"Yang-Baxter equation (YBE)"의 두 판 사이의 차이
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− | <h5 style="line-height: 3.428em; margin | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">introduction</h5> |
* exact solvability of many models is explained by commuting transfer matrices<br> | * exact solvability of many models is explained by commuting transfer matrices<br> | ||
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− | <h5 style="line-height: 2em; margin | + | <h5 style="line-height: 2em; margin: 0px;">Yang and Baxter</h5> |
* '''[Yang1967]''' [[interacting particles with potential]]<br> | * '''[Yang1967]''' [[interacting particles with potential]]<br> | ||
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* [[Bethe ansatz]] amplitude<br> | * [[Bethe ansatz]] amplitude<br> | ||
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− | <h5 style="line-height: 2em; margin | + | <h5 style="line-height: 2em; margin: 0px;">integrability of a model</h5> |
* 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<br> | * 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<br> | ||
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− | <h5 style="line-height: 2em; margin | + | <h5 style="line-height: 2em; margin: 0px;">transfer matrix</h5> |
* borrowed from [[transfer matrix in statistical mechanics]]<br> | * borrowed from [[transfer matrix in statistical mechanics]]<br> | ||
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* we make a matrix from the Boltzmann weights<br> | * we make a matrix from the Boltzmann weights<br> | ||
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− | <h5 style="line-height: 2em; margin | + | <h5 style="line-height: 2em; margin: 0px;">YBE for vertex models</h5> |
* Yang-Baxter equation<br> | * Yang-Baxter equation<br> | ||
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* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
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* http://en.wikipedia.org/wiki/ | * http://en.wikipedia.org/wiki/ | ||
* http://en.wikipedia.org/wiki/ | * http://en.wikipedia.org/wiki/ | ||
− | * Princeton companion to mathematics(첨부파일로 올릴것) | + | * Princeton companion to mathematics(첨부파일로 올릴것) |
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* [http://gigapedia.com/items:links?id=71502 Knots and physics]<br> | * [http://gigapedia.com/items:links?id=71502 Knots and physics]<br> | ||
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− | * | + | * Introduction to the Yang-Baxter equation<br> |
+ | ** Jimb | ||
* '''[Baxter1995]'''[http://dx.doi.org/10.1007/BF02183337 Solvable models in statistical mechanics, from Onsager onward]<br> | * '''[Baxter1995]'''[http://dx.doi.org/10.1007/BF02183337 Solvable models in statistical mechanics, from Onsager onward]<br> | ||
** Baxter, Journal of Statistical Physics, Volume 78, Numbers 1-2, 1995 | ** Baxter, Journal of Statistical Physics, Volume 78, Numbers 1-2, 1995 | ||
* '''[Baxter1972]'''[http://dx.doi.org/10.1006/aphy.2000.6010 Partition Function of the Eight-Vertex Lattice Model]<br> | * '''[Baxter1972]'''[http://dx.doi.org/10.1006/aphy.2000.6010 Partition Function of the Eight-Vertex Lattice Model]<br> | ||
** Baxter, Rodney , J. Publication: Annals of Physics, 70, Issue 1, p.193-228, 1972<br> | ** Baxter, Rodney , J. Publication: Annals of Physics, 70, Issue 1, p.193-228, 1972<br> | ||
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* '''[Yang1967]'''[http://dx.doi.org/10.1103/PhysRevLett.19.1312 Some exact results for the many-body problem in one dimension with repulsive delta-function interaction]<br> | * '''[Yang1967]'''[http://dx.doi.org/10.1103/PhysRevLett.19.1312 Some exact results for the many-body problem in one dimension with repulsive delta-function interaction]<br> | ||
** C.N. Yang, Phys. Rev. Lett. 19 (1967), 1312-1315 | ** C.N. Yang, Phys. Rev. Lett. 19 (1967), 1312-1315 | ||
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* http://www.pdf-search.org/ | * http://www.pdf-search.org/ | ||
* http://pythagoras0.springnote.com/ | * http://pythagoras0.springnote.com/ | ||
− | * http://math.berkeley.edu/~reb/papers/index.html | + | * [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html] |
* http://dx.doi.org/10.1007/BF02183337 | * http://dx.doi.org/10.1007/BF02183337 | ||
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* http://math.ucr.edu/home/baez/braids/node4.html | * http://math.ucr.edu/home/baez/braids/node4.html | ||
* 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q= | * 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q= | ||
* 트렌비 블로그 검색 http://www.trenb.com/search.qst?q= | * 트렌비 블로그 검색 http://www.trenb.com/search.qst?q= |
2010년 8월 10일 (화) 11:57 판
introduction
- exact solvability of many models is explained by commuting transfer matrices
- manifestations of Yang-Baxter equation
- factorizable S-matrix
- 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
- Bethe ansatz gave rise to an equation
- [Baxter1972] considered the problem of eight-vertex model and quantum XYZ model
- commutation of transfer matrices
- 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
- 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
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
- Jimb
- [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
- 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