Quantum Integrable Models

Annotation

Two wide classes of integrable models are mainly discussed in this course. The first one includes quantum spin chains and Gaudin models, and the second class consists of quantum many-body problems. For spin chains we describe basic ideas of the Quantum Inverse Scattering Method based on R-matrix quantization and Yang-Baxter relations. Then we proceed to the Bethe ansatz method for calculating the eigenvectors. The scalar products of Bethe vectors are also briefly discussed, resulting in the calculation of correlation functions. Next, we describe solutions to quantum many-body systems of the Calogero-Ruijsenaars-Macdonald type. Solutions in the class of symmetric functions are obtained and solutions from a more general class are given by integral representations. Relations between the two classes of integrable models are described through the Knizhnik-Zamolodchikov equations. Finally, we also discuss some other examples of integrable systems including 2d dimensional and/or stochastic models.

Course plan

1. Classical spin chains and quadratic r-matrix structure.
2. Quantum spin chains quantum R-matrices, Yang-Baxter equation.
3. Quantum algebras, quantum groups and RTT relations.
4. Algebraic Bethe ansatz for 6-vertex models.
5. Algebraic Bethe ansatz for spin chain and Gaudin model.
6. Scalar products of Bethe vectors.
7. Knizhnik-Zamolodchikov equations.
8. Relation of KZ equations to quantum many-body problems.
9. Symmetric polynomials and Calogero-Sutherland models.
10. Ruijsenaars operators and symmetric Macdonald functions.
11. Dunkl-Cherednik operators and Hecke algebras.
12. Integrable 2d models, integrable probabilities, stochastic models.

Literature

1. Faddeev, L.D., L.A. Takhtajan, “Hamiltonian methods in the theory of solitons”, Springer, 2007.
2. Arutyunov, G., “Elements of Classical and Quantum Integrable Systems”, Springer, 2019.
3. Faddeev, L.D., “How Algebraic Bethe Ansatz works for integrable model”, Les-Houches lectures 1996; arXiv:hep-th/9605187.
4. Slavnov, N., “Introduction to the nested algebraic Bethe ansatz”, SciPost Physics Lecture Notes 19, 2020.
5. Macdonald, I.G., “Symmetric functions and Hall polynomials”, Oxford Classic Texts in the Physical Sciences, 2nd Edition, 1998.