Abstract This chapter introduces a select number of models of strongly interacting quantum many-particle physics and examines their basic properties. These models represent Bosonic and Fermionic systems as well as systems where magnetic moments, i.e. spins, interact. The main selection criterion has been the existence of a variant of the model that is quantum integrable using Bethe ansatz methods. After studying the Bose fluid, the Landau Fermi liquid, and the one-dimensional concept of the Luttinger liquid, it reviews some of the major models of condensed matter theory, including the Hubbard model describing itinerant magnetism, the Heisenberg model describing localized magnetism, and the Kondo model describing the interaction of a magnetic impurity and band electrons. It also presents the Rabi model and some of its descendants in order to describe the interaction of light and quantum matter in quantum optics.
Abstract This chapter introduces a select number of models of strongly interacting quantum many-particle physics and examines their basic properties. These models represent Bosonic and Fermionic systems as well as systems where magnetic moments, i.e. spins, interact. The main selection criterion has been the existence of a variant of the model that is quantum integrable using Bethe ansatz methods. After studying the Bose fluid, the Landau Fermi liquid, and the one-dimensional concept of the Luttinger liquid, it reviews some of the major models of condensed matter theory, including the Hubbard model describing itinerant magnetism, the Heisenberg model describing localized magnetism, and the Kondo model describing the interaction of a magnetic impurity and band electrons. It also presents the Rabi model and some of its descendants in order to describe the interaction of light and quantum matter in quantum optics.
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.
Introduction to integrable many-body systems I This is the first volume of a three-volume introductory course about integrable (exactly solvable) systems of interacting bodies. The aim of the course is to derive and analyze, on an elementary mathematical and physical level, the Bethe ansatz solutions, ground-state properties and the thermodynamics of integrable many-body systems in many domains of physics: Nonrelativistic one-dimensional continuum Fermi and Bose gases; One-dimensional quantum models of condensed matter physics like the Heisenberg, Hubbard and Kondo models; Relativistic models of the (1+1)-dimensional Quantum Field Theory like the Luttinger model, the sine-Gordon model and its fermionic analog the Thirring model; Two-dimensional classical models, especially the symmetric Coulomb gas. In the first part of this volume, we deal with nonrelativistic one-dimensional continuum Fermi and Bose quantum gases of spinless (identical) particles with specific types of pairwise interactions like the short-range δ-function and hard-core interactions, and the long-range 1/
This chapter gives an introduction to the physics of interacting quantum systems, both bosonic and fermionic. Section 1.2 introduces the physics of quantum particles in periodic lattices. Section 1.3 examines, for the case of bosons, how the combined effects of lattice and interaction can turn the system into an insulator, the so-called Mott insulator, and discusses the corresponding physics. Section 1.4 discusses what happens when the system is one-dimensional. Section 1.5 moves on to the case of fermions. It discusses first Fermi-liquid theory and the concept of quasi-particles, Landau’s description of the low-energy excitations of interacting fermion systems. Section 1.6 shows how, like the method for bosons, the combination of a lattice and strong interactions can turn a Fermi liquid into a Mott insulator. Section 1.7 looks at the properties of one-dimensional fermions, shows how Fermi liquid theory fails because low-energy excitations are now collective modes instead of quasi-particles, and examines the corresponding physics for both the conducting and insulating phases. Finally, Section 1.8 draws some conclusions and give some perspectives.
Using eigen-functional bosonization method, we study quantum many-particle systems, and show that the quantum many-particle problems end in to solve the differential equation of the phase fields which represent the particle correlation strength. Thus, the physical properties of these systems are completely determined by the differential equation of the phase fields. We mainly focus on the study of D-dimensional electron gas with/without transverse gauge fields, two-dimensional electron gas under an external magnetic field, D-dimensional boson systems, a D-dimensional Heisenberg model and a one-band Hubbard model on a square lattice, and give their exact (accurate for Heisenberg model) functional expressions of the ground state energy and action, and the eigen-functional wave functions of the fermions/bosons. With them, we can calculate a variety of correlation functions of the systems, such as single particle Green's functions and their ground state wave functions. In present theoretical framework, we can unifiably represent the Landau Fermi liquid, non-Fermi liquid (D≥2) and Tomonaga-Luttinger liquid.
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.
This chapter gives an introduction to the physics of interacting quantum systems, both bosonic and fermionic. Section 1.2 introduces the physics of quantum particles in periodic lattices. Section 1.3 examines, for the case of bosons, how the combined effects of lattice and interaction can turn the system into an insulator, the so-called Mott insulator, and discusses the corresponding physics. Section 1.4 discusses what happens when the system is one-dimensional. Section 1.5 moves on to the case of fermions. It discusses first Fermi-liquid theory and the concept of quasi-particles, Landau’s description of the low-energy excitations of interacting fermion systems. Section 1.6 shows how, like the method for bosons, the combination of a lattice and strong interactions can turn a Fermi liquid into a Mott insulator. Section 1.7 looks at the properties of one-dimensional fermions, shows how Fermi liquid theory fails because low-energy excitations are now collective modes instead of quasi-particles, and examines the corresponding physics for both the conducting and insulating phases. Finally, Section 1.8 draws some conclusions and give some perspectives.