Yang-Baxter equation (YBE)

introduction

• exact solvability of many models is explained by commuting transfer matrices
  
• manifestations of Yang-Baxter equation
  
  • factorizable S-matrix
    

• \(R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\)
  
• 1966 Yang and Yang
  

 

 

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

• transfer matrix is builtup from matrices of  Boltzmann weights
  
• we need the trasfer 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
  
  
  

 

Bethe ansatz

• Bethe ansatz
  

 

 

YBE for vertex models

• Yang-Baxter equation
  
• conditions satisfied by the Boltzmann weights of vertex models
  

 

 

 

표준적인 도서 및 추천도서

• 찾아볼 수학책
• 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=

 

 

참고할만한 자료

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

 

 

논문검색

• Integrable theories, Yang-Baxter algebras and quantum groups: An overview
• Yang-Baxter algebras, integrable theories and quantum groups
  
  • H. J. De Vega
• Yang-Baxter algebras, integrable theories and Betre Ansatz
• http://www.zentralblatt-math.org/zmath/en/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html

• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=

 

 

 

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• http://math.ucr.edu/home/baez/braids/node4.html
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