Yang-Baxter equation (YBE)

introduction

• most important roles in Integrable systems and solvable models
• at the heart of quantum groups
• exact solvability of many models is explained by commuting transfer matrices
• in 1+1D S-matrix theory, the YBE is the condition for consistent factorization of the multiparticle S-matrix into two-particle factors
  • see Exact S-matrices in ATFT
• \(R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\)
• for vertex models, YBE becomes the star-triangle relation
• see [Baxter1995] for a historical account

Yang and Baxter

• [Yang1967] interacting particles with potential
  • Bethe ansatz gave rise to an equation
• [Baxter1972] considered the problem of eight-vertex model and quantum XYZ model
  • commutation of transfer matrices

Bethe ansatz

• Bethe ansatz amplitude

integrability of a model

• in the space of couplings a submanifold exists, such as that the transfer matrices corresponding to any two points P and P' on it commute
• characterized by a set of equations on the Boltzmann weights
  • this set of equations is called the Yang-Baxter equation
• solutions to Yang-Baxter equation can lead to a construction of integrable models

transfer matrix

• borrowed from transfer matrix in statistical mechanics
• transfer matrix is builtup from matrices of Boltzmann weights
• we need the transfer matrices coming from different set of Boltzman weights commute
• partition function = trace of power of transfer matrices
• so the problem of solving the model is reduced to the computation of this trace

R-matrix

• we make a matrix from the Boltzmann weights
• if we can find an R-matrix, then it implies the existence of a set of Boltzmann weights which give exactly solvable models
• that is why we care about the quantum groups
• spectral parameters
• anistropy parameters
• with an R-matrix satisfying the YBE, we obtain a representation of the Braid group, which then gives a link invariant in Knot theory
• R-matrix

YBE for vertex models

• Yang-Baxter equation
• conditions satisfied by the Boltzmann weights of vertex models
• has been called the star-triangle relation

classical YBE

$$ [X_{12}(u_1-u_2),X_{13}(u_1-u_3)]+[X_{13}(u_1-u_3),X_{23}(u_2-u_3)]+[X_{12}(u_1-u_2),X_{23}(u_2-u_3)]=0 $$

related items

• quantum groups
• Belavin-Drinfeld theory
• Yangian
• Proofs and Confirmation

 

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Yang–Baxter_equation

books

• Louis H. Kauffman, Knots and physics
• Quantum Groups in Two-Dimensional Physics
• Yang-Baxter Equations, Conformal Invariance And Integrability In Statistical Mechanics And Field Theory
• knots+physics
• two-dimensional+physics

 

expositions

• http://math.ucr.edu/home/baez/braids/node4.html
• Perk, Jacques H. H., and Helen Au-Yang. 2006. “Yang-Baxter Equations.” arXiv:math-ph/0606053 (June 20). http://arxiv.org/abs/math-ph/0606053.
• Jimbo, Introduction to the Yang-Baxter equation http://www.math.polytechnique.fr/~renard/Jimbo.pdf
• [Baxter1995] BaxterSolvable models in statistical mechanics, from Onsager onward, Journal of Statistical Physics, Volume 78, Numbers 1-2, 1995

 

articles

• Belavin, A. A., and V. G. Drinfel’d. 1982. “Solutions of the Classical Yang - Baxter Equation for Simple Lie Algebras.” Functional Analysis and Its Applications 16 (3) (July 1): 159–180. doi:10.1007/BF01081585.
• Kulish, P. P., N. Yu Reshetikhin, and E. K. Sklyanin. 1981. “Yang-Baxter Equation and Representation Theory: I.” Letters in Mathematical Physics 5 (5) (September 1): 393–403. doi:10.1007/BF02285311. http://dx.doi.org/10.1007/BF02285311
• [Baxter1972]Partition Function of the Eight-Vertex Lattice Model
  
  • Baxter, Rodney , J. Publication: Annals of Physics, 70, Issue 1, p.193-228, 1972
    
• [Yang1967]Some exact results for the many-body problem in one dimension with repulsive delta-function interaction
  
  • C.N. Yang, Phys. Rev. Lett. 19 (1967), 1312-1315

questions

• http://mathoverflow.net/questions/5103/solutions-of-the-quantum-yang-baxter-equation