Exactly Solved Models in Statistical Mechanics
The course intends to provide an introduction to the theory of integrable lattice models. Basic examples are the two-dimensional Ising model in a zero magnetic field, the six-vertex model, as well as related two-dimensional models and spin chains.
It is planned to explain with simple model examples the concept of matrix transfer, duality between high and low temperatures, the concept of lattice operators of order and disorder, the star-triangle equation, the Yang-Baxter equation, and connected algebraic structures. One of the main topics I would like to try to explain is diagonalization of the Hamiltonian (matrix transfer) in the ansatz Bethe.
It is planned to explain with simple model examples the concept of matrix transfer, duality between high and low temperatures, the concept of lattice operators of order and disorder, the star-triangle equation, the Yang-Baxter equation, and connected algebraic structures. One of the main topics I would like to try to explain is diagonalization of the Hamiltonian (matrix transfer) in the ansatz Bethe.
Lecturer
Date
11th October, 2023 ~ 5th January, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday,Friday | 13:30 - 15:05 | A3-4-101 | ZOOM 05 | 293 812 9202 | BIMSA |
Prerequisite
Knowledge of basic concepts of linear algebra and mathematical analysis is required. Knowledge of general physics and basic statistical mechanics would be helpful, but the minimum necessary information will be given in the course.The familiarity with the basics of quantum mechanics(Hamiltonian, wavefunctions, Pauli matrices) will be useful throughout the whole course.
Syllabus
1. Introduction. Ising model D=1:
1) Basic Statistical Mechanics.
2) Solution and critical behaviour of one dimentional Ising model.
2. Ising model D=2 on Bethe lattice:
1) Ising model D=2 on Bethe lattice.
2) Star-triangle equation for planar Ising model.
3) Ising model D=2 on square lattice.
3. Ice-type model(6-vertex model):
1) The transfer matrix.
2) Three phases.
3) Baxter Q-operator method.
4. 8-vertex model:
1) Star - triangle relation.
2) Baxter Q-operator method.
3) Connection to Ising model.
4) Relation to XYZ model.
5. Potts and Ashkin-Teller Models.
1) Basic Statistical Mechanics.
2) Solution and critical behaviour of one dimentional Ising model.
2. Ising model D=2 on Bethe lattice:
1) Ising model D=2 on Bethe lattice.
2) Star-triangle equation for planar Ising model.
3) Ising model D=2 on square lattice.
3. Ice-type model(6-vertex model):
1) The transfer matrix.
2) Three phases.
3) Baxter Q-operator method.
4. 8-vertex model:
1) Star - triangle relation.
2) Baxter Q-operator method.
3) Connection to Ising model.
4) Relation to XYZ model.
5. Potts and Ashkin-Teller Models.
Reference
+ The basic book
- R. J. Baxter, Exactly solved models in statistical mechanics (1982)
+ Good books on integrating models
- 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)
- Essler, F. H. L.; Frahm, H., Goehmann, F., Kluemper, A., & Korepin, V. E., The One-Dimensional Hubbard Model. (2005)
+ A simple exposition of some issues of exactly solvable lattice models
- Giuseppe Mussardo, Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics. (2020)
+ A good textbook on stat mechanics and CFT
- С. Itzykson, J.M. Drouffe. Statistical field theory. (1989)
+ A simple and concise presentation of the physics of phase transitions
- M. Yeomans, Statistical mechanics of phase transitions. (1992)
- R. J. Baxter, Exactly solved models in statistical mechanics (1982)
+ Good books on integrating models
- 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)
- Essler, F. H. L.; Frahm, H., Goehmann, F., Kluemper, A., & Korepin, V. E., The One-Dimensional Hubbard Model. (2005)
+ A simple exposition of some issues of exactly solvable lattice models
- Giuseppe Mussardo, Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics. (2020)
+ A good textbook on stat mechanics and CFT
- С. Itzykson, J.M. Drouffe. Statistical field theory. (1989)
+ A simple and concise presentation of the physics of phase transitions
- M. Yeomans, Statistical mechanics of phase transitions. (1992)
Audience
Advanced Undergraduate
, Graduate
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.