Yang-Baxter equation (YBE)

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  • 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
  • \(R_{12}(u)R_{13}(u+v)R_{23}(v)=R_{23}(v)R_{13}(u+v)R_{12}(u)\)
  • for vertex models, YBE becomes the star-triangle relation
  • see [Baxter1995] for a historical account

Yang and Baxter

Bethe ansatz

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


  • 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

computational resource





  • 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




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