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Author: Fabio Franchini Publisher: Springer ISBN: 3319484877 Category : Science Languages : en Pages : 186
Book Description
This book introduces the reader to basic notions of integrable techniques for one-dimensional quantum systems. In a pedagogical way, a few examples of exactly solvable models are worked out to go from the coordinate approach to the Algebraic Bethe Ansatz, with some discussion on the finite temperature thermodynamics. The aim is to provide the instruments to approach more advanced books or to allow for a critical reading of research articles and the extraction of useful information from them. We describe the solution of the anisotropic XY spin chain; of the Lieb-Liniger model of bosons with contact interaction at zero and finite temperature; and of the XXZ spin chain, first in the coordinate and then in the algebraic approach. To establish the connection between the latter and the solution of two dimensional classical models, we also introduce and solve the 6-vertex model. Finally, the low energy physics of these integrable models is mapped into the corresponding conformal field theory. Through its style and the choice of topics, this book tries to touch all fundamental ideas behind integrability and is meant for students and researchers interested either in an introduction to later delve in the advance aspects of Bethe Ansatz or in an overview of the topic for broadening their culture.
Author: Fabio Franchini Publisher: Springer ISBN: 3319484877 Category : Science Languages : en Pages : 186
Book Description
This book introduces the reader to basic notions of integrable techniques for one-dimensional quantum systems. In a pedagogical way, a few examples of exactly solvable models are worked out to go from the coordinate approach to the Algebraic Bethe Ansatz, with some discussion on the finite temperature thermodynamics. The aim is to provide the instruments to approach more advanced books or to allow for a critical reading of research articles and the extraction of useful information from them. We describe the solution of the anisotropic XY spin chain; of the Lieb-Liniger model of bosons with contact interaction at zero and finite temperature; and of the XXZ spin chain, first in the coordinate and then in the algebraic approach. To establish the connection between the latter and the solution of two dimensional classical models, we also introduce and solve the 6-vertex model. Finally, the low energy physics of these integrable models is mapped into the corresponding conformal field theory. Through its style and the choice of topics, this book tries to touch all fundamental ideas behind integrability and is meant for students and researchers interested either in an introduction to later delve in the advance aspects of Bethe Ansatz or in an overview of the topic for broadening their culture.
Author: Hans-Peter Eckle Publisher: Oxford University Press ISBN: 0191668044 Category : Science Languages : en Pages : 752
Book Description
An important task of theoretical quantum physics is the building of idealized mathematical models to describe the properties of quantum matter. This book provides an introduction to the arguably most important method for obtaining exact results for strongly interacting models of quantum matter - the Bethe ansatz. It introduces and discusses the physical concepts and mathematical tools used to construct realistic models for a variety of different fields, including condensed matter physics and quantum optics. The various forms of the Bethe ansatz - algebraic, coordinate, multicomponent, and thermodynamic Bethe ansatz, and Bethe ansatz for finite systems - are then explained in depth and employed to find exact solutions for the physical properties of the integrable forms of strongly interacting quantum systems. The Bethe ansatz is one of the very few methodologies which can calculate physical properties non-perturbatively. Arguably, it is the only such method we have which is exact. This means, once the model has been set up, no further approximations or assumptions are necessary, and the relevant physical properties of the model can be computed exactly. Furthermore, an infinite set of conserved quantities can be obtained. The quantum mechanical model under consideration is fully integrable. This makes the search for quantum models which are amenable to an exact solution by the Bethe ansatz, and which are quantum integrable, so important and rewarding. The exact solution will provide benchmarks for other models, which do not admit an exact solution. Bethe ansatz techniques provide valuable insight into the physics of strongly correlated quantum matter.
Author: Decio Levi Publisher: Springer ISBN: 3319566660 Category : Science Languages : en Pages : 441
Book Description
This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference equations. Difference equations are playing an increasingly important role in the natural sciences. Indeed, many phenomena are inherently discrete and thus naturally described by difference equations. More fundamentally, in subatomic physics, space-time may actually be discrete. Differential equations would then just be approximations of more basic discrete ones. Moreover, when using differential equations to analyze continuous processes, it is often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference ones. Each of the nine peer-reviewed chapters in this volume serves as a self-contained treatment of a topic, containing introductory material as well as the latest research results and exercises. Each chapter is presented by one or more early career researchers in the specific field of their expertise and, in turn, written for early career researchers. As a survey of the current state of the art, this book will serve as a valuable reference and is particularly well suited as an introduction to the field of symmetries and integrability of difference equations. Therefore, the book will be welcomed by advanced undergraduate and graduate students as well as by more advanced researchers.
Author: Guillaume Lang Publisher: Springer ISBN: 3030052850 Category : Science Languages : en Pages : 193
Book Description
The book addresses several aspects of thermodynamics and correlations in the strongly-interacting regime of one-dimensional bosons, a topic at the forefront of current theoretical and experimental studies. Strongly correlated systems of one-dimensional bosons have a long history of theoretical study. Their experimental realisation in ultracold atom experiments is the subject of current research, which took off in the early 2000s. Yet these experiments raise new theoretical questions, just begging to be answered. Correlation functions are readily available for experimental measurements. In this book, they are tackled by means of sophisticated theoretical methods developed in condensed matter physics and mathematical physics, such as bosonization, the Bethe Ansatz and conformal field theory. Readers are introduced to these techniques, which are subsequently used to investigate many-body static and dynamical correlation functions.
