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Research Seminar "Quantum Integrable Systems in Formulas and Pictures"

2019/2020
Учебный год
ENG
Обучение ведется на английском языке
3
Кредиты
Статус:
Дисциплина общефакультетского пула
Когда читается:
1, 2 модуль

Course Syllabus

Abstract

Within the framework of the theory of quantum integrable systems, many powerful methods have been developed to study their underlying mathematical structures and possible physical implications. They are quite sophisticated and, basically, they can be divided into three sections: algebraic, analytical and graphical. Here, the algebraic methods are based on the quantum group structures with which the integrable model is associated; the analytical methods imply analysis of properties of the integrability objects as functions of the spectral parameter, their asymptotic behavior with respect to parameters such as an external magnetic field, temperature, or other parameters of this type; the graphical methods stem from the famous triangle representation of the Yang–Baxter equation. As a rule, the most effective is a certain combination of these methods, when each of them is applied to solve a specific task to achieve a common goal. We will present a systematic discussion of these three main approaches to the investigation of quantum integrable systems. Various useful objects and equations will be given in both algebraic and graphical forms. As the basic example, we will specialize our discussion to quantum integrable systems associated with the quantum loop algebra of lowest rank, on which the «classical» six-vertex model and the corresponding XXZ spin chain are based.
Learning Objectives

Learning Objectives

  • To extend and deepen knowledge about basic structures of quantum integrability
  • To study different methods used for the investigation of vertex models
Expected Learning Outcomes

Expected Learning Outcomes

  • Mastering powerful methods of quantum groups to study quantum integrable systems
Course Contents

Course Contents

  • Loop algebras. Quantum groups. Universal R-matrix. Representations of quantum loop algebras
  • Universal integrability objects. Transfer- and Q-operators. Commutativity
  • Functional relations between integrability objects. TQ-, TT-and QQ-relations
  • R-operators. Unitarity and crossing. Graphical description of open chains
  • Basic example in detail: one quantum loop algebra
Assessment Elements

Assessment Elements

  • non-blocking Cumulative grade
  • non-blocking Final exam
Interim Assessment

Interim Assessment

  • Interim assessment (2 module)
    You will collect cumulative score C (estimated by 10 points). If C >= 8, then the resulting mark S = C. If C < 8, then the written exam must be presented (estimated from 10 points), and the resulting mark S = (C + E)/2, where E is the exam mark, rounded off by the standard rules.
Bibliography

Bibliography

Recommended Core Bibliography

  • Nirov, K. S., & Razumov, A. V. (2018). Vertex Models and Spin Chains in Formulas and Pictures. https://doi.org/10.3842/SIGMA.2019.068
  • Nirov, K. S., & Razumov, A. V. (2018). Vertex models and spin chains in formulas and pictures. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.E070C80B

Recommended Additional 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
  • Baxter, R. J. (2007). Exactly Solved Models in Statistical Mechanics (Vol. Dover ed). Mineola, N.Y.: Dover Publications. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1152951