The six-vertex model and related topics
his course explores the exact solution and the underlying mathematical structure of the classical two-dimensional six-vertex model. Starting from its origins in statistical mechanics (ice model), the course develops the toolbox of quantum integrability, focusing on the Transfer Matrix, the Yang-Baxter Equation, and the Algebraic Bethe Ansatz to derive the model’s exact free energy and spectrum. Relation to other statistical models, applications to geometrical critical phenomena and some new developments will be considered.
讲师
Ivan Kostov
日期
2025年10月23日 至 12月12日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周四,周五 | 00:00 - 00:00 | - | - | - |
修课要求
Basic knowledge of statistical mechanics and quantum fields
课程大纲
1. Definition, basic properties and integrability
Vertex configurations and ice rule.
Boundary conditions and partition function.
Phase diagram: Ferroelectric, anti-ferroelectric and disordered phases.
R-matrix and Yang-Baxter equation.
Commuting transfer matrices.
2. Exact solution
The Algebraic Bethe Ansatz. Bethe-Yang equations.
Relation to the XXZ Heisenberg spin chain.
T-Q, Q-Q relations and Quantum Spectral Curve.
The partition function on the torus. Thermodynamical limit.
3. Relation to other solvable statistical models
Loop expansion. Relation to the O(n), SOS and A-D-E lattice models.
Coulomb gas mapping.
Thermodynamical limit: O(n) and A-D-E conformal field theories.
4. Domain Wall Boundary Conditions (DWBC)
Izergin determinant formulas for DWBC.
Relation to classical integrability (Toda lattice) and random matrices.
Thermodynamical limit with DWBC. Limit shape phenomenon.
Determinant formulas for partial DWBC and scalar product of Bethe vectors.
5. The 6-vertex model in a light-cone lattice.
Non-linear integral equation.
Continuum limit: the sine-Gordon model.
6. The inhomogeneous (staggered) 6-vertex model
as a discretisation if the two-dimensional Euclidean black-hole sigma model.
7. Exact solution of the 6-vertex model on planar graphs and thermodynamical limit.
Vertex configurations and ice rule.
Boundary conditions and partition function.
Phase diagram: Ferroelectric, anti-ferroelectric and disordered phases.
R-matrix and Yang-Baxter equation.
Commuting transfer matrices.
2. Exact solution
The Algebraic Bethe Ansatz. Bethe-Yang equations.
Relation to the XXZ Heisenberg spin chain.
T-Q, Q-Q relations and Quantum Spectral Curve.
The partition function on the torus. Thermodynamical limit.
3. Relation to other solvable statistical models
Loop expansion. Relation to the O(n), SOS and A-D-E lattice models.
Coulomb gas mapping.
Thermodynamical limit: O(n) and A-D-E conformal field theories.
4. Domain Wall Boundary Conditions (DWBC)
Izergin determinant formulas for DWBC.
Relation to classical integrability (Toda lattice) and random matrices.
Thermodynamical limit with DWBC. Limit shape phenomenon.
Determinant formulas for partial DWBC and scalar product of Bethe vectors.
5. The 6-vertex model in a light-cone lattice.
Non-linear integral equation.
Continuum limit: the sine-Gordon model.
6. The inhomogeneous (staggered) 6-vertex model
as a discretisation if the two-dimensional Euclidean black-hole sigma model.
7. Exact solution of the 6-vertex model on planar graphs and thermodynamical limit.
参考资料
1-2. Baxter: Exactly Solved Models in Statistical Mechanics. Academic Press (1982). Especially sections 8.1 to 8.5; M. Gaudin. The Bethe Wavefunction. St Ives, Cambridge University Press, 2014.
3. B. Nienhuis. Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Stat. Phys., 34:731–761, 1984; V. Pasquier. Etiology of IRF models. Communications in Mathematical Physics, 118, 09 1988
4. N. Reshetikhin. Lectures on the integrability of the 6-vertex model. ArXiv e-prints, Oct. 2010; P. Zinn-Justin. Six-Vertex, Loop and Tiling models: Integrability and Combinatorics. ArXiv e-prints, Jan. 2009.
5. C. Destri and H. J. De Vega. Unified approach to thermodynamic Bethe Ansatz and finite size corrections for lattice models and field theories. Nucl. Phys., B438:413–454, 1995.
6. V. V. Bazhanov, G. A. Kotousov, S. M. Koval, and S. L. Lukyanov. Some Algebraic Aspects of the Inhomogeneous Six-Vertex Model. SIGMA, 17:025, 2021.
7. I. Kostov. Exact solution of the six-vertex model on a random lattice. Nucl. Phys., B575:513–534, 2000.
3. B. Nienhuis. Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Stat. Phys., 34:731–761, 1984; V. Pasquier. Etiology of IRF models. Communications in Mathematical Physics, 118, 09 1988
4. N. Reshetikhin. Lectures on the integrability of the 6-vertex model. ArXiv e-prints, Oct. 2010; P. Zinn-Justin. Six-Vertex, Loop and Tiling models: Integrability and Combinatorics. ArXiv e-prints, Jan. 2009.
5. C. Destri and H. J. De Vega. Unified approach to thermodynamic Bethe Ansatz and finite size corrections for lattice models and field theories. Nucl. Phys., B438:413–454, 1995.
6. V. V. Bazhanov, G. A. Kotousov, S. M. Koval, and S. L. Lukyanov. Some Algebraic Aspects of the Inhomogeneous Six-Vertex Model. SIGMA, 17:025, 2021.
7. I. Kostov. Exact solution of the six-vertex model on a random lattice. Nucl. Phys., B575:513–534, 2000.
听众
Advanced Undergraduate
, Graduate
, 博士后
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
Ivan Kostov obtained his PhD in 1982 from the Moscow State University, with scientific advisers Vladimir Feinberg and Alexander Migdal. Then he worked in the group of Ivan Todorov at the INRNE Sofia, and since 1990 as a CNRS researcher at the IPhT, CEA-Saclay, France. Currently he is emeritus DR CNRS at IPhT and a visiting professor at UFES, Vitoria, Brazil.