Author: Herbert Spohn Publisher: World Scientific ISBN: 9811283540 Category : Science Languages : en Pages : 255
Book Description
This book provides a broad introduction to integrable systems with many degrees of freedom. Within a much larger orbit, discussed are models such as the classical Toda lattice, Calogero fluid, and Ablowitz-Ladik discretized nonlinear Schrödinger equation. On the quantum mechanical side, featured are the Lieb-Liniger delta-Bose gas and the quantum Toda lattice. As a genuinely novel twist, the study deals with random initial data described by generalized Gibbs ensembles with parameters of slow spatial variation. This is the hydrodynamic scale, in spirit similar to the ballistic Euler scale of nonintegrable simple fluids. While integrable microscopic particle models are very diverse, the central theme of this book is to elucidate their structural similarity on hydrodynamic scales.
Author: Tobias Wirth Publisher: Sudwestdeutscher Verlag Fur Hochschulschriften AG ISBN: 9783838123875 Category : Languages : en Pages : 124
Book Description
The framework of the Quantum Inverse Scattering Method is used to study the hamiltonian of the XXX and XXZ spin chain with general boundary fields. Key ingredient is the underlying algebraic structure which is a combination of the Yang-Baxter algebra and a so-called Reflection algebra including boundary fields of arbitrary direction and strength. For spin chains with diagonal boundary fields this setup has been well studied using algebraic Bethe ansatz and the inverse problem was solved by Kitanine for infinite chain lengths. These results are picked up and generalized to arbitrary lengths using non-linear integral equations. In the case of non-diagonal boundary fields the lack of a reference state or pseudo vacuum prohibits the solution by algebraic Bethe ansatz. The method of separation of variables is not constrained in that sense and is applied to the XXX chain and a spin-boson model. Finally a different approach to the case of non-diagonal boundary conditions is studied. Starting from the so-called fusion hierarchy non-linear integral equations are derived bearing the possibility to extract information about an eigenvalue of a specific state.
Author: Gleb Arutyunov Publisher: Springer ISBN: 303024198X Category : Science Languages : en Pages : 414
Book Description
Integrable models have a fascinating history with many important discoveries that dates back to the famous Kepler problem of planetary motion. Nowadays it is well recognised that integrable systems play a ubiquitous role in many research areas ranging from quantum field theory, string theory, solvable models of statistical mechanics, black hole physics, quantum chaos and the AdS/CFT correspondence, to pure mathematics, such as representation theory, harmonic analysis, random matrix theory and complex geometry. Starting with the Liouville theorem and finite-dimensional integrable models, this book covers the basic concepts of integrability including elements of the modern geometric approach based on Poisson reduction, classical and quantum factorised scattering and various incarnations of the Bethe Ansatz. Applications of integrability methods are illustrated in vast detail on the concrete examples of the Calogero-Moser-Sutherland and Ruijsenaars-Schneider models, the Heisenberg spin chain and the one-dimensional Bose gas interacting via a delta-function potential. This book has intermediate and advanced topics with details to make them clearly comprehensible.
Author: Nikita Slavnov Publisher: World Scientific ISBN: 9811254273 Category : Science Languages : en Pages : 399
Book Description
It is unlikely that today there is a specialist in theoretical physics who has not heard anything about the algebraic Bethe ansatz. Over the past few years, this method has been actively used in quantum statistical physics models, condensed matter physics, gauge field theories, and string theory.This book presents the state-of-the-art research in the field of algebraic Bethe ansatz. Along with the results that have already become classic, the book also contains the results obtained in recent years. The reader will get acquainted with the solution of the spectral problem and more complex problems that are solved using this method. Various methods for calculating scalar products and form factors are described in detail. Special attention is paid to applying the algebraic Bethe ansatz to the calculation of the correlation functions of quantum integrable models. The book also elaborates on multiple integral representations for correlation functions and examples of calculating the long-distance asymptotics of correlations.This text is intended for advanced undergraduate and postgraduate students, and specialists interested in the mathematical methods of studying physical systems that allow them to obtain exact results.
Author: Abolfazl Bayat Publisher: Springer Nature ISBN: 303103998X Category : Science Languages : en Pages : 549
Book Description
This book covers recent developments in the understanding, quantification, and exploitation of entanglement in spin chain models from both condensed matter and quantum information perspectives. Spin chain models are at the foundation of condensed matter physics and quantum information technologies and elucidate many fundamental phenomena such as information scrambling, quantum phase transitions, and many-body localization. Moreover, many quantum materials and emerging quantum devices are well described by spin chains. Comprising accessible, self-contained chapters written by leading researchers, this book is essential reading for graduate students and researchers in quantum materials and quantum information. The coverage is comprehensive, from the fundamental entanglement aspects of quantum criticality, non-equilibrium dynamics, classical and quantum simulation of spin chains through to their experimental realizations, and beyond into machine learning applications.
Author: Fabio Franchini
Author: Fabio Franchini
Author: Nathan Keenan
Author: Hans-Peter Eckle
Author: Decio Levi
Author: Guillaume Lang
Author: Herbert Spohn
Author: Tobias Wirth
Author: Gleb Arutyunov
Author: Nikita Slavnov
Author: Abolfazl Bayat