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
• \(R_{12}(u)R_{13}(u+v)R_{23}(v)=R_{23}(v)R_{13}(u+v)R_{12}(u)\)
• 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

• Belavin-Drinfeld theory
• Quantum groups
• Yangian
• Sklyanin algebra
• Proofs and Confirmation

computational resource

• R-matrix
• https://docs.google.com/file/d/0B8XXo8Tve1cxdDk5Rm9DQy1nelk/edit

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

• Nichita, Florin F. “Yang-Baxter Equations, Computational Methods and Applications.” arXiv:1506.03610 [math-Ph], June 11, 2015. http://arxiv.org/abs/1506.03610.
• Yang-Baxter equation in Physics, 안창림
• 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.
• [Baxter1995] BaxterSolvable models in statistical mechanics, from Onsager onward, Journal of Statistical Physics, Volume 78, Numbers 1-2, 1995
• Hietarinta, J. “The Complete Solution to the Constant Quantum Yang-Baxter Equation in Two Dimensions.” In Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, edited by Peter A. Clarkson, 149–54. NATO ASI Series 413. Springer Netherlands, 1993. http://link.springer.com/chapter/10.1007/978-94-011-2082-1_15.
• Jimbo, Michio. 1989. “Introduction to the Yang-Baxter Equation.” International Journal of Modern Physics A. Particles and Fields. Gravitation. Cosmology. Nuclear Physics 4 (15): 3759–3777. doi:10.1142/S0217751X89001503. http://www.worldscientific.com/doi/abs/10.1142/S0217751X89001503

articles

• Yamazaki, Masahito, and Wenbin Yan. ‘Integrability from 2d N=(2,2) Dualities’. arXiv:1504.05540 [hep-Th, Physics:math-Ph], 21 April 2015. http://arxiv.org/abs/1504.05540.
• Chicherin, D., and S. Derkachov. ‘Matrix Factorization for Solutions of the Yang-Baxter Equation’. arXiv:1502.07923 [hep-Th, Physics:math-Ph], 27 February 2015. http://arxiv.org/abs/1502.07923.
• Chicherin, D., S. E. Derkachov, and V. P. Spiridonov. “New Elliptic Solutions of the Yang-Baxter Equation.” arXiv:1412.3383 [hep-Th, Physics:math-Ph], December 10, 2014. http://arxiv.org/abs/1412.3383.
• Chicherin, D., S. E. Derkachov, and V. P. Spiridonov. “From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation.” arXiv:1411.7595 [hep-Th, Physics:math-Ph], November 27, 2014. http://arxiv.org/abs/1411.7595.
• Hietarinta, Jarmo. “Solving the Two‐dimensional Constant Quantum Yang–Baxter Equation.” Journal of Mathematical Physics 34, no. 5 (May 1, 1993): 1725–56. doi:10.1063/1.530185.
• Hietarinta, Jarmo. “All Solutions to the Constant Quantum Yang-Baxter Equation in Two Dimensions.” Physics Letters A 165, no. 3 (May 18, 1992): 245–51. doi:10.1016/0375-9601(92)90044-M.
• 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

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