Abstract
In these lecture notes we aim for a pedagogical introduction to both classical and quantum integrability. Starting from Liouville integrability and passing through Lax pair and r-matrix we discuss the construction of the conserved charges for classical integrable models taking as example the harmonic oscillator. The construction of these charges for 2D integrable field theories is also discussed using a Lax connection and the sine-Gordon model as example. On the quantum side, the XXZ spin chain is used to explain the systematic construction of the conserved charges starting from a quantum R-matrix, solution of the quantum Yang–Baxter equation. The diagonalization of these charges is performed using the algebraic Bethe ansatz. At the end, the interpretation of the R-matrix as an S-matrix in a scattering process is also presented. These notes were written for the lectures delivered at the school ‘Integrability, Dualities and Deformations’, that ran from 23 to 27 August 2021 in Santiago de Compostela and virtually.
Highlights
Introduction to classical and quantum integrabilityAbstractIn these lecture notes we aim for a pedagogical introduction to both classical and quantum integrability
Three aspects of integrable models are explored in the Integrability, dualities and deformations school : the basic concepts and techniques on both classical and quantum integrability presented in these lecture notes, integrable deformations of sigma models in Ben Hoare’s lecture notes [23] and 4-dimensional Chern-Simons theory and integrable field theories in Sylvain Lacroix’s lecture notes [24]
With the introduction of the Lax pair and the classical Yang-Baxter equation, the systematic construction of several new integrable models became possible, including integrable hierarchies associated to some algebras
Summary
Integrable models play a role in many areas of physics ranging from condensed matter, to string theory, passing through Temperley-Lieb and Hecke algebras, quantum groups and Yangians, the bootstrap program, AdS/CFT, sigma models, quantum computing, statistical mechanics and many others [1–22]. With the introduction of the Lax pair and the classical Yang-Baxter equation, the systematic construction of several new integrable models became possible, including integrable hierarchies associated to some algebras. One very interesting application is in spin chains, which are discrete quantum spin systems which have applications ranging from N = 4 Super Yang-Mills (N = 4 SYM) theory to condensed matter In these lecture notes we discuss a famous example of a spin chain, the XXZ model, which can be understood as a toy model to study magnetism. In Appendix A we provide a proof for equation (3.27), while Appendices B and C are dedicated to introduce integrable hierarchies and to explain a systematic method to find new solutions of the qYBE, respectively Along these lecture notes we included many constructive exercises that we believe can help in the understanding of the concepts and techniques
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Journal of Physics A: Mathematical and Theoretical
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
