Yang-Baxter equation (YBE)

introduction

• roles in the following fields
  • integrable systems
  • exactly solvable statistical models
  • quantum groups
  • conformal field theory
  • topological quantum field theory
  • braid groups
• exact solvability of many models is explained by commuting transfer matrices
  
• manifestations of Yang-Baxter equation
  
  • factorizable S-matrix
    
• \(R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\)
  
• 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
  

 

 

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Yang–Baxter_equation
• http://en.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/
• Princeton companion to mathematics(첨부파일로 올릴것)

 

 

 

books

• Knots and physics
  
  • Louis H. Kauffman
• Quantum Groups in Two-Dimensional Physics
• Yang-Baxter Equations, Conformal Invariance And Integrability In Statistical Mechanics And Field Theory
• http://gigapedia.info/1/knots+physics
• http://gigapedia.info/1/two-dimensional+physics
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

articles

• Introduction to the Yang-Baxter equation
  
  • Jimbo
• [Baxter1995]Solvable models in statistical mechanics, from Onsager onward
  
  • Baxter, Journal of Statistical Physics, Volume 78, Numbers 1-2, 1995
• [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

• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://arxiv.org/
• http://www.pdf-search.org/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://dx.doi.org/10.1007/BF02183337

 

 

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