the sine-Gordon model in a finite volume, is pointed out in Sect. 27. The obtained results are applied in Sect. 28. to the derivation of the exact thermodynamics for the (symmetric) two-component Coulomb gas; this is the first classical two-dimensional fluid with exactly solvable thermodynamics.