After reviewing the basic ideas of the Bethe Ansatz using the simplest possible model—the one-dimensional interacting boson system of Leib and Liniger—we discuss the magnetic systems for which it has been used successfully. These include spin one-half chains, the Hubbard model, and the Kondo problem.
Abstract The Lieb–Liniger model describes one-dimensional bosons with contact interactions. This many-body system admits an exact solution in terms of the Bethe ansatz. Some of the exact and perturbative results for this model are reviewed. Particular attention is devoted to the explicit evaluation, in terms of the interaction parameter, of physical quantities that can be formally exactly extracted from the Bethe ansatz solution. Another goal of this review is to stress exact relations between various quantities. The technical developments are explained in detail. The most relevant experimental realisations of the studied problems are eventually discussed. This review also contains several new results such as the study of convergence of the ground-state energy series at strong interactions, the excitation spectrum at high energies, and the evaluation of the boundary energy.
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.
In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example (`basic' model) of this class of models: the antiferromagnetic two-chain spin-1/2 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate-incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the `quantum transfer matrix') for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behaviour of non-integrable (less constrained) multi-chain quantum models is discussed.
Low dimensional magnetic models.-The one-dimensional Heisenberg model is known to be soluble by the Bethe Ansatz.The spectrum of a ferromagnet consists of magnons and bound state of magnons.The problem of finding the bound state of a macroscopically large number of magnons poses a special problem.These states are described by macroscopically long string solutions of the Bethe Ansatz equations.To find such solutions numerically we developed a program by which even chains of several hundred sites can be treated.Our results show that, as expected, the structure of the macroscopically long strings is not of the Bethe-type.Recently several new materials have been discovered in which the localized magnetic moments are situated in a chain like structure.The energy spectrum and the behaviour of these systems depends strongly on the value of the spin and the number of coupled chains.We started to study these models using the density matrix renormalization group.One-dimensional fermionic models.-The Hubbard model is the simplest model of the interacting electron system.It is usually formulated on a discrete lattice.Sometimes it is more appropriate to work in the continuum limit.We have shown, that the continuum limit of the Hubbard chain, in which the particle number per site is kept constant leads to a relativistic model describing both finite and zero mass particles.By constructing the relativistic limit of the half-filled band we have shown, that also the relativistic particles are described by Bethe-Ansatz type equations, while in the relativistic limit of the non-half-filled chain this is true only for the massive particles, and the equations of the massless particles are of somewhat different structure.Theory of dissipative motion of heavy particles.-We studied the low energy fixed point structure of an N f -flavor two level system (TLS) model, where the spin of the conduction electrons is replaced by a flavor quantum number, f= 1, ... , N f .Using the multiplicative renormalization group we constructed the (1/N f ) 2 order scaling equations and we analyzed their fixed point structure.We showed that the fixed point structure of this Hamiltonian is equivalent to that of the N f -channel Kondo model and we found that the exponent of the leading irrelevant operators is also the same.However, for a TLS an infinite number of leading irrelevant operators appear which give a contribution to the different physical quantities but do not change their critical exponents.
Abstract The Bethe ansatz can be generalized to problems where particles have internal degrees of freedom. The generalized method can be viewed as two Bethe ansätze executed one after the other: nested Bethe ansatz. Electronic systems are the most relevant examples for condensed matter physics. Prominent electronic many-particle systems in one dimension solvable by a nested Bethe ansatz are the one-dimensional δ-Fermi gas, the one-dimensional Hubbard model, and the Kondo model. The major difference to the Bethe ansatz for one component systems is a second, spin, eigenvalue problem, which has the same form in all cases and is solvable by a second Bethe ansatz, e.g. an algebraic Bethe ansatz. A quantum dot tuned to Kondo resonance and coupled to an isolated metallic ring presents an application of the coupled sets of Bethe ansatz equations of the nested Bethe ansatz.
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.