Algebraic Bethe Ansatz

2025/2026

Academic Year

ENG

Instruction in English

6

ECTS credits

Delivered by:

Faculty of Mathematics

Type:

Optional course (faculty)

When:

1, 2 module

Instructors

Zabrodin, Anton

Прокофьев Вадим Вячеславович

Course Syllabus

Abstract

The course is devoted to the theory of quantum integrable systems with many (and even infinitely many) degrees of freedom, which is a key part of modern mathematical physics. Integrable models are of great interest and importance from both physical and mathematical points of view. As a rule, there are reach algebraic structures behind them. A deep understanding of these structures has leaded to some new branches of modern mathematics such as, for example, quantum groups and algebras. We will discuss the main method of solving the related problems - the celebrated Bethe ansatz in its coordinate and algebraic versions. The secret of universality and effectiveness of Bethe ansatz hasn't been fully revealed yet. In his time, Nobel winner Richard Feynman pointed out that revealing these secrets is one of the main problems in future theoretical and mathematical physics.

Learning Objectives

• Ознакомить слушателей с техниками Координатного и алгебраического анзецев Бете и с помощью них получить современные результаты из теории интегрируемых систем.

Expected Learning Outcomes

• Derived Bethe equation from ABA for models with elliptic and trigonometric R matrices
• Can write commutation relation for Sklyanin algebra
• Obtain formulas for scalar product of Bethe vectors
• Explain connection between generalized spin chain and Ruijsenaars-Schneider system

Course Contents

• Coordinate Bethe ansatz
• Vertex models of statistical mechanics on two-dimensional lattice
• Algebraic Bethe ansatz
• Scalar products of Bethe vectors
• Quantum-classical duality

Assessment Elements

• Контрольные работы
  Контрольные работы с задачами по пройденным материалам
• Проверка домашних задач
  Проверка задач, которые не успели разобрать на семинарах

Interim Assessment

• 2025/2026 2nd module
  0.3 * Контрольные работы + 0.4 * Проверка домашних задач + 0.3 * Проверка домашних задач

Bibliography

Recommended Core Bibliography

• Baxter, R. J. (1982). Exactly Solved Models in Statistical Mechanics. [Place of publication not identified]: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1258304

Recommended Additional Bibliography

• Korepin, V. E., Izergin, A. G., & Bogoliubov, N. M. (1993). Quantum Inverse Scattering Method and Correlation Functions.

Authors

• Prokofev Vadim Viacheslavovich
• ZABRODIN ANTON VLADIMIROVICH
• Klimenko Aleksei Vladimirovich