This chapter focuses on the physical basis of some integrable spin models in condensed matter physics and the useful physical insights derived from the solution of these models. The antiferromagnetic ground state serves as the physical vacuum for the creation of elementary excitations. The study of integrable models constitutes an important area of theoretical physics. Integrable models in condensed matter physics describe interacting many-particle systems. The most prominent examples are interacting spin and electron systems which include several real materials of interest. Integrable models, because of their exact solvability, provide a complete and unambiguous understanding of the variety of phenomena exhibited by real systems. Integrable models are sometimes more general than what is necessary to describe real systems. A major portion of the review is devoted to physical systems which exhibit rich phenomena, like the systems with both spin and orbital degrees of freedom and undoped and doped spin ladder systems, where the need for integrable systems is particularly strong.
This chapter focuses on the physical basis of some integrable spin models in condensed matter physics and the useful physical insights derived from the solution of these models. The antiferromagnetic ground state serves as the physical vacuum for the creation of elementary excitations. The study of integrable models constitutes an important area of theoretical physics. Integrable models in condensed matter physics describe interacting many-particle systems. The most prominent examples are interacting spin and electron systems which include several real materials of interest. Integrable models, because of their exact solvability, provide a complete and unambiguous understanding of the variety of phenomena exhibited by real systems. Integrable models are sometimes more general than what is necessary to describe real systems. A major portion of the review is devoted to physical systems which exhibit rich phenomena, like the systems with both spin and orbital degrees of freedom and undoped and doped spin ladder systems, where the need for integrable systems is particularly strong.
Abstract This book focuses on the theory of quantum matter, strongly interacting systems of quantum many–particle physics, particularly on their study using exactly solvable and quantum integrable models with Bethe ansatz methods. Part 1 explores the fundamental methods of statistical physics and quantum many–particle physics required for an understanding of quantum matter. It also presents a selection of the most important model systems to describe quantum matter ranging from the Hubbard model of condensed matter physics to the Rabi model of quantum optics. The remaining five parts of the book examines appropriate special cases of these models with respect to their exact solutions using Bethe ansatz methods for the ground state, finite–size, and finite temperature properties. They also demonstrate the quantum integrability of an exemplary model, the Heisenberg quantum spin chain, within the framework of the quantum inverse scattering method and through the algebraic Bethe ansatz. Further models, whose Bethe ansatz solutions are derived and examined, include the Bose and Fermi gases in one dimension, the one–dimensional Hubbard model, the Kondo model, and the quantum Tavis–Cummings model, the latter a model descendent from the Rabi model.
Abstract The Yang–Baxter equation has long been recognised as the masterkey to integrability, providing the basis for exactly solved models which capture the fundamental physics of a number of realistic classical and quantum systems. In this article we provide an introductory survey of the impact of Yang–Baxter integrable models on experiments in condensed matter physics and ultracold atoms. A number of prominent examples are covered, including the hard-hexagon model, the Heisenberg spin chain, the transverse quantum Ising chain, a spin ladder model, the Lieb–Liniger Bose gas, the Gaudin–Yang Fermi gas and the two-site Bose–Hubbard model. The review concludes by pointing to some other recent developments with promise for further progress.
There are several interacting spin systems which have a gap in their spin excitation spectrum. The gap does not occur due to anisotropies present in the system but is quantum mechanical in origin. We give a brief overview on the different types of spin gap (SG) antiferromagnets, the models proposed to describe their physical properties and experimental realizations of such systems. Our special focus is on exactly-solvable models and rigorous theories which provide the correct physical picture for the novel phenomena exhibited by SG systems
Lattice models in theoretical physics A major part of theoretical physics involves the construction and systematic analysis of mathematical models as a description of the physical world. For microscopic phenomena, in particular, the reference to models becomes quite explicit. The phenomena are often complex, and any theory that attempts to include every detail soon becomes intractable. It is more fruitful to develop models which ignore irrelevant details but, hopefully, capture the essential physics of the phenomena of interest. Thus we have the Heisenberg model of magnetic order, the Bardeen–Cooper– Schrieffer (BCS) model of superconductivity, and so on. It is important to note, at the outset, that the models we shall be discussing describe strongly interacting, and therefore highly correlated, systems of particles. These are difficult and interesting problems. Where interactions are absent, or weak, elementary treatments are possible and the resulting phenomena are generally unspectacular. Why lattice models? In solid-state phenomena there is usually an underlying lattice structure, and the symmetry properties of this lattice play an important role in the analysis. Even the process of electrical conduction in metals or semiconductors can be equally well described in terms of localized quantum states or in terms of the more usual continuum picture. In quantum field theory, which is formulated in a space–time continuum, a lattice is often introduced for computational purposes. One can think of this in two ways: as an approximation to the continuum, which is then recovered as a limit at the end of the calculations, or as a necessary means of regulating the theory (i.e. controlling divergences) in calculating the Feynman path integral.
Based on an established course and covering the fundamentals, central areas and contemporary topics of this diverse field, Fundamentals of Condensed Matter Physics is a much-needed textbook for graduate students. The book begins with an introduction to the modern conceptual models of a solid from the points of view of interacting atoms and elementary excitations. It then provides students with a thorough grounding in electronic structure and many-body interactions as a starting point to understand many properties of condensed matter systems - electronic, structural, vibrational, thermal, optical, transport, magnetic and superconducting - and methods to calculate them. Taking readers through the concepts and techniques, the text gives both theoretically and experimentally inclined students the knowledge needed for research and teaching careers in this field. It features 246 illustrations, 9 tables and 100 homework problems, as well as numerous worked examples, for students to test their understanding. Solutions to the problems for instructors are available at www.cambridge.org/cohenlouie.