Heisenberg spin chain
Quantum algebras and the theory of their representations appear in many contexts of theoretical physics. This course will consider the Heisenberg integral chain one of their most famous examples. In the first part of the course, we will focus on describing the spin chain, its integrability, and its connection to other contexts. In the second part of the course, we will explore this model and its physical observables using the Bethe ansatz method.
Lecturer
Date
21st March ~ 16th June, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Friday | 15:20 - 16:55 | Online | - | - | - |
Prerequisite
The knowledge of linear algebra and basic methods of analysis (integral calculus, theory of the function of a complex variable, determinant identities) is required (but I will spend some time on recalling this material). The knowledge of representation theory of sl_2 is welcome. The familiarity with the basics of quantum mechanics(Hamiltonian, wavefunctions, Pauli matrices) will be useful throughout the whole course.
Syllabus
* Part I: Integrability structures
** Lecture 1 - Introduction
+ Spin chains
+ Quantum integrability
+ Physic phenomena
** Lecture 2 - N=3 XXX Heisenberg spin chain
- Notation: tensor product, etc.
- Integrability
- Solution: eigenvalues and eigenvectors
** Lecture 3-4 - 6-vertex model
- Yang-Baxter equation
- Transfer matrix
- Boundary conditions
- Domain wall boundary conditions
** Lecture 5-6 - RTT-relation and Integrability. The connection between the Heisenberg model and the 6-vertex model.
- Inhomogeneous and homogeneous cases
- Transfer matrix of homogeneous 6v-model and Heisenberg XXZ model
- Yang-Baxter equation and RTT-relation
- Symmetries of spin chains. Higher integrals of motion
* Part II: Algebraic Bethe ansatz
** Lecture 7-8 - Algebraic Bethe ansatz
- Commutation relations
- Algebraic Bethe ansatz
- Eigenvalues and Bethe equation
** Lecture 9 - Bethe equations
- Physical and non-physical solutions
- String hypothesis
** Lecture 10-11 - Scalar products of Bethe vectors
- Off-shell -- off-shell scalar products (Izergin-Korepin sum formula)
- Izergin determinant
- On-shell -- off-shell scalar product (Slavnov formula)
- Norm of on-shell Bethe vector (Gaudin formula)
** Lecture 12-13 - Form-factors
- global form-factors
- Zero modes and QISP
- local form-factors
** Lecture 14-15 - Correlation functions
- Form-factor expansion
- The generating functional
- Asymptotic of correlation functions
** Lecture 16 - ABACUS and the picture
- Numerical solution of Bethe equations
- ABACUS "modules" of solutions
** Lecture 1 - Introduction
+ Spin chains
+ Quantum integrability
+ Physic phenomena
** Lecture 2 - N=3 XXX Heisenberg spin chain
- Notation: tensor product, etc.
- Integrability
- Solution: eigenvalues and eigenvectors
** Lecture 3-4 - 6-vertex model
- Yang-Baxter equation
- Transfer matrix
- Boundary conditions
- Domain wall boundary conditions
** Lecture 5-6 - RTT-relation and Integrability. The connection between the Heisenberg model and the 6-vertex model.
- Inhomogeneous and homogeneous cases
- Transfer matrix of homogeneous 6v-model and Heisenberg XXZ model
- Yang-Baxter equation and RTT-relation
- Symmetries of spin chains. Higher integrals of motion
* Part II: Algebraic Bethe ansatz
** Lecture 7-8 - Algebraic Bethe ansatz
- Commutation relations
- Algebraic Bethe ansatz
- Eigenvalues and Bethe equation
** Lecture 9 - Bethe equations
- Physical and non-physical solutions
- String hypothesis
** Lecture 10-11 - Scalar products of Bethe vectors
- Off-shell -- off-shell scalar products (Izergin-Korepin sum formula)
- Izergin determinant
- On-shell -- off-shell scalar product (Slavnov formula)
- Norm of on-shell Bethe vector (Gaudin formula)
** Lecture 12-13 - Form-factors
- global form-factors
- Zero modes and QISP
- local form-factors
** Lecture 14-15 - Correlation functions
- Form-factor expansion
- The generating functional
- Asymptotic of correlation functions
** Lecture 16 - ABACUS and the picture
- Numerical solution of Bethe equations
- ABACUS "modules" of solutions
Reference
* Other Lectures
N. A. Slavnov "Algebraic Bethe ansatz" https://arxiv.org/abs/1804.07350
N. Reshetikhin "Lectures on the integrability of the 6-vertex model" https://arxiv.org/abs/1010.5031.pdf
L. D. Faddeev "How Algebraic Bethe Ansatz works for integrable model" https://arxiv.org/pdf/hep-th/9605187.pdf
* Books
R. J. Baxter, (1982), Exactly solved models in statistical mechanics (1982)
M. Gaudin, The Bethe wavefunction. (1983 in French, 2014 in English)
V. E. Korepin, N. M. Bogoliubov, A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions. (1993)
N. A. Slavnov "Algebraic Bethe ansatz" https://arxiv.org/abs/1804.07350
N. Reshetikhin "Lectures on the integrability of the 6-vertex model" https://arxiv.org/abs/1010.5031.pdf
L. D. Faddeev "How Algebraic Bethe Ansatz works for integrable model" https://arxiv.org/pdf/hep-th/9605187.pdf
* Books
R. J. Baxter, (1982), Exactly solved models in statistical mechanics (1982)
M. Gaudin, The Bethe wavefunction. (1983 in French, 2014 in English)
V. E. Korepin, N. M. Bogoliubov, A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions. (1993)
Audience
Undergraduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Andrii Liashyk is a researcher in the field of integrated systems, mainly quantum ones. He received his degree from the Center for Advanced Study at Skoltech in 2020. In 2022 he joined BIMSA as a Assistant Professor